NUREG/CR-6940 

PNNL-16396 

Combined Estimation of 

Hydrogeologic Conceptual 

Model, Parameter, and Scenario 

Uncertainty with Application to 

Uranium Transport at the Hanford 

Site 300 Area 

Pacific Northwest National Laboratory 

U.S. Nuclear Regulatory Commission 

Office of Nuclear Regulatory Research 

Washington, DC 20555-0001 
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 NUREG/CR-6940 

 PNNL-16396 

 

Combined Estimation of 

Hydrogeologic Conceptual 

Model, Parameter, and Scenario 

Uncertainty with Application to 

Uranium Transport at the Hanford 

Site 300 Area 

Manuscript Completed: June 2007 

Date Published: July 2007 

Prepared by 

P.D. Meyer, M. Ye (DRI), M.L. Rockhold, S.P. Neuman (UA) 

K.J. Cantrell 

Pacific Northwest National Laboratory 

P.O. Box 999 

Richland, WA 99352 

Desert Research Institute 

Las Vegas, NV 89119 

(Currently at Florida State University, Tallahassee, FL 32306) 

University of Arizona 

Tucson, AZ 85721 

T.J. Nicholson, NRC Project Manager 

Prepared for 

Division of Fuel, Engineering and Radiological Research 

Office of Nuclear Regulatory Research 

U.S. Nuclear Regulatory Commission 

Washington, DC 20555-0001 

NRC Job Code Y6465 
iii

ABSTRACT

We describe the development and application of a 

methodology to systematically and quantitatively assess predictive uncertainty in groundwater flow and 

transport modeling. The methodology considers the 

combined impact of hydrogeologic uncertainties associated with the conceptual-mathematical basis of a 

model, model parameters, and the scenario to which 

the model is applied. The methodology is based on an 

extension of a Maximum Likelihood implementation of 

Bayesian Model Averaging. Model uncertainty is represented by postulating a discrete set of alternative 

conceptual models for a site with associated prior 

model probabilities. The prior model probabilities reflect a subjective belief about the relative plausibility 

of each model based on its apparent consistency with 

available knowledge and data. Posterior model probabilities are computed and parameter uncertainty is 

estimated by calibrating each model to observed system behavior. Posterior model probabilities are modifications of the subjective prior values based on an objective evaluation of each model’s consistency with 

available data. Prior parameter estimates are optionally 

included. Scenario uncertainty is represented as a discrete set of alternative future conditions affecting 

boundary conditions, source/sink terms, or other aspects of the models. The associated prior scenario 

probabilities reflect a subjective belief about the relative plausibility of the alternative scenarios. A joint 

assessment of uncertainty results from combining 

model predictions computed under each scenario using 

as weights the posterior model and prior scenario probabilities. The computed model predictions incorporate 

parameter uncertainties using, for example, Monte 

Carlo simulation. The uncertainty methodology was 

applied to modeling of groundwater flow and uranium 

transport at the Hanford Site 300 Area. Eight alternative models representing uncertainty in the hydrogeologic and geochemical properties as well as the 

temporal variability were considered. Two scenarios 

representing alternative future behavior of the Columbia River adjacent to the site were considered. The scenario alternatives were implemented in the models 

through the boundary conditions. Alternative models 

were calibrated using hydraulic head and uranium concentration observations over a seven-year period. Uranium concentrations under each scenario were predicted over a 20-year period. Results demonstrate the 

feasibility of applying a comprehensive uncertainty 

assessment to large-scale, detailed groundwater flow 

and transport modeling. Results also illustrate the ability of the methodology to provide better estimates of 

predictive uncertainty, quantitative results for use in 

assessing risk, and an improved understanding of the 

system behavior and the limitations of the models. 

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Public Protection Notification 

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information collection requirement unless the requesting document displays a currently valid OMB control number. 
v

FOREWORD

This report was prepared by Pacific Northwest National Laboratory (PNNL) under a U.S. Department of Energy 

(DOE) Interagency Work Order (JCN Y6465) with the U.S. Nuclear Regulatory Commission (NRC). This research 

report describes a systematic, quantitative approach for combining estimations of uncertainties in the data, conceptual models, parameters, and scenarios related to hydrogeology (i.e., the occurrence and distribution of underground 

water). This is important because the ground-water pathway is a key consideration in assessing the potential for 

radionuclide transport and possible exposure scenarios to the public and, consequently, it is important to evaluate 

and understand the related uncertainties. Toward that end, this report provides details on the primary factors that 

contribute to those uncertainties, and how to quantify them through comparative modeling and analyses of site characterization and monitoring data. 

The combined uncertainties approach described in this report uses a statistical method, known as maximum likelihood Bayesian model averaging, to assess predictive uncertainty in ground-water flow and transport modeling. To 

do so, this approach compares alternative models and assesses their combined predictive uncertainty. This approach 

is demonstrated by application to ground-water flow and uranium transport modeling at the 300-Area of the DOE 

Hanford Site. Toward that end, eight alternative models were considered representing uncertainty in hydrogeologic 

and geochemical properties, as well as the temporal variability. In addition, two scenarios representing alternative 

future behavior of the Columbia River adjacent to the site were also considered. Results demonstrate the feasibility 

of applying a comprehensive uncertainty assessment to large-scale, detailed ground-water flow and transport modeling. In addition, the results illustrate the practical benefits of the approach to provide better estimates of predictive 

uncertainty, quantitative results for use in assessing risk, and an improved understanding of the system behavior and 

limitations of the models. Although the approach and its applications were designed for reviews of radionuclide 

transport at complex decommissioning sites, it is also useful for assessing nuclear facility siting, designing groundwater monitoring programs, remediating ground water, and identifying and selecting strategies to preclude offsite 

migration of abnormal radionuclide releases. 

This approach is consistent with the NRC’s strategic performance goal of making the agency’s activities and decisions more effective, efficient, and realistic by identifying and estimating uncertainties. Toward that end, this report 

demonstrates, using examples relevant to decommissioning analyses, that sources of uncertainty can be identified, 

quantified, and integrated using a comparative model analysis approach. This report also illustrates the effectiveness 

of the combined uncertainty approach to estimate uncertainty in model predictions arising from conceptual, parameter and scenario uncertainties. This information is assisting NRC licensing staff and regional inspectors, Agreement 

State regulators, and licensees in their decision-making by identifying and quantifying overall uncertainties in performance assessment models. 

This report is not a substitute for NRC regulations, and compliance is not required. Consequently, the approaches 

and methods described in this report are provided for information only, and publication of this report does not necessarily constitute NRC approval or agreement with the information contained herein. Similarly, use of product or 

trade names is for identification purposes only and does not constitute endorsement by either the NRC or PNNL. 

Brian W. Sheron, Director 

Office of Nuclear Regulatory Research 

U.S. Nuclear Regulatory Commission
vii

CONTENTS 

Abstract ........................................................................................................................................................................iii

Foreword........................................................................................................................................................................v

Executive Summary....................................................................................................................................................xiii

Acknowledgments .....................................................................................................................................................xvii

1 Introduction ...........................................................................................................................................................1

2 Quantification of Parameter, Conceptual Model, and Scenario Uncertainties.....................................................11

2.1 Parameter Uncertainty ..............................................................................................................................11

2.1.1 Analysis of Parameter Uncertainty ......................................................................................................12

2.2 Conceptual Model Uncertainty.................................................................................................................14

2.2.1 Analysis of Conceptual Model Uncertainty.........................................................................................15

2.3 Scenario Uncertainty ................................................................................................................................15

2.3.1 Analysis of Scenario Uncertainty ........................................................................................................17

3 Joint Estimation of Model, Parameter, and Scenario Uncertainties.....................................................................21

3.1 Bayesian Model Averaging – Combining Conceptual Model and Parameter Uncertainty.......................21

3.1.1 Maximum Likelihood Bayesian Model Averaging (MLBMA)...........................................................22

3.1.1.1 Applicability of MLBMA ...............................................................................................................24

3.2 Incorporation of Scenario Uncertainty .....................................................................................................24

3.2.1 Bayesian Model Averaging Conditioned on a Specific Scenario ........................................................25

3.2.2 Scenario Averaging..............................................................................................................................25

3.3 Summary of Uncertainty Assessment Methodology ................................................................................26

4 Example Application ...........................................................................................................................................29

4.1 Available Site Data and Related Information ...........................................................................................29

4.1.1 Application Field Site Background......................................................................................................29

4.1.2 Hanford 300 Area Data Sources ..........................................................................................................36

4.1.2.1 Surface and Borehole Geophysics ..................................................................................................36

4.1.2.2 Physical and Hydraulic Properties ..................................................................................................37

4.1.2.3 Geochemistry, Uranium Sorption, and Transport ...........................................................................39

4.1.2.4 Source Terms and Boundary Conditions ........................................................................................43
viii

4.1.2.5 Hydraulic Head and Uranium Concentration Data .........................................................................43

4.2 Postulate Alternative Conceptual Models.................................................................................................48

4.2.1 Hydraulic Property Heterogeneity .......................................................................................................52

4.2.2 River Boundary Transience .................................................................................................................52

4.2.3 Adsorption ...........................................................................................................................................52

4.2.4 Alternative Conceptual Model Summary ............................................................................................53

4.3 Alternative Model Implementations .........................................................................................................53

4.3.1 Simulation Domain..............................................................................................................................53

4.3.2 Discretization.......................................................................................................................................54

4.3.3 Hydrogeologic Zonation and Hydraulic Parameters............................................................................54

4.3.4 Initial Conditions .................................................................................................................................55

4.3.5 Geochemical Modeling........................................................................................................................55

4.4 Prior Model Probability............................................................................................................................55

4.5 Prior Parameter Probability ......................................................................................................................59

4.5.1 Modeling Adsorption in Contaminant Transport Models....................................................................59

4.5.2 Hanford Site 300 Area Uranium Example...........................................................................................60

4.5.2.1 No Site-Specific Data Case.............................................................................................................60

4.5.2.2 Known Soil Texture and pH Case...................................................................................................60

4.5.2.3 Available Regional Batch Kd Values ..............................................................................................61

4.5.2.4 Available Site-Specific Batch Kd Values ........................................................................................61

4.5.2.5 Available Site-Specific, Mechanistically-Based, Adsorption Model Case .....................................62

4.5.3 Prior Values of Kd for 300 Area Model Calibration.............................................................................63

4.6 Alternative Hydrologic Scenarios.............................................................................................................64

4.7 Prior Scenario Probability ........................................................................................................................65

4.8 Model Calibration.....................................................................................................................................65

4.8.1 Selection of Calibration Data...............................................................................................................65

4.8.2 Weights Associated with Calibration Data ..........................................................................................65

4.8.3 Joint Model Calibration and Calibrated Model Parameters .................................................................67

4.9 Model Calibration Results........................................................................................................................69

4.10 Posterior Model Probabilities...................................................................................................................73
ix

4.11 Prediction..................................................................................................................................................77

4.11.1 Predictive Simulation Description .......................................................................................................77

4.11.1.1 Modeled Parameter Uncertainty.................................................................................................78

4.11.2 Baseline Scenario.................................................................................................................................78

4.11.3 Alternative Scenario ............................................................................................................................81

4.11.3.1 Scenario-Averaged Prediction....................................................................................................82

5 Conclusions .........................................................................................................................................................85

6 References ...........................................................................................................................................................89

Appendix A: Derivation of Posterior Mean and Variance Considering Model Structure and Parameter Uncertainty 

for a Given Scenario..................................................................................................................................................A.1 

Appendix B: Derivation of Posterior Mean and Variance Considering Model Structure, Parameter, and Scenario 

Uncertainties.............................................................................................................................................................. B.1 

Appendix C: On Evaluation of Recharge Model Uncertainty: A Priori and A Posteriori ......................................... C.1 

Figures 

Figure 1-1. Framework for application of hydrogeologic models to regulatory decision making.................................2

Figure 1-2. Illustration of possible results of a decommissioning analysis using a sensitivity approach. Best-estimate 

predicted peak dose is shown in blue. Sensitivity cases are shown in red.............................................................4

Figure 1-3. Example results of a decommissioning analysis: the probability density function of peak dose results 

from the inclusion of parameter uncertainty..........................................................................................................4

Figure 1-4. Example results of a decommissioning analysis with three conceptual model alternatives: the probability 

density functions of peak dose result from the inclusion of parameter uncertainty for each model ......................5

Figure 1-5. Model averaged probability density function for the three-model example of Figure 1-4 with model 

probabilities (see legend) used as weights.............................................................................................................6

Figure 1-6. Results for two alternative scenarios (top/middle), each of which includes the impact of conceptual 

model and parameter uncertainties. Model-averaged results for each scenario and scenario-averaged result 

assuming probabilities of 0.7 and 0.3 for Scenarios 1 and 3, respectively (bottom)..............................................8

Figure 2-1. Photographs of (left) a trench face from an excavation in the 200 Area (photograph by John Selker, 

Oregon State University) and (right) sediments excavated from a trench beneath the South Process Pond in the 

300 Area (from Bjornstad 2003), both on the Hanford Site, Washington ...........................................................11

Figure 2-2. Ratio of estimated to true parameter values for variance and correlation length of transmissivity for 

seven different inverse methods. Results from Test Problem 1 of Zimmerman et al. (1998). (FF=Fast Fourier 

Transform, FS=Fractal Simulation, LC=Linearized Cokriging, LS=Linearized Semianalytical, ML=Maximum 

Likelihood, PP=Pilot Point, SS=Sequential Self-Calibration).............................................................................12

Figure 2-3. Conceptual response of parameter uncertainty to the quantity and quality of data and the level of 

conditioning.........................................................................................................................................................13

Figure 2-4. Framework for hydrogeologic modeling with scenario uncertainty represented by three alternative 

scenarios ..............................................................................................................................................................18
x

Figure 3-1. The effect of ΔKIC on posterior model probability given as a fraction of the probability for the best 

model ...................................................................................................................................................................23

Figure 3-2. Flowchart for the combined estimation of model, parameter, and scenario uncertainties using Maximum 

Likelihood Bayesian Model Averaging. Data boxes contain duplicated data/information and are shown 

separately to clarify the application of the data/information. ..............................................................................28

Figure 4-1. The Hanford Site and its location within Washington State. The 300 Area is in the southeastern part of 

the site adjacent to the Columbia River. ..............................................................................................................30

Figure 4-2. Plan view of the 300 Area primary liquid waste discharge sites, boundaries of the 300 Area proper and 

the 300-FF-5 Operable Unit, and the water table elevation in March 2003 (NAVD88, m) with inferred 

groundwater flow directions (from Hartman et al. 2004) ....................................................................................31

Figure 4-3. Stratigraphy of the 300 Area. Red lines indicate the boundary of the 300 Area, the North and South 

Process Ponds, and the 316-5 Process Trenches, vertical yellow lines indicated the location of boreholes, u1 

indicates the Hanford Formation, and u5-u9 indicate units of the Ringold Formation (vertical exaggeration 

unknown).............................................................................................................................................................32

Figure 4-4. East-west cross-sections through the 300 Area at (top to bottom) Northing coordinates 117000, 116500, 

116000, 115500, and 115000 m (refer to Figure 4-17 for coordinate locations) ................................................33

Figure 4-5. Average uranium concentrations in groundwater (top of unconfined aquifer) during low river stage 

(December 2002 - top) and during high river stage (June 2003 – bottom) at the Hanford Site 300 Area (from 

Hartman et al. 2004) ............................................................................................................................................34

Figure 4-6. Water level and uranium concentration (top of aquifer) in well 399-1-17A near the southern end of the 

316-5 Process Trenches. Contaminated soil was removed starting in 1991. Discharge of uranium-free water 

continued until December 1994...........................................................................................................................35

Figure 4-7. Conceptual model of uranium remobilization in the 300 Area during high river stages (from Lindberg 

and Chou 2001) ...................................................................................................................................................36

Figure 4-8. Photos of sediments from the location of samples obtained by Serne et al. (2002) for geochemical 

analyses at the south side (top) and the northeast corner (bottom) of the North Process Pond ...........................40

Figure 4-9. Photograph of a pit excavated in the base of the South Process Pond at the Hanford Site 300 Area and 

close-up photos of sediment at discrete depths where samples were taken (from Bjornstad 2003) ....................42

Figure 4-10. Estimates of liquid volume (top) and U-238 discharged to the 316-1 South and 316-2 North Process 

Ponds, the 316-5 Process Trenches, and the 316-3 trenches................................................................................44

Figure 4-11. Monthly discharge as measured below Priest Rapids Dam, the nearest dam upriver from the Hanford 

Site 300 Area .......................................................................................................................................................45

Figure 4-12. Hourly, daily average, and monthly average river stage at the 300 Area in 1996...................................45

Figure 4-13. Inferred water table in the 300 Area during (left) average river stage and (right) high river stage. 

Groundwater movement in the uranium source region is indicated by the red arrow. (from Lindberg and Chou 

2001)....................................................................................................................................................................46

Figure 4-14. Available data from the 300 Area: (top) hydraulic head and (bottom) uranium concentrations in 

groundwater.........................................................................................................................................................47

Figure 4-15. Uranium concentration (μg/l) contours interpreted from groundwater measurements in 1959 made at 

the wells indicated in green .................................................................................................................................48
xi

Figure 4-16. Uranium concentration contours derived from measurements in the year and quarter indicated; 

measurement locations shown in green, contours at 10, 30, 50...150 μg/l. (A) 1977-1985................................49

Figure 4-17. Simulation domain and locations of major features of the 300 Area simulation model .........................56

Figure 4-18. Discretization of three-dimensional simulation domain. Cells are inactive within the Columbia River 

(shown in blue). Hydrogeologic units are also shown in the figure as follows: u1 (teal), u5 (green), u6 (grey), 

u7 (yellow), u8 (red). The black lines indicate an adsorption zone near the river. ..............................................57

Figure 4-19. Monthly average river stage at the 300 Area simulated by Waichler et al. (2005) and used as a 

boundary condition in the groundwater flow models ..........................................................................................58

Figure 4-20. Contours of (left) observed uranium concentration in groundwater in August 1997 and (right) modified 

values used as the initial condition in layer 1 of the simulation models..............................................................58

Figure 4-21. Ranges of Kd values for the Hanford Site 300 Area derived using a hierarchy of data ..........................64

Figure 4-22. Distribution of observation wells from which calibration data were selected.........................................66

Figure 4-23. Distribution of calibration data in time for (top) head and (bottom) concentration observations ...........67

Figure 4-24. Plot of total analytical error with observation of uranium concentration at well 399-1-17A..................68

Figure 4-25. Weighted concentration residuals versus weighted simulated concentrations for Model 4 using (top) 

concentration weights inversely proportional to observed concentrations and (bottom) equal concentration 

weights.................................................................................................................................................................68

Figure 4-26. Estimated means (top) and standard deviations (bottom) of the elementary effects of Model 4 computed 

using the method of Morris (1991). Boxplots represent the distribution over all head and concentration 

measurements used in the calibration. Median, 10, 25, 75, and 90 percentiles shown; outliers are outside 5 and 

95 percentiles.......................................................................................................................................................70

Figure 4-27. Comparison of final calibrated parameter values: (top) hydraulic conductivity parameters, (middle) 

adsorption parameters, and (bottom) dispersivity................................................................................................72

Figure 4-28. Simulated versus observed (top) heads and (bottom) uranium concentrations for Model 4 ...................73

Figure 4-29. Calibration results for (top two plots) well 399-2-2 and (bottom two plots) well 399-1-17A ................74

Figure 4-30. Calibration results for (top two plots) well 399-3-10 and (bottom two plots) well 399-4-11 .................75

Figure 4-31. Model 4 simulated uranium concentrations in December 2004 (color) overlain by contours based on 

observed uranium concentration (μg/l) in the fourth quarter of 2004. Color levels indicate the 10, 30, 50, 70, 90, 

and 110 μg/l locations. Dark blue indicates the Columbia River. .......................................................................76

Figure 4-32. Sample histograms, cumulative distribution functions (mean +/- one standard deviation shown), and 

scatterplot for Model 4 parameters Kh_1 and alphaL resulting from 200 Latin hypercube realizations 

(correlation = 0.27)..............................................................................................................................................79

Figure 4-33. Monte Carlo simulation results for Model 4 in January 2010: (left) concentration mean (μg/l) and 

(right) concentration variance..............................................................................................................................80

Figure 4-34. Monte Carlo simulation results for Model 4 under the baseline scenario: uranium concentration (μg/l) 

at well 399-1-1 for 200 realizations and average concentration ..........................................................................80

Figure 4-35. Probability distributions for the predicted uranium concentration at well 399-1-1 on 1/1/2025: (top) 

empirical probability density functions for the four model alternatives and the model average and (bottom) 

empirical cumulative distribution functions ........................................................................................................81
xii

Figure 4-36. Monte Carlo simulation results for Model 4 under the alternative scenario: uranium concentration (μg/l) 

at well 399-1-1 for 200 realizations and average concentration ..........................................................................82

Figure 4-37. Probability distributions for the predicted uranium concentration at well 399-1-1 on 1/1/2025: (top) 

empirical probability density functions for Model 4 under the two scenarios and the scenario average and 

(bottom) empirical cumulative distribution functions .........................................................................................84

Tables 

Table 1-1. Attribution of primary errors in hydrogeologic model applications (after Bredehoeft 2005) ......................3

Table 1-2. Statistics of individual models and model-averaged results shown in Figure 1-5 using model probabilities 

of 0.5, 0.25, and 0.25 for Models 1, 2, and 3, respectively....................................................................................7

Table 1-3. Statistics of individual scenarios (model-average results) and the scenario-average results shown in 

Figure 1-6 (lower plot) using probabilities of 0.7 and 0.3 for Scenarios 1 and 2, respectively .............................9

Table 2-1. Example formulation of mutually exclusive scenarios from three scenario-characterizing events. Marginal 

probabilities for the three events and resulting scenario probabilities assuming independence between events 

are given. .............................................................................................................................................................19

Table 4-1. Estimated hydraulic conductivities (m/d) for the hydrogeologic units underlying the Hanford 300 Area.39

Table 4-2. Model alternatives considered in the 300 Area application of uncertainty assessment..............................53

Table 4-3. Hierarchical application of data to determine best estimates and uncertainty in Kd values........................60

Table 4-4. Textural characteristics and pH for sediment sample NPP-1 obtained 16 ft. below ground surface from the 

North Process Pond (Zachara et al. 2005) ...........................................................................................................61

Table 4-5. 300 Area groundwater compositions (μg/L) used to calculate Kd (L/kg) values using the SCM, along with 

the calculated Kd values. Units for alkalinity are in μg/L as CaCO3. Included at the bottom are the average, 

standard deviation, and coefficient of variation (COV) for each of the chemical constituents and Kd values for 

all the wells..........................................................................................................................................................63

Table 4-6. Kd values estimated for near-river wells using a surface complexation model ..........................................64

Table 4-7. Prior estimates and calibration weight for the log of distribution coefficient parameters ..........................64

Table 4-8. Initial values and upper and lower limits (in brackets) for parameters estimated in the calibration. Empty 

spaces indicate that the parameter was not present in a model. Log-transformed parameters were calibrated for 

all parameters except alphaL. ..............................................................................................................................71

Table 4-9. Values of parameters that were fixed in the calibrations............................................................................71

Table 4-10. Calibrated parameter values and 95% upper and lower linear confidence limits (in brackets) ................71

Table 4-11. Calibration objective function components for each alternative model ...................................................72

Table 4-12. Calibration results and computation of model probabilities using KIC....................................................77

Table 4-13. Mean, variance, and standard deviation of individual scenarios and the scenario average for the 

predicted concentration at well 399-1-1 on 1/1/2025 ..........................................................................................83
xiii

EXECUTIVE SUMMARY

Regulatory and design applications of hydrogeologic 

models of flow and contaminant transport often involve 

using the models to make predictions of future system 

behavior. Factors contributing to the uncertainty of 

hydrogeologic model predictions include incomplete 

knowledge of the system, variability in system properties, randomness in system stresses, measurement and 

sampling errors, disparity among sampling, simulation, 

and actual scales of the system, long time frames over 

which predictions must be made (e.g., 1000 years or 

more), and multiple exposure pathways or transport 

media. In addition, regulatory criteria may be close to 

contaminant background concentrations potentially 

requiring the precise prediction of small effects. 

The potential benefits of an explicit consideration of 

uncertainty in hydrogeologic modeling include improving decision making by quantifying risk, providing a 

means to systematically weight and combine different 

information sources to estimate the precision of the 

predicted values, providing guidance on what additional data to collect, reducing predictive bias, improving model development by defining the appropriate 

level of detail for model components, and documenting 

and communicating the limitations and assumptions of 

the models. 

A methodology to assess uncertainty in hydrogeologic 

modeling should have several important characteristics. 

The methodology should be comprehensive in the 

sense that all types of uncertainty can be included. The 

methodology should also be quantitative so that results 

can be compared to regulatory criteria. Finally, the 

methodology should be systematic so that it can be 

applied to a wide range of sites and objectives and to 

enable the common application of computer codes and 

methods. We describe the development and application 

of such a methodology to systematically and quantitatively assess predictive uncertainty in groundwater 

flow and transport modeling. The methodology considers the combined impact of hydrogeologic uncertainties 

associated with the conceptual-mathematical basis of a 

model, model parameters, and the scenarios to which 

the model is applied. 

A common approach to uncertainty assessment in hydrogeologic modeling is to assume the conceptual 

model and scenario are known and to address parameter uncertainty only, often using only a sensitivity 

analysis and/or conservative approach. The problem 

with a sensitivity approach to assessing the impact of 

parameter uncertainty is that the probabilities of predicted outcomes are not estimated. There is thus no 

way to quantitatively estimate the risk involved in a 

particular decision. The same holds true for a conservative analysis, with one possible consequence being 

adoption of an overly conservative model. Of potentially more importance, however, is an outcome in 

which the model results are not as conservative as believed. Even when parameter uncertainty is fully addressed (e.g., with a Monte Carlo simulation), it is possible, or even likely, that predictive uncertainty will be 

under-estimated when conceptual and scenario uncertainties are ignored. Published results from hydrogeologic model post-audits demonstrate these outcomes. 

While the potential importance of conceptual model 

uncertainty has been accepted for some time, practical 

methods to assess the impact of model uncertainty on 

prediction have not yet found their way into widespread practice. The methodology presented here is 

based on an extension of a Maximum Likelihood implementation of Bayesian Model Averaging 

(MLBMA). Model uncertainty is represented by postulating a discrete set of alternative conceptual models 

for a site with associated prior model probabilities. 

These prior probabilities reflect a belief about the relative plausibility of each model based on its apparent 

consistency with available knowledge and data. Posterior model probabilities are computed and parameter 

uncertainty is estimated by calibrating each model to 

observed system behavior using maximum likelihood. 

Under commonly assumed conditions, maximum likelihood reduces to generalized least squares parameter 

estimation. In this case, available codes such as PEST 

and UCODE can be used in the model calibrations, 

reducing the effort required to implement the methodology. 

Whereas prior model probabilities are subjective, the 

posterior model probabilities are modifications of these 

subjective values based on an objective evaluation of 

each model’s consistency with available data. Posterior 

model probabilities are valid only in a comparative, not 

in an absolute, sense. They are conditional on the 

choice of models (in addition to being conditional on 

the data) and may be sensitive to the choice of prior 

model probabilities. This sensitivity is expected to diminish with increased level of conditioning on data. As 

computed in this report, posterior model probabilities 

incorporate a measure of model fit (a better fit to the 

data contributes to a higher model probability), a 

measure of parsimony (given an equal fit to the data, a 

model with fewer parameters will have a higher probability), and a factor that, all else being equal, favors a 
xiv

model for which the data is less informative in estimating parameter values (a bias toward maximum parameter uncertainty). 

The MLBMA methodology is extended here to include 

scenario uncertainty, which is represented as a discrete 

set of alternative future conditions affecting boundary 

conditions, source/sink terms, or other aspects of the 

models. Each scenario is associated with a prior scenario probability related to the probability of occurrence of the scenario. A joint assessment of uncertainty 

results from combining model predictions computed 

under each scenario using as weights the posterior 

model and prior scenario probabilities. Because the 

scenario uncertainty applies only to future conditions, 

the model calibrations do not need to be repeated in 

order to estimate the combined impact of scenario, 

conceptual model, and parameter uncertainties. While 

computationally demanding, the methodology is thus 

practicable for application to field-scale modeling. 

To implement the uncertainty assessment methodology 

the following steps are followed. 

1. Postulate alternative conceptual-mathematical 

models for a site and assign a prior probability to 

each model. Optionally assign prior probabilities 

to the parameters of each model. 

2. Postulate alternative scenarios affecting the future 

hydrologic conditions at the site and assign prior 

probabilities to each scenario. 

3. Obtain posterior parameter estimates and the parameter estimation covariance for each model by 

model calibration. Calculate a posterior probability 

for each model using the model calibration results 

and the prior model probabilities. 

4. Predict quantities of interest using each model and 

assess prediction uncertainty for each model using 

Monte Carlo simulation, stochastic moment methods, or first-order approximation. Weight predictions and uncertainties by the corresponding posterior model probabilities and sum the results over 

all models. 

5. Repeat step 4 for each alternative scenario. (Prior 

model probabilities, and consequently posterior 

model probabilities also, may be modified for each 

scenario.) Weight the results for each scenario by 

the appropriate scenario probability and sum the 

results over all scenarios. 

The methodology can be conducted in an iterative fashion; as additional data become available, they can be 

included in the data set and steps 3-5 repeated. When 

additional data indicate that other models or scenarios 

should be considered, or that models should be modified, the entire process may be repeated. 

The uncertainty assessment methodology was applied 

to modeling of groundwater flow and uranium transport at the 300 Area of the U.S. Department of Energy 

Hanford Site. Located on the site are former nuclear 

fuel fabrication facilities, fuels research laboratories, 

several solid waste burial grounds, and liquid effluent 

disposal sites to which uranium and other contaminants 

were formerly discharged. Discharges of uraniumladen liquid wastes occurred from 1944-1984. Although source-removal activities have been conducted 

at the primary waste sites, uranium concentrations in 

groundwater have remained above the drinking water 

standard over a large area, likely due to the presence of 

uranium above the water table. The unconfined aquifer 

at the site is composed primarily of high permeability 

fluvial sediments. Groundwater heads are strongly influenced by the elevation of the Columbia River adjacent to the site. River elevations are controlled by dam 

operations and can vary more than a meter during the 

course of a day. Characterization data indicate the importance of (possibly nonequilibrium) geochemistry in 

determining uranium occurrence and transport. 

Model uncertainty was represented using alternative 

representations of the hydrogeology (either homogeneous saturated hydraulic conductivity or zones of hydraulic conductivity based on identified hydrogeologic 

units) and alternative representations of uranium adsorption (spatially homogeneous adsorption or zones 

based on average water chemistry). Models using a 

steady-state Columbia River elevation were considered 

also, but were rejected as overwhelmingly inferior to 

models using transient river elevations. The alternative 

conceptualizations were implemented using 

MODFLOW and MT3DMS as three-dimensional, unconfined, saturated flow and transport models. The 

period of September 1997 to December 2004 was selected as the simulation period for calibration. Because 

this period was after the period of active waste discharges at the site, the impact on the model calibrations 

of the significant uncertainty in the magnitude and timing of these discharges was reduced. This greatly simplified the model calibrations for the purposes of the 

application. 

Parameter uncertainties for each model were based on 

the estimated parameter covariances resulting from the 

joint calibration of each alternative model to 222 observations of hydraulic head and 208 uranium concentration measurements obtained from 21 wells over the 

seven-year calibration period. Calibrated parameters 

included saturated hydraulic conductivities, dispersiv-
xv

ity, and linear equilibrium adsorption coefficients. 

Overall, the parameter estimates and simulated results 

appeared reasonable. However, a systematic bias in 

simulated heads for the calibrated models appeared to 

be due to the use of a steady inland boundary head 

where actual heads are influenced by the river stage. 

Underestimation of the largest uranium concentrations 

was also apparent and can be attributed to unresolved 

conceptual errors in the models. 

Using equal prior model probabilities, the calibration 

results were used to compute the posterior model probabilities. Of the four alternative models, one had a posterior probability of 98.5% indicating that this model 

should be preferred to the near exclusion of the others. 

This result was somewhat unexpected as the apparent 

differences between calibrated models were small. It 

indicates that additional conceptual models are needed 

at this site and that additional research methods for 

estimating model probabilities is warranted. 

Although the result that one model dominated the posterior probability means that predictions need be made 

with one model only, all four alternative models were 

used to make predictions of flow and transport to illustrate the full application of the methodology. There is 

an independent reason to apply even low-probability 

models in prediction, however. If the conditions of the 

predictive period are significantly different than the 

calibration period, differences between models that 

were not apparent in the calibration may appear during 

prediction. Applying even low-probability models to 

the prediction may bring these differences to light. At 

that point, the conditions of the predictive and calibration periods and the data used in the calibration can all 

be reviewed to determine whether the model probabilities should be reevaluated. Additional applications of 

the methodology may provide better information on 

when these issues are important in hydrogeologic modeling. 

Two scenarios representing alternative future behaviors 

of the Columbia River adjacent to the site were considered. The baseline scenario represented river behavior 

similar to what has been observed over the last 30 

years. An alternative scenario represented river behavior similar to when there was less control of river elevations by dam operations. Predictive simulations were 

carried out using all four calibrated models to compute 

groundwater flow and uranium transport over the period of 2005-2025 under each scenario. The simulation 

results illustrate how model- and scenario-averaged 

predictions are computed and how results can be displayed to clearly indicate the individual contributions 

to predictive uncertainty of the model, parameter, and 

scenario uncertainties. The application demonstrates 

the practicability of applying a comprehensive uncertainty assessment to large-scale, detailed groundwater 

flow and transport modeling. 
xvii

ACKNOWLEDGMENTS

The authors would like to thank Paul Thorne, PNNL, 

for providing the unit designations from PNNL's 

EarthVision geologic model of the 300 Area. Jason 

Keller and Terri Gilbride, PNNL, provided technical 

and editorial reviews of this report. 

Mary Hill, U.S. Geological Survey, and Albert 

Valocchi, Abhishek Singh, and Yonas Demissie, all at 

the University of Illinois, graciously agreed to provide 

comments on this report. 

The authors also gratefully acknowledge the financial 

assistance of the Office of Nuclear Regulatory 

Research of the U.S. Nuclear Regulatory Commission 

(NRC) and the guidance and support of the NRC 

Project Manager, Thomas J. Nicholson. 
1

1 INTRODUCTION

Regulatory and design applications of hydrogeologic 

models of flow and contaminant transport often involve 

using the models to make predictions of future system 

behavior. For example, the primary U.S. Nuclear Regulatory Commission (NRC) criterion for license termination requires an estimate of maximum dose for a 

period up to 1000 years from the time of decommissioning. This dose estimate can be made using a simulation model of the transport of residual radionuclide 

contaminants through the environment. Similarly, the 

design of a subsurface contaminant remediation system 

and a network to monitor performance of that system 

can be facilitated using a model that predicts system 

behavior under the conditions of the remediation. 

A variety of factors conspire to render the predictions 

of hydrogeologic models uncertain, including 

• incomplete knowledge of the system, 

• variability in system properties, 

• randomness in the system stresses, 

• measurement and sampling errors, and 

• disparity among sampling, simulation, and actual 

scales of the system. 

Other factors that may affect the degree of uncertainty 

inherent in these predictions are the potentially long 

time frame over which predictions may be made (e.g., 

1000 years or more), the simulation of transport 

through multiple exposure pathways and media, and 

regulatory criteria that may be close to contaminant 

background concentrations (requiring the prediction of 

small effects). 

Comprehensive and quantitative assessment of the impact of the factors listed above on the uncertainty in 

predictions of hydrogeologic models may require a 

significant investment in time for analysis and data 

collection. Given this, what benefits would justify this 

investment? Morgan and Henrion (1990) considered 

this question and concluded that explicit consideration 

of uncertainty is important for the following reasons. 

• When attitudes toward risk are important, e.g., 

when stakeholders are risk averse (as is commonly 

the case for people living or working near an environmentally contaminated site), considering uncertainty can improve decision making by quantifying 

risk. 

• When multiple, uncertain sources of information 

must be combined (a defining characteristic of environmental transport modeling), uncertainty assessment provides a means to systematically 

weight and combine the information sources to estimate the precision of the predicted value(s). 

• When one of the possible actions is to collect additional data, consideration of uncertainty provides 

guidance on what data to collect and the potential 

benefit in reduced uncertainty. This issue is particularly important in hydrogeologic modeling, 

which is generally constrained by limited characterization data. 

• Empirical evidence suggests that “best estimate” 

answers are regularly biased. Thinking about the 

uncertainty associated with an analysis can reduce 

this bias. 

• Analysis of uncertainty can help with model development, defining the appropriate level of detail 

for model components. 

• A documented uncertainty analysis helps users of 

model results evaluate conclusions and limitations, 

particularly over time as different users arise and 

project objectives evolve. 

• Attempts to characterize and address important 

uncertainties help analysts fulfill a professional 

and ethical responsibility to communicate the implications and limitations of their work. 

It thus appears that the potential benefits of uncertainty 

analysis can justify the expense. The main question is 

how this analysis should proceed. In any case, there are 

several desirable characteristics that should be cultivated. A methodology for uncertainty assessment 

should be comprehensive in the sense that all types of 

uncertainty can be included. The methodology should 

also be quantitative so that results can be compared to 

regulatory criteria. Finally, the methodology should be 

systematic so that it can be applied to a wide range of 

sites and objectives and to enable the common application of computer codes and methods. 

Before presenting the uncertainty assessment methodology adopted in this work, it will be useful to discuss 

a framework for the application of hydrogeologic models to regulatory decision making. As illustrated in 

Figure 1-1, the time domain over which model simulations are conducted can be viewed as two distinct periods. The history-matching period consists of the time 

over which observations of the system are available (as 

indicated by the diamond symbols in Figure 1-1). 

Model building and model evaluation take place within 

the context of the data available during the historymatching period. The predictive period consists of the 

time over which the behavior of the system is to be 

predicted for site management purposes. For many 
2 

Time

Concentration

History-Matching Period Predictive Period

~ 10 years ~ 1000 years

problems, the predictive period will be much longer 

than the history-matching period. 

In general, specific factors resulting in model predictive uncertainty are assessed in the history-matching 

period and then propagated into the predictive period. 

Parameter uncertainty in hydrogeologic models is typically addressed in this fashion. Some factors, however, 

only apply in the predictive period: randomness in future rainfall is an example. As part of the model building and evaluation process, uncertainty in the historymatching period can be reduced by collecting additional data. For example, a pump test could be conducted to reduce uncertainty about the hydraulic conductivity at a specific location. In contrast, uncertainties that apply only to the predictive period cannot be 

reduced by collecting data. For example, the annual 

rainfall 10 years in the future is uncertain due to natural 

variability. Although past and current measurements of 

annual rainfall can better characterize that variability, 

the essential randomness of rainfall ten years hence 

remains. 

Although the analysis described here is limited to hydrogeologic uncertainty, it is comprehensive in the 

sense that the primary uncertainties in most hydrogeologic modeling applications can be included under 

this framework. Three broad types of uncertainty are 

considered. Uncertainties are manifested in a hydrogeologic modeling application as uncertainty in model 

conceptualization, model parameters, and modeling 

scenarios. The model conceptual basis can be thought 

of as a hypothesis about the behavior of the system 

being modeled and the relationships between the components of the system. This conceptualization is typically represented mathematically to render quantitative 

predictions; thus it is appropriate to talk about a conceptual-mathematical model (sometimes referred to as 

model structure). The model parameters are the quantities required to obtain a solution from the model (and 

thus are model-specific). A scenario is defined here as 

a future state or condition assumed for a system, with 

the emphasis on those aspects of a scenario that affect 

the system hydrology (e.g., future irrigation schemes, 

ground-water extraction, natural recharge). With reference to the framework of Figure 1-1, conceptual model 

and parameter uncertainties are assessed in the historymatching period and applied in the predictive period. 

Scenario uncertainty applies to the predictive period 

only. 

What evidence is there for the relative importance of 

conceptual model, parameter, and scenario uncertainties in modeling practice? Published results from hydrogeologic model post-audits were reviewed to attribute the primary modeling errors in these applications to 

conceptual, parameter, or scenario uncertainties. Six 

additional modeling applications described in Bredehoeft (2005) were included in this review. Results are 

shown in Table 1-1 and demonstrate the importance of 

conceptual and scenario uncertainties in contributing to 

model predictive errors. In 9 of the 16 applications, 

conceptual model errors were most significant. Model 

scenario errors were the most significant in 4 of the 16 

applications. Parameter errors were most significant in 

three of the applications. This (limited) review suggests 

that a comprehensive approach to uncertainty assessment in hydrogeologic modeling should not be limited 

to parameter uncertainties, but must also consider the 

potential for significant conceptual model and scenario 

uncertainties if a realistic estimate of predictive uncertainty is desired. 

Given our knowledge of the potential importance of 

hydrogeologic uncertainties and their categorization as 

conceptual model, parameter, or scenario uncertainty, 

Figure 1-1. Framework for application of hydrogeologic models to regulatory decision making
3 

Model Application (Reference) Comments Error 

Phoenix (Konikow 1986) Assumed past groundwater pumping would continue 

in future 

Scenario/ 

Conceptual 

Cross Bar Ranch Wellfield (Stewart and 

Langevin 1999) 

Assumed a 75-day, no-recharge scenario would represent long-term maximum drawdown Scenario/ 

Conceptual 

Arkansas Valley (Konikow and Person 

1985) 

Needed a longer period of calibration Scenario/ 

Parameter 

Coachella Valley (Konikow and Swain 

1990) 

Recharge events unanticipated Scenario 

INEL (Lewis and Goldstein 1982) Dispersivities poorly estimated Parameter 

Milan Army Plant (Andersen and Lu 2003) Extrapolated localized pump test results to larger area Parameter 

Blue River (Alley and Emery 1986) Storativity poorly estimated Parameter/ 

Conceptual 

Houston (Jorgensen 1981) Including subsidence in model improved predictions Conceptual 

HYDROCOIN (Konikow et al. 1997) Boundary condition modeled poorly Conceptual 

Ontario Uranium Tailings (Flavelle et al. 

1991) 

Inadequate chemical reaction model Conceptual 

Los Alamos (Bredehoeft 2005) Flow through unsaturated zone not understood Conceptual 

Los Angeles (Bredehoeft 2005) Flow vectors 90° off in model Conceptual 

Summitville (Bredehoeft 2005) Seeps on mountain unaccounted for Conceptual 

Santa Barbara (Bredehoeft 2005) Fault zone flow unaccounted for Conceptual 

WIPP (Bredehoeft 2005) Assumed salt had no mobile interstitial brine Conceptual 

Fractured Rock Waste Disposal (Bredehoeft 2005) Preferential flow in unsaturated zone unaccounted for Conceptual 

what are the analysis options? One option is to assume 

the conceptual model and scenario are known and to 

address parameter uncertainty only. This has been and 

remains the most common approach to uncertainty 

analysis in hydrogeologic modeling. One way to assess 

the impact of parameter uncertainty is by using a sensitivity approach. An example of this approach is to develop best-estimates of model parameter values using 

data in the history-matching period and then use these 

to compute a best-estimate predicted value in the predictive period. Parameter values are then perturbed 

from their best estimates to determine potential uncertainty in the predicted value. For a decommissioning 

analysis, the results of this sensitivity approach could 

be displayed as in Figure 1-2, which compares the predicted best-estimate peak annual dose (in blue) and 

four sensitivity cases (in red) with the regulatory criterion of 25 mrem/yr. The vertical axis in this figure is 

the probability of peak dose, representing the degree of 

plausibility of the model result (Jaynes 2003). The 

question marks indicate that the actual values of the 

probabilities are unknown, although statements about 

the relative values may be possible (e.g., the bestestimate result is presumably more plausible than the 

sensitivity cases). A variation of this approach is a 

bounding (conservative) analysis in which the desired 

predicted value represents the worst plausible behavior 

of the system (e.g., the right-most sensitivity case in 

Figure 1-2). 

The problem with a sensitivity approach to assessing 

the impact of parameter uncertainty is that, as indicated, the probabilities of predicted outcomes are not 

estimated. There is thus no way to quantitatively estimate the risk involved in a particular decision. In the 

example of Figure 1-2, the risk corresponds to the 

probability of failure in meeting the regulatory criterion, that is, the probability that the peak dose is greater 

than 25 mrem/yr. For a bounding analysis, the significance of the bounding case needs to be assessed to 

avoid being overly conservative. In Figure 1-2, the 

significance of the sensitivity case that violates the 

regulatory criterion rises with its probability, but there 

is no way to know whether that probability is 1% or 

20%. 

A quantitative assessment of parameter uncertainty can 

be completed by computing the probability density 

function of the desired predicted value. This can be 

accomplished by assigning a joint probability distribution to model parameters and propagating this through 

the model, using a Monte Carlo simulation for example. The joint probability distribution is a measure of 

the degree of plausibility of the values of model parameters and is assigned based on site information and 

Table 1-1. Attribution of primary errors in hydrogeologic model applications (after Bredehoeft 2005) 
4 

p(

Δ)

0 20 40 60 80

Δ = Peak Dose (mrem/yr)

?

?

? ?

?

Regulatory Criterion

data available in the history-matching period (using, 

for example, expert judgment or inverse modeling). 

Assessing the impact of parameter uncertainty on 

model predictions in this manner is accepted in policy 

(EPA 1997; NRC 2003) and is not uncommon in practice. For the decommissioning example, the simulation 

result could be displayed as shown in Figure 1-3 in 

which the probability density function of the desired 

predicted value, peak dose, is compared to the regulatory criterion of 25 mrem/yr. As in Figure 1-2, the vertical axis (probability density) represents the degree of 

plausibility associated with the predicted value. Unlike 

the sensitivity approach of Figure 1-2, a quantitative 

estimate of risk is available here. The probability of 

exceeding the regulatory criterion is easily calculated 

from the density function as the area under that portion 

of the curve exceeding 25 mrem/yr, as indicated by the 

hatched area in Figure 1-3. 

As stated above, the examples considered so far have 

assumed that the conceptual model and scenario are 

known. That is, it is implicit in any assessment that 

considers only parameter uncertainty that the resulting 

predictive uncertainty is conditional on the structure of 

the model. It is generally recognized, however, that a 

0.10

0.08

0.06

0.04

0.02

0.00

p(

Δ)

0 20 40 60 80

Δ = Peak Dose (mrem/yr)

Regulatory Criterion

Figure 1-2. Illustration of possible results of a decommissioning analysis using a sensitivity approach. Bestestimate predicted peak dose is shown in blue. Sensitivity cases are shown in red. 

Figure 1-3. Example results of a decommissioning analysis: the probability density function of peak dose results from the inclusion of parameter uncertainty 
5 

hydrogeologic model of a site is invariably an approximation of the actual system. As a consequence, it 

may be possible to postulate more than one conceptual 

model for a site that is consistent with site characterization data and observed system behavior, as evaluated in 

the history-matching period. Although the potential 

importance of conceptual model uncertainty has been 

accepted for some time (Apostolakis 1990; Mosleh et 

al. 1994), practical methods to assess the impact of 

model uncertainty on prediction have not found their 

way into widespread practice. 

In a quantitative uncertainty analysis, parameter uncertainty is typically characterized using continuous probability distributions. When characterizing conceptual 

uncertainty in hydrogeologic modeling, specifying a 

continuum of conceptual model possibilities is likely to 

be infeasible. Instead, it is generally more appropriate 

to postulate a discrete set of alternative conceptual 

models (Neuman and Wierenga 2003). This suggests a 

sensitivity approach to conceptual model uncertainty 

analogous to the sensitivity approach to addressing 

parameter uncertainty. Namely, each model alternative 

is used to simulate the desired predicted value, producing a result that might look like the example in Figure 

1-4. In this figure there are three alternative conceptual 

models, each predicting the same quantity, peak dose. 

Each model result is represented as a probability distribution because the parameters of each model are uncertain. There is no requirement that the models have a 

common set of parameters or that parameters common 

to more than one model have the same value (or probability distribution). 

A sensitivity approach to assessing model uncertainty 

has the same drawback as the parameter sensitivity 

approach discussed with reference to Figure 1-2. Without a quantitative measure of the degree of plausibility 

of model alternatives, it is impossible to determine the 

risk of a decision based on the model predictions. Similarly, a conservative approach to model uncertainty 

relies on an implied belief that the most conservative 

model has a non-negligible degree of plausibility. A 

conservative approach in this case would select Model 

3 for the comparison with the regulatory criterion; this 

is easier to justify if the three models are equally plausible than if Models 1 and 2 are significantly more 

plausible than Model 3. Such a justification requires a 

quantitative measure of model plausibility. 

A quantitative assessment of the combined effects of 

parameter and conceptual model uncertainty can be 

achieved by assigning a discrete probability distribution to the model alternatives. The model predictions 

are then combined using a weighted average with the 

weight for each model’s prediction consisting of that 

model’s probability (e.g., Apostolakis 1990). Analogous to the interpretation of parameter probability, the 

discrete model probability distribution represents the 

degree of plausibility of the model alternatives. Shown 

in Figure 1-5 is the model averaged result for the threemodel decommissioning example with model probabilities of 0.5 for Model 1 and 0.25 for the other two 

models. In this example, Model 1 is thus twice as plausible as Model 2 or 3, with the latter two models being 

equally plausible. The resulting model-averaged probability density function is properly interpreted as a 

0.10

0.08

0.06

0.04

0.02

0.00

p(

Δ)

0 20 40 60 80

Δ = Peak Dose (mrem/yr)

 Model 1

 Model 2

 Model 3

Regulatory Criterion

Figure 1-4. Example results of a decommissioning analysis with three conceptual model alternatives: the 

probability density functions of peak dose result from the inclusion of parameter uncertainty for 

each model 
6 

measure of the degree of plausibility of peak dose that 

takes into consideration the joint effect of parameter 

and model uncertainties. 

The model-averaging approach has been criticized on 

the basis that there is only one model corresponding to 

the physical reality and therefore an average over several models has no physical interpretation (e.g., 

Abramson 1994). Even if one postulates the existence 

of a model corresponding exactly to reality, the inherent complexity of the hydrogeologic environment 

makes it unlikely that this model will be contained in 

any realistic set of alternatives. Rather, although the 

models are physically based, it is likely that all models 

considered will be an approximation to physical reality. 

When appropriately formulated, each model alternative 

will have some merit in reproducing aspects of the 

physical system, this merit being quantified by each 

model’s probability. As articulated in Ye et al. (2004), 

model probability is interpreted as a relative measure 

with respect to the other model alternatives considered. 

Thus if an additional model (Model 4) were to be 

added to the example of Figure 1-5, the individual 

model probabilities would change to accommodate 

this, but the probability of Model 1 would still be twice 

that of Models 2 and 3. A model-averaged consequence 

has an intuitive and consistent meaning under this interpretation of model probability. Note that we are not 

averaging the underlying physics, but the predicted 

consequences of the physics as rendered by the models. 

Model averaging has also been criticized for masking 

information essential to the decision maker (Abramson 

1994). This criticism would be valid if the only quantity the decision maker had available was the modelaveraged probability density function (the black curve 

in Figure 1-5) or some measure derived from it, such as 

the mean peak dose. If, however, the individual model 

results are presented along with the model-average 

results and the model probabilities (as in Figure 1-5 

and Table 1-2), a fully informed decision can be made. 

For example, clear differences in the predicted values 

of models (e.g., Models 1 and 3 in Figure 1-5) suggest 

that a decision maker might justifiably request additional data/information to better discriminate between 

these models (i.e., to modify the model probabilities 

until one model dominates the others). In addition, if 

some conservatism is to be built into the decision, the 

model probabilities can provide a basis for selecting 

the most conservative model (Model 3 in this case) that 

carries a significant degree of plausibility. 

Conservatism could also be included using the modelaveraged results, for example, by comparing the regulatory criterion to a high percentile (e.g., the 90th percentile) of the model-averaged probability distribution, 

or by using a probability of exceedance regulatory criterion instead of a deterministic criterion. As given in 

Table 1-2, the model-averaged mean peak dose is 21.2 

mrem/yr (satisfying the deterministic criterion), the 90th

percentile is 48.5 mrem/yr, and the probability of exceeding 25 mrem/yr is 34%. These factors all suggest 

that a conservative regulatory action may be preferred 

in this case, but based on a fully informed consideration of model and parameter uncertainty (i.e., risk), 

rather than on adoption of the most conservative 

model. 

0.10

0.08

0.06

0.04

0.02

0.00

p(

Δ)

0 20 40 60 80

Δ = Peak Dose (mrem/yr)

 Model 1 (Prob = 0.5)

 Model 2 (Prob = 0.25)

 Model 3 (Prob = 0.25)

 Model-Averaged Result

Regulatory Criterion

Figure 1-5. Model averaged probability density function for the three-model example of Figure 1-4 with 

model probabilities (see legend) used as weights 
7 

Mean 

Dose 

Prob 

(Dose > 25) 90%ile 

Model 1 10.0 8.2 23.0 

Model 2 20.0 23.9 32.7 

Model 3 45.0 97.7 57.8 

Model Average 21.2 34.5 48.5 

Portraying model results in the manner of Figure 1-5 

does not mask essential information, but rather clearly 

illuminates the effect of model uncertainty and the importance of a consistent and defensible means to estimate model probabilities. Model averaging reduces the 

risk of relying on an overly conservative model and 

provides a consistent and quantitative way to address 

model uncertainty in the context of a regulatory decision. 

Parameter and model probabilities, being measures of 

the degree of plausibility of parameter values and 

model alternatives, are based on information and data 

available in the history-matching period. That is, they 

are estimated as part of the process of model development and evaluation. The characterizations of parameter and model uncertainty can be projected into the 

predictive period. As mentioned previously, there is an 

additional source of uncertainty that applies in the predictive period, namely the scenario uncertainty. A scenario is a description of the future conditions under 

which a model is applied. While scenario development 

is commonly associated with radioactive waste disposal performance assessment (NEA 2001), the concept applies to any modeling application in which prediction of future system behavior is made. Scenarios 

are inherently uncertain since they describe conditions 

in the (uncertain) future. 

Similar to the representation of conceptual model uncertainty, the uncertainty in future site conditions can 

be represented as a set of alternative scenarios. In general, only those scenarios that are minimally plausible 

and have a significant potential impact on the predicted 

consequence are included in the set of alternatives. In 

the case where future scenarios are characterized by 

changes in model inputs such as boundary or source 

terms (e.g., changes in surface recharge or pumping 

rates), it may also be useful to represent scenario uncertainties by treating these inputs as random. As in the 

case of model uncertainty, considering discrete scenario alternatives allows for a systematic specification 

of scenario uncertainty. 

Once the alternative scenarios are defined, the impact 

of scenario uncertainty can be addressed using a sensitivity approach in which the desired predictions are 

computed with each model alternative under each alternative scenario. Results for the decommissioning 

example with two alternative scenarios are shown in 

Figure 1-6. In this example, Scenario 1 is identical to 

Figure 1-5. Scenario 2 is an alternative that, in general, 

increases the expected peak dose by about 5 mrem/yr 

for each model (the impact on Model 3 is somewhat 

greater). Although the example is hypothetical, one 

might think of Scenario 2 as an alternative land use 

scenario that increases the expected recharge and, as a 

result, the expected peak dose from residual contamination. For Scenario 2, the model-averaged mean peak 

dose is 27.3 mrem/yr (violating the deterministic criterion), the 90th percentile is 58.5 mrem/yr, and the probability of exceeding 25 mrem/yr is 45%. 

For some applications, a sensitivity approach to scenario uncertainty may be the most appropriate point at 

which to terminate the analysis. The predicted impacts 

under each scenario can be qualitatively assessed to 

determine the course of action. Analogous to the characterization of conceptual model uncertainty, it may be 

possible, however, to assign a discrete probability distribution to the scenario alternatives. In that case, a 

scenario-averaged probability distribution of the desired predicted value can be computed as a weighted 

average of the individual scenario (model-averaged) 

results with the weights being the scenario probabilities. Sample results for the decommissioning example 

are shown in Figure 1-6 (bottom) using probabilities of 

0.7 and 0.3 for the two scenarios. The model-averaged 

distributions include the impacts of parameter and conceptual model uncertainties under each scenario. The 

scenario-averaged distribution includes the combined 

impacts of parameter, conceptual model, and scenario 

uncertainties. Relevant statistics can be directly determined from the scenario-averaged result. For the example in Figure 1-6, the scenario-averaged mean peak 

dose is 23.0 mrem/yr, the 90th percentile is 52.1 

mrem/yr, and the probability of exceeding 25 mrem/yr 

is 38%. Summary statistics for the individual scenarios 

and the scenario average are listed in Table 1-3. 

As with model probabilities, the scenario probabilities 

should be interpreted as relative values conditional to 

the alternatives considered. If an additional scenario 

alternative were to be added to the two considered in 

Figure 1-6, the probability of Scenario 1 would still be 

2.33 times the probability of Scenario 2, reflecting the 

judgment that Scenario 1 is more than two times as 

Table 1-2. Statistics of individual models and 

model-averaged results shown in 

Figure 1-5 using model probabilities of 

0.5, 0.25, and 0.25 for Models 1, 2, and 3, 

respectively 
8 

0.10

0.08

0.06

0.04

0.02

0.00

p(

Δ)

0 20 40 60 80

Δ = Peak Dose (mrem/yr)

Scenario 1

 Model 1 (Prob = 0.5)

 Model 2 (Prob = 0.25)

 Model 3 (Prob = 0.25)

 Model-Averaged Result

Regulatory Threshold

0.10

0.08

0.06

0.04

0.02

0.00

p(

Δ)

0 20 40 60 80

Δ = Peak Dose (mrem/yr)

Scenario 2

 Model 1 (Prob = 0.5)

 Model 2 (Prob = 0.25)

 Model 3 (Prob = 0.25)

 Model-Averaged Result

Regulatory Threshold

0.05

0.04

0.03

0.02

0.01

0.00

p(

Δ)

0 20 40 60 80

Δ = Peak Dose (mrem/yr)

 Scenario 1 (Prob = 0.7)

 Scenario 2 (Prob = 0.3)

 Scenario-Averaged Result

Regulatory Threshold

Figure 1-6. Results for two alternative scenarios (top/middle), each of which includes the impact of conceptual 

model and parameter uncertainties. Model-averaged results for each scenario and scenarioaveraged result assuming probabilities of 0.7 and 0.3 for Scenarios 1 and 3, respectively (bottom).
9 

plausible as Scenario 2. Because the scenario probabilities define a discrete distribution for scenario uncertainty, the sum of all scenario probabilities will be 1.0 

for any number of scenario alternatives. 

Mean 

Dose 

Prob 

(Dose > 25) 90%ile 

Scenario 1 

(model-average) 

21.2 34.5 48.5 

Scenario 2 

(model-average) 

27.3 44.8 58.5 

Scenario Average 23.0 37.6 52.1 

Table 1-3. Statistics of individual scenarios (modelaverage results) and the scenario-average 

results shown in Figure 1-6 (lower plot) 

using probabilities of 0.7 and 0.3 for Scenarios 1 and 2, respectively 
11

2 QUANTIFICATION OF PARAMETER, CONCEPTUAL MODEL, AND 

SCENARIO UNCERTAINTIES

2.1 Parameter Uncertainty

One of the primary factors that contribute to hydrogeologic uncertainty is the natural variability of the 

subsurface. Examples of this variability are illustrated 

in Figure 2-1, two photos from excavations at the Hanford Site in Washington State. On the left is a trench 

face from an excavation in the 200 Area. A large variation in soil particle size can be seen, ranging from fine 

silts to very coarse gravels. The profile shows a layered 

structure with evidence of cross-bedding; the scale of 

the structures is on the order of a few centimeters. This 

variation results in hydraulic and transport properties 

that may vary over several orders of magnitude on this 

same small scale. Laboratory and some field measurements are likely to be made on a somewhat larger 

scale, perhaps 10 cm or more. Exhaustive sampling to 

determine the exact nature of the subsurface at this 

scale will be impossible, thus requiring interpolation 

between measurements and other indirect methods to 

estimate properties at unmeasured locations. In addition, the simulation scale for most practical applications (and thus the scale of the parameters) is likely to 

be significantly larger than the measurement scale, 

from a few tens of centimeters to many meters. 

On the right of Figure 2-1 is a photo of sediments excavated from a trench beneath the South Process Pond 

in the 300 Area. A large fraction of these sediments is 

made up of cobble-sized material, which makes it difficult to obtain representative samples for laboratory 

analysis and to conduct field measurements (e.g., of 

Figure 2-1. Photographs of (left) a trench face from an excavation in the 200 Area (photograph by 

John Selker, Oregon State University) and (right) sediments excavated from a trench beneath the South Process Pond in the 300 Area (from Bjornstad 2003), both on the Hanford 

Site, Washington 
12 

hydraulic conductivity). Also present in these sediments are large clasts composed of semi-consolidated, 

fine-grained sediments eroded upstream and deposited 

in the 300 Area during flood events (Bjornstad 2003). 

These are indicated in Figure 2-1 by the white ellipses. 

Characterization of these clasts, their preponderance, 

location, and effect on flow and transport is unlikely to 

be successful in a deterministic framework. 

Measurement errors are relatively easy to quantify 

compared to other sources of uncertainty. While their 

impact on parameter uncertainty may sometimes be 

relatively small, Holt et al. (2002) provide evidence 

that relatively simple measurement errors can introduce 

significant parameter uncertainties. In their simulations, they also observed that the measurement errors 

produced spurious parameter correlations, an effect that 

has likely been poorly appreciated in most applications. 

An additional source of parameter uncertainty that has 

likely not been fully appreciated can be illustrated using results presented in Zimmerman et al. (1998). They 

compared results from seven models calibrated on the 

same set of data by different participant groups using 

different inverse methods. The ratio of estimated to 

true parameter values for the variance and correlation 

length of the transmissivity are shown in Figure 2-2 for 

each of the inverse methods used. The true transmissivity field was synthetically generated. An exponential 

model was fit to the average empirical variogram for a 

set of realizations obtained from each inverse method. 

The results shown are for Test Problem 1, the simplest 

transmissivity model used (an isotropic, exponential 

variogram). Parameter errors reflect two sources of 

uncertainty: use of different inverse methods and applications by different experts. The latter resulted in differences in model conceptualization for each inverse 

method, which contributed to the significant differences in parameter estimates. Interestingly, one of the 

conclusions of this study was that the inverse methods 

applied did not adequately assess the prediction uncertainty. “The total uncertainty could therefore be better 

described by the results of the ensemble of several 

methods, as any one single method in general tends to 

underestimate the uncertainty.” (Zimmerman et al. 

1998, pg. 1405) This observation is consistent with the 

thesis of this report, that including conceptual model 

uncertainty (in this case, using an ensemble of inverse 

methods) improves estimates of predictive uncertainty. 

2.1.1 Analysis of Parameter Uncertainty 

The analysis of parameter uncertainty has received 

much attention in the literature. Helton (1993) and 

Transmissivity Variogram

0

0.5

1

1.5

2

2.5

FF FS LC LS ML PP SS

Inverse Method

Est./True Parameter Value

Variance

Correl. Length

Figure 2-2. Ratio of estimated to true parameter values for variance and correlation length of transmissivity 

for seven different inverse methods. Results from Test Problem 1 of Zimmerman et al. (1998). 

(FF=Fast Fourier Transform, FS=Fractal Simulation, LC=Linearized Cokriging, LS=Linearized 

Semianalytical, ML=Maximum Likelihood, PP=Pilot Point, SS=Sequential Self-Calibration) 
13 

McKay (1995) provide discussions of parameter uncertainty that are particularly relevant to dose assessment 

modeling. See also the recent review of Helton et al. 

(2006) and the book of Hill and Tiedeman (2007). The 

primary steps involved in addressing uncertainty in 

model parameters are 

• characterization of parameter uncertainty, 

• propagation of parameter uncertainty into model 

output uncertainty, and 

• parameter sensitivity analysis. 

To the extent that the parameters considered represent 

what is unknown about the system and important to 

predictions, uncertainty analysis based on parameters 

becomes more useful. 

Parameter estimation, including the characterization of 

parameter uncertainty, is driven by the available data 

and information. Figure 2-3 is a conceptual representation of the relative parameter uncertainty as a function 

of the quantity and quality of the data used in the parameter estimation process and the level of conditioning. Conceptually, we expect parameter uncertainty to 

be reduced as the quantity/quality of data increases and 

as the level of conditioning increases. No conditioning 

corresponds to the use of prior parameter estimates. 

Meyer and Gee (1999) discuss data sources for characterizing prior hydrogeologic parameter uncertainty in 

the context of dose assessment modeling for license 

termination decisions. In data-limited applications, 

prior parameter probability distributions can also be 

based on the subjective opinions of one or more experts 

(Morgan and Henrion 1990). 

Meyer et al. (1997) demonstrated the use of information from national-scale databases to specify prior parameter distributions and the subsequent updating of 

these distributions using site-specific parameter data in 

a Bayesian approach. This is an example of conditioning on parameter measurements (see Figure 2-3). Another example of this level of conditioning is the application of kriging to interpolate from a set of hydraulic 

conductivity (or other parameter) measurements. 

When observations of state variables (e.g., hydraulic 

head, radionuclide concentration) are available at a site, 

formal calibration methods can be used to improve 

parameter estimates and characterize the uncertainty of 

these estimates (Carrera and Neuman 1986a; Hill 1998; 

Hill and Tiedeman 2007). Such inverse modeling is a 

means of conditioning on system behavior, which can 

be carried out with or without the inclusion of parameter measurements. Calibrated parameter estimates represent the application of the maximum amount of 

data/information and yield parameters with the minimum uncertainty. An application to unsaturated flow 

presented in Wang et al. (2003) illustrates the relationships between the data used in parameter estimation 

and the resulting prediction uncertainty. 

Note that parameter estimates obtained without condi 

tioning or that are conditioned only on parameter 

Figure 2-3. Conceptual response of parameter uncertainty to the quantity and quality of data and the level of 

conditioning 

Low

High 

Quantity/Quality of Data

Limited/Poor Abundant/Good 

Conditioning 

None

on Parameter 

Measurements

on System 

Behavior

Relative Parameter Uncertainty

on System 

Behavior & 

Param. Meas.
14 

measurements may be independent of a model. As discussed in Meyer and Gee (1999), however, there must 

be a correspondence between the estimates and the 

parameters assigned those estimates, e.g., a model that 

has a single value of a parameter representing a site 

must be assigned a value that represents a mean. Similarly, the uncertainty in that parameter value must represent uncertainty in the mean. Because they rely on an 

inverse model, calibrated parameter estimates are 

model-dependent and can be expected to change if the 

underlying flow or transport model is modified. In fact, 

most calibration methods assume the model is correct. 

Errors thus represent the uncertainty in parameters 

given that the model is correct. This will underestimate 

parameter uncertainty. Hill (1998) and Hill and Tiedeman (2007) discuss these and related issues. 

Zimmerman et al. (1998) evaluated a variety of calibration methods using a set of hypothetical (generated) 

data based on the Waste Isolation Pilot Plant site. 

Transmissivity fields for two-dimensional groundwater 

flow models were calibrated on four test problems. One 

of their conclusions was that the calibrated models 

consistently underestimated the “true” variability in 

transport. The maximum likelihood (Carrera and Neuman 1986a) and sequential self-calibration (GomezHernandez et al. 1997) methods were consistently 

ranked higher than the other methods. The sequential 

self-calibration method offers the advantage of producing spatially variable transmissivity fields that honor 

the spatial statistics of the transmissivity field. A calibrated, stochastic groundwater simulation can be carried out using a set of these fields in a Monte Carlo 

simulation. Calibration must be carried out for each 

realization, however. The maximum likelihood method 

is more general and can be applied to the calibration of 

a wide variety of parameters, including statistical parameters. Other alternatives include direct calibration 

of stochastic moment equations, recently demonstrated 

by Hernandez et al. (2006), and the gradual deformation method of Hu et al. (2001). 

Computer codes that can be adapted to the calibration 

of any simulation model are available (Poeter et al. 

2005; Doherty 2004). A method for calibrating geostatistically-simulated parameter fields (similar to the sequential self-calibration method) has recently been 

demonstrated using PEST (Doherty 2003). 

A variety of methods for propagating parameter uncertainty are available, including Monte Carlo simulation, 

the first-order, second-moment method (Kunstmann et 

al. 2002; Vecchia and Cooley 1987), the stochastic 

response surface method (Isukapalli et al. 1998), and 

stochastic moment methods (Dagan and Neuman 1997; 

Zhang 2001). Monte Carlo simulation is the most generally applicable method. The stochastic moment 

methods are appealing because of their potential computational advantage over Monte Carlo simulation. 

Recent progress in handling conditions that introduce 

nonstationarities (Zhang 2001) have made these methods more generally applicable. 

Uncertainties must be defined on a site-specific basis 

and the importance of individual sources may vary site 

by site or even with different objectives at the same 

site. Determination of the parameters that are most 

important to the prediction uncertainty is the final element of an assessment of parameter uncertainty. This is 

generally carried out through the implementation of 

sensitivity analysis (Saltelli et al. 2000a, 2004; Helton 

1993). Sensitivity measures may also be obtained during the calibration procedure (Hill 1998; Tiedeman et 

al. 2003; Hill and Tiedeman 2007). Global sensitivity 

methods (Borgonovo et al. 2003; Saltelli et al. 2000b, 

2004; McKay 1995; Hill and Tiedeman 2007) partition 

the total prediction variance according to the contribution of each parameter and also determine the contribution to prediction variance due to interactions between 

parameters. 

2.2 Conceptual Model Uncertainty

As discussed in the previous chapter, hydrogeologic 

uncertainty may result in valid alternative model structures or conceptualizations. When multiple model conceptualizations are consistent with the available data, it 

may not be justifiable to rely on a single model structure. Relying on a single conceptual representation of a 

system has two potential pitfalls: the rejection by omission of valid alternatives, and reliance on an invalid 

representation by failing to adequately test it. The potential consequences are underestimation of uncertainty 

by under-sampling model space and biased results by 

relying on an invalid model. 

As mentioned previously, conceptual model uncertainty refers here to uncertainty in both the conceptualization of the system and the mathematical implementation of that conceptualization in a model. Conceptual model alternatives are based on the available 

site data and other relevant information; each represents a distinct conceptualization of system characterization or behavior. For example, alternative conceptual 

models might be represented by the presence and absence of leakage from an underlying aquifer; or the 

presence and absence of matrix-fracture interaction in a 

fractured rock. Each conceptualization may be implemented in more than one way: for example, a fractured 

rock may be represented as an equivalent porous medium or as a discrete network of fractures. The process 

of conceptual-mathematical model development may 
15

be iterative as additional site data becomes available 

and conceptual models are updated. 

2.2.1 Analysis of Conceptual Model 

Uncertainty 

Methods for the quantification of conceptual model 

uncertainty are much less well established than those 

addressing parameter uncertainty. Mosleh et al. (1994) 

provide a good introduction to the issues involved. 

Neuman and Wierenga (2003) discuss a wide variety of 

issues related to hydrogeologic conceptual model uncertainty, including many instances of its practical importance. 

While it is generally possible to specify a reasonable 

probability distribution representing the complete set of 

possibilities for the value of a parameter, it is not generally possible to specify the complete set of possible 

conceptual model alternatives. As a result, conceptual 

model uncertainty has generally been represented as a 

discrete distribution, with a small number of model 

alternatives taken as the complete set of possibilities. 

Any approach based on evaluation of a discrete set of 

alternative models will only be as good as the set of 

alternatives. That is, if the set of alternatives does not 

represent the full range of possibilities, conceptual 

model uncertainty will be underestimated. In Neuman 

and Wierenga’s (2003) extensive discussion of conceptual model uncertainty, they provide some advice on 

the generation of alternatives, summarized as follows. 

• From the assembled database of site-specific data 

and other relevant information, consider alternative representations of space-time scales, number 

and type of hydrogeologic units, flow and transport property characterization, system boundaries, 

initial conditions, fast flow paths, controlling 

transport phenomena, etc. 

• Each conceptual model alternative should be supported by key data. 

• Minimize inconsistencies, anomalies, and ambiguities. 

• Apply the principle of Occam’s window (Jefferys 

and Berger 1992; Madigan and Raftery 1994) according to which one considers only a relatively 

small set of the most parsimonious models among 

those which, a priori, appear to be hydrologically 

most plausible in light of all knowledge and data 

relevant to the purpose of the model and, a posteriori, explain the data in an acceptable manner. 

• Maximize the number of experts involved in the 

generation of alternative conceptualizations. 

• Articulate uncertainties associated with each alternative conceptualization. 

Because the set of alternative conceptual models is 

unlikely to represent the full range of possibilities, 

evaluations of model uncertainty should be viewed as 

relative comparisons. That is, they may be used to conclude that one model is better than another for the intended purpose, but they cannot necessarily be used to 

conclude that any model is a good model. In addition, 

as stated above, the contribution of model uncertainty 

to overall prediction uncertainty will be underestimated. 

Having defined the set of alternatives, the options for 

addressing conceptual model uncertainty include the 

following. 

• Evaluate each alternative and select the best 

model. This may be carried out through an informal comparison (James and Oldenburg 1997; Cole 

et al. 2001b) or through evaluation of a formal 

model selection criterion (Burnham and Anderson 

2002). As mentioned, selection of a single model 

may not always be justifiable. 

• Evaluate each alternative and combine the results 

using some weighting scheme, such as the likelihood-based weighting of Beven and Freer (2001), 

the multimodel ensemble approach of Krishnamurti et al. (2000), the model likelihood weighting 

of Burnham and Anderson (2002), and the model 

probability weighting of Draper (1995). 

Neuman (2003) and Ye et al. (2004) reviewed these 

and other approaches that have been used to address 

conceptual model uncertainty. As described in the previous chapter and presented in detail in the following 

chapter, the method used here is based on model probability weighting. 

2.3 Scenario Uncertainty

The concept of a scenario has been much discussed in 

the literature related to performance assessment of nuclear waste disposal facilities. Definitions provided by 

practitioners working in this area emphasize the notion 

of scenarios as a set of alternative future conditions for 

assessing facility performance. Scenario development 

generally takes place within an iterative process of 

modeling a system (NEA 2001). The system in this 

case is the waste disposal facility, the surrounding 

natural environment, and any external factors acting on 

these. 

In the nuclear waste disposal arena, scenario development is intimately linked with the assessment of features, events, and processes (FEPs) for a system. These 

terms generally refer to characteristics of the system 

(features), factors acting on the system (events), and 
16

phenomena governing the behavior of the system 

(processes). NEA (2000) provides a database of generic FEPs from which site-specific scenarios can be 

developed through a process of screening and adjustment based on site-specific information. Scenario development and the use of the FEP process are not limited to the assessment of proposed waste disposal facilities. An application of this process for a site-wide 

assessment of impacts from (numerous) currently contaminated areas and future waste disposal operations at 

the Hanford Site is given by Solar et al. (2001). In this 

application, generic FEPs were excluded based on a 

low probability of occurrence and/or a lack of consequences. Thus only those FEPs that have an impact on 

the performance criteria were included in the scenarios. 

Consideration of FEPs was deferred when there was 

insufficient information to make a decision about inclusion in or exclusion from the analysis. A similar 

procedure was followed by Moschandreas and Karuchit (2002). 

In the context of dose assessment modeling for decommissioning, the U.S. Nuclear Regulatory Commission (NRC) refers to “exposure scenarios,” which it 

defines as “reasonable sets of human activities related 

to the future use of the site. Therefore, scenarios provide a description of future land uses, human activities, 

and behavior of the natural system” (NRC 2000, p. 

C26). A number of exposure scenarios are defined in 

NRC (2000). 

As indicated by the above discussion, the term scenario 

is used to indicate a general statement about possible 

future conditions. For example, a climate change scenario is a general statement describing a possible 

change in climate. A residential farmer exposure scenario is a general description of the pathways leading 

to possible exposure. To evaluate the effect of a scenario, however, the specific characteristics of the scenario must be specified. Climate change, for example, 

may involve changes to the average and extreme values 

of temperature and to the timing, spatial distribution, 

and total amount of precipitation and recharge. A residential farmer scenario involves specification of the 

extent and rate of irrigation and the location and rate of 

possible groundwater extraction. In this report we are 

primarily concerned with the specific characteristics of 

scenarios. In addition, our interest is limited to the effect of a scenario on the hydrologic aspects of a system. Scenario elements that may affect the hydrologic 

characteristics of a site include: geological events (e.g., 

earthquake, landslide), climatic events (e.g., flood, 

change in precipitation or temperature), changes in 

engineered components (e.g., surface or subsurface 

barriers whose hydrologic properties change over 

time), and human activities (e.g., excavation, well drilling, land use changes that affect the rate and chemistry 

of recharge). Hydrologic effects of a scenario could 

include slow changes over time (e.g., a gradual increase in precipitation), or sudden events (e.g., a flood). 

We refer to the future hydrologic state of a system as a 

hydrologic scenario (which we will also refer to simply 

as a scenario). Regardless of how large or small a system of interest may be, it is always part of a larger system (which makes it open) and made up of smaller 

systems (which render it complex). To render any reasonable hydrologic predictions for an open, complex 

system, we must be able to describe its present and 

future features, hydrologic processes operating within 

it, and events that drive them. The system’s features 

include the geometric boundaries, geology, hydrogeologic properties, nature and distribution of permeating fluids, topography, physiography, and climate. Hydrologic processes include fluid flow, advective and 

diffusive chemical and energy transport, phase transitions such as evaporation and condensation, geochemical and biological alterations, and radioactive decay. 

Events include driving forces acting on the system’s 

external boundaries (e.g., precipitation, evaporation, 

transpiration, infiltration, chemical spills) and acting 

internally (e.g., pumping and injection through wells, 

contamination from nonaqueous liquid sources at 

depth, contamination through wells or other openings). 

No matter what anthropogenic or natural changes occur 

in a system in the predictable (though uncertain) future, 

such actions will at most modify one or more of the 

above three system elements (features, events, processes) in a predictable (though generally uncertain) 

manner. To postulate hydrologic scenarios for a system 

thus translates to evaluating the possible future features, events, and processes of the system. To determine the effect of hydrologic scenarios on exposure it 

is necessary that the features, events, and processes be 

amenable to mathematical descriptions, i.e., a mathematical model. Typically the features will be expressed 

in terms of a parameterized description of the system’s 

geometry and its properties, the processes in terms of 

governing equations that include these parameters, and 

the events in terms of initial, boundary, and source 

terms constraining or entering into the governing equations. 

If a system is defined (arbitrarily) such that the soil 

surface is one of its boundaries, then a scenario with 

conditions acting at or near this surface (climatic conditions, precipitation, evapotranspiration, surface or 

near surface contamination) would manifest itself in 

the form of boundary conditions. If a river is modeled 

(arbitrarily) as a boundary, a scenario defining conditions in the river (stage, contaminant levels) would also 
17

be reflected in the boundary conditions. (It may be better to make the river part of the system being modeled, 

in which case conditions upstream would form the 

boundary). In general, the water table would be part of 

the system and so computed from the system model. If 

it is treated (arbitrarily) as a boundary, however, then a 

scenario affecting conditions at the water table would 

enter as part of the boundary conditions. Similarly, 

water pH would generally be computed using the system model, but if it is prescribed then a scenario that 

affects water pH would enter the analysis as a modification to the system features. If a scenario specifies 

that water is pumped or contamination takes place 

through a well, this is part of an internal source/sink 

term. If a scenario specifies that a farmer irrigates a 

field with contaminated water, this is a boundary condition. If the system is subject to tectonic events such 

as faulting (often associated with earthquakes), the 

occurrence of such a scenario would change the system 

features and thus its parametric description. If a dyke 

or volcano intrudes into the system, this will again 

change the system features and its parametric description. If heat generated at a repository causes water to 

boil near the repository and to condense at some distance from it, this would be reflected in the governing 

equations. 

In summary, a hydrologic scenario can be reduced to a 

set of conditions described by the three elements of any 

simulation model: geometry and parameterization (features), structure of governing equations (processes), 

and driving forces (events). Since we presently know 

(imperfectly) how to deal with parameter uncertainty, 

uncertainty in the model structure, and uncertainty in 

the forcing terms, we know mathematically how to deal 

with scenario uncertainty. Most of the time a scenario 

will impact mainly forcing terms, which are often easier to deal with. 

2.3.1 Analysis of Scenario Uncertainty 

Uncertainty in scenarios can be considered on several 

levels. One approach is to admit the possibility that the 

future exposure scenario is unknown. Given that the 

future occurrence of a particular exposure scenario 

depends entirely on human behavior, however, specifying probabilities of occurrence for exposure scenarios 

is a speculative proposition and averaging the consequence over multiple exposure scenarios appears unjustifiable. A conservative approach may be more appropriate in this case. 

Given a particular exposure scenario, however, there 

may still be many uncertain elements associated with 

that exposure scenario. For example, the future infiltration rate depends on future precipitation rates and patterns, whether or not the land is irrigated, and the rate 

of irrigation if it is used. Future uncertainty in the infiltration rate may affect exposures through pathways 

involving groundwater migration. These uncertainties 

in the future hydrologic conditions are more amenable 

to representation using probability theory and are the 

aspects of scenario uncertainty to which the methods 

discussed here are applicable. 

The primary considerations in an analysis of hydrologic scenario uncertainty are (1) determining those 

future hydrologic conditions that are most important to 

include in the analysis, (2) characterizing the uncertainty of those conditions, and (3) evaluating hydrologic scenario uncertainty jointly with model and parameter uncertainty. 

Identification of future hydrologic conditions that have 

potentially significant impacts on exposure are best 

undertaken as part of a model building process that 

includes exposure scenario development. Established 

methodologies, such as that described in NEA (2001), 

can provide guidance on the process. In addition to the 

use of the International FEP database, NRC (2000) 

suggests the use of Kennedy and Strenge (1992), Shipers (1989), and Shipers and Harlan (1989) for identifying appropriate exposure pathways for decommissioning sites. 

Upon identifying the exposure pathways, hydrologic 

conditions affecting contaminant transport along the 

pathways can be specified. Consideration of conceptual 

model, parameter, and model forcing uncertainties during the model building process (i.e., in the history 

matching period of Figure 1-1) will help to identify 

uncertainties in the predictive period of model application. Principles articulated in Neuman and Wierenga 

(2003) for generating alternative conceptual models 

may be useful in postulating alternative hydrologic 

scenarios. Among these are the principle of Occam’s 

window according to which one would consider only a 

relatively small set of the most parsimonious scenarios 

among those which, a priori, appear to be hydrologically most plausible in light of all knowledge and data 

relevant to the purpose of the model. Figure 2-4 illustrates the modeling framework with scenario uncertainty represented as a set of three discrete scenarios. 

An additional principle valuable in postulating alternative scenarios is to maximize the number of experts 

involved in their generation. 

Since scenarios describe future conditions, the probability of occurrence must be assigned to each scenario 

based on a subjective understanding and judgment of 

the plausibility of the scenario. This does not mean that 

scenario probabilities need be speculative. It does 
18 

Time

Concentration

History-Matching Period Predictive Period

Scenario 1

Scenario 3

Scenario 2

~ 10 years ~ 1000 years

mean, however, that there is no opportunity to condition one’s belief about the probability of occurrence of 

a given scenario on observations. In Bayesian terms, 

scenario uncertainty can be characterized by prior 

probabilities only. As discussed in the previous chapter, prior probabilities are appropriately interpreted as a 

subjective measure of the degree of plausibility based 

on current knowledge, experience, and judgment. Thus 

the evidence available to establish the probability of a 

scenario will depend on the specific scenario. For example, established methods are available to estimate 

the probability of a specific flood occurring within a 

1000-year period. No such methods are available for 

estimating the probability that irrigated agriculture will 

occur at a site. However, the probability of irrigated 

agriculture need not be complete speculation. Evaluation of the fraction of nearby land with similar soil and 

climatic attributes that is under irrigation would be a 

reasonable basis for establishing the probability of irrigated agriculture occurring at a site, assuming that 

other factors affecting the occurrence of irrigated agriculture (e.g., climatic, economic, and demographic 

conditions) remain relatively unchanged. 

As discussed in the following chapter, using the methodology presented in this report requires that the alternative scenarios must be mutually exclusive and collectively exhaustive. Although it is likely impossible to 

prove that a set of scenarios is collectively exhaustive, 

a relatively small set of scenarios may adequately represent the primary sources of uncertainty in future hydrologic conditions, particularly if the scenarios can be 

expressed in a general way. An example is a climate 

change scenario, which may have several impacts on 

the models. By specifying the scenario general way we 

avoid having to consider each of the individual impacts 

separately. Because we require that the set of alternative scenarios is collectively exhaustive, scenario probabilities should be interpreted as relative probabilities 

(i.e., relative to the other scenarios in the set). 

Alternative scenarios are often likely to be characterized as discrete events. Climate change, floods, and 

introduction of irrigated agriculture are all examples of 

discrete events affecting the hydrologic conditions at a 

site. Such events are often not mutually exclusive (e.g., 

the occurrence of irrigated agriculture does not preclude the occurrence of climate change). By defining 

scenarios as possible combinations of alternative 

events, the scenarios can be made mutually exclusive. 

An example for three events is shown in Table 2-1. A 

“1” in the table signifies the occurrence of the event in 

a scenario and a “0” indicates the absence of that event. 

Scenario 1 in Table 2-1 has none of the events occurring and might be referred to as a reference scenario, 

perhaps characterized by the continuation of current 

hydrologic conditions into the future. For n events, this 

procedure will result in 2n

 scenarios; some of these 

scenarios may be discarded because of an insignificant 

probability or because they are not of regulatory concern. 

If the scenarios are enumerated from a set of events 

such as in Table 2-1, the scenario probabilities can be 

determined from estimates of the marginal and conditional probabilities of the events. Procedures to ensure 

that the results are consistent with probability theory 

are available (e.g., De Kluyver and Moskowitz 1984; 

Brauers and Weber 1988). If the events are independent, the scenario probabilities can be easily computed 

from the marginal probabilities of the events, as illustrated in Table 2-1. Note that the marginal probabilities 

for the events characterizing the scenarios may sum to 

more than 1.0, but the scenario probabilities must total 

1.0. 

Figure 2-4. Framework for hydrogeologic modeling with scenario uncertainty represented by three alternative scenarios 
19

Although this method has been described using a characterization of scenario uncertainty as a set of discrete 

alternatives, it is possible to consider additional parameter uncertainty specific to a scenario. Consider for 

example, a scenario that involves a change in land use 

to irrigated agriculture with a consequent change in the 

water flux to the soil surface, which represents the 

boundary of the system. The scenario could specify a 

constant irrigation rate. Alternatively, the irrigation rate 

in this scenario could be modeled as a random variable 

and included, for example, as a sampled parameter in a 

Monte Carlo simulation along with any uncertain parameters evaluated in the history matching period. 

The following chapter presents a quantitative methodology for assessing the combined impact of parameter, 

conceptual model, and scenario uncertainties.

Events Characterizing Scenarios 

Climate 

Change 

(p=0.3) 

Flood 

(p=0.2) 

Irrigated 

Agriculture 

(p=0.6) 

1 0 0 0 0.224 

2 1 0 0 0.096 

3 0 1 0 0.056 

4 1 1 0 0.024 

5 0 0 1 0.336 

6 1 0 1 0.144 

7 0 1 1 0.084 

Scenarios 

8 1 1 1 0.036 

Scenario Probability 

Table 2-1. Example formulation of mutually exclusive scenarios from three scenario-characterizing events. 

Marginal probabilities for the three events and resulting scenario probabilities assuming independence between events are given. 
21

3 JOINT ESTIMATION OF MODEL, PARAMETER, AND SCENARIO 

UNCERTAINTIES

A method is presented here to provide an optimal way 

of combining the predictions of several alternative 

models and assessing their joint predictive uncertainty, 

with consideration of parameter, conceptual model, and 

scenario uncertainties. This method relies on the specification of a set of alternative models and scenarios and 

weights the alternative model results by a measure of 

the model probabilities and the alternative scenarios by 

a measure of the scenario probabilities. The method 

closely follows the presentation of Draper (1995). 

3.1 Bayesian Model Averaging –

Combining Conceptual Model and 

Parameter Uncertainty

A formal method of evaluating prediction uncertainty 

with consideration of model and parameter uncertainty 

is Bayesian Model Averaging (BMA) (Draper 1995; 

Hoeting et al. 1999). Using the notation of Hoeting et 

al. (1999), if Δ is the predicted quantity, its posterior 

distribution given a set of data D is 

( ) ( ) ( )

1

,

K

k k

k

p p M pM

=

Δ= Δ D DD ∑ (1) 

where M = ( ) 1,..., M MK is the set of all models considered, ( ) , k p MΔ D is the posterior distribution of Δ

for model Mk, and ( ) k p M D is the posterior model 

probability for model Mk. Referring back to Figure 1-5, 

p(Δ D) is the solid black curve (the model-averaged 

result of interest) while the individual model results, 

( ) , k p MΔ D , are shown as the dashed curves. Model 

probabilities, ( ) k p M D , are given in the legend of 

Figure 1-5. 

Model uncertainty is represented in (1) by the discrete 

set of models M. Parameter uncertainty enters (1) as 

the random contribution to 

( )

( )

,

,, ( ,)

k

k k kk k

p M

pM p M d

Δ =

Δ ∫

D

D D θ θ θ

 (2) 

where θ k is the vector of parameters associated with 

model Mk and ( ) , k k p M θ D is the posterior probability density of θ k given Mk and D. Given the joint 

parameter probability distribution, ( ) , k k p M θ D , (2) 

could be solved using, for example, Monte Carlo simulation. 

Posterior model probability is given by Bayes’ theorem, 

( ) ( ) ( )

( )

( ) ( )

( ) ( )

1

k k

k

k k

K

l l

l

p M pM p M

p

p M pM

p M pM

=

=

=

∑

D

D

D

D

D

 (3) 

where p M (D k ) is the likelihood of model Mk and 

( ) k p M is the prior probability of model Mk . The 

model likelihood can be expressed as 

p M p M p Md ( k kk kk k ) = ( , ) ( ) ∫ D D θ θθ (4) 

where p M (θk k ) is the prior probability density of 

θ k under model Mk , and ( ) , k k p M D θ is the joint 

likelihood of model Mk and its parameters θ k . 

The model-averaged values of the posterior mean and 

variance of Δ are (Draper 1995) 

( ) 1

,

K

k k

k

E E M pM =⎡Δ ⎤ = ⎡Δ ⎤ ⎣⎦ ⎣ ⎦ DD D ∑ (5) 

( )

( ) ( )

1

2

1

,

,

K

k k

k

K

k k

k

Var Var M p M

E M E pM

=

=

⎡Δ ⎤ = ⎡Δ ⎤ ⎣⎦ ⎣ ⎦

+ ⎡Δ ⎤ − ⎡Δ ⎤ ⎣ ⎦⎣⎦

∑

∑

D DD

D DD

(6) 

In (6), the first term on the right-hand side represents 

within-model variance; the second term represents between-model variance. Note that the predictive probabilities (1) and leading moments (5) and (6) are 

weighted by the posterior probabilities of the individual 

models. 

To apply BMA, one formally requires that the prior 

model probabilities sum up to one, 
22

( ) 1

1

K

k

k

p M

=

∑ = . (7) 

This implies that all possible models of relevance are 

included in M (i.e., the set is collectively exhaustive), 

and that all models in M differ from each other sufficiently to be considered mutually exclusive (i.e., the 

joint probability of any two models is zero). In practice, it may be impossible to demonstrate that the set of 

models is collectively exhaustive. In this case, model 

uncertainty may be underestimated and model probability must be interpreted as relative to the other models in M, as implied by the fact that all probabilities 

computed using BMA are conditional on M. 

As discussed in Ye et al. (2004) and Meyer et al. 

(2004), basing the analysis on a set of model alternatives that do not encompass all possibilities implies a 

relative comparison between models. We thus interpret 

prior model probabilities to be subjective values reflecting a belief about the relative plausibility of each 

model based on its apparent (qualitative, a priori) consistency with available knowledge and data. Ye et al. 

(2005) discuss the use of maximum entropy and expert 

judgment to determine prior model probabilities. 

Whereas prior model probabilities are subjective, the 

posterior model probabilities are modifications of these 

subjective values based on an objective evaluation of 

each model’s consistency with available data. Hence, 

the posterior probabilities are valid only in a comparative, not in an absolute, sense. They are conditional on 

the choice of models (in addition to being conditional 

on the data) and may be sensitive to the choice of prior 

model probabilities (see Ye et al. 2004, 2005 for additional discussion of sensitivity to prior model probabilities). This sensitivity is expected to diminish with increased level of conditioning on data. 

3.1.1 Maximum Likelihood Bayesian Model 

Averaging (MLBMA) 

Computational difficulties in the BMA approach include the calculation of ( ) , k p MΔ D in (1) and 

( ) k p M D in (3), which may require exhaustive Monte 

Carlo simulations of the prior parameter space θ k for 

each model. This may be computationally and hydrologically very demanding. MLBMA (Neuman 

2003; Ye et al. 2004; Meyer et al. 2004) uses two approximations to simplify the application of BMA in 

hydrogeological modeling. First, approximating 

( ) , k p MΔ D by ( ) ˆ, , k k p M Δ θ D , where ˆ

θk is the 

maximum likelihood (ML) estimate of θ k based on the 

likelihood p M (D θk k , ) , was suggested by Taplin 

(1993) and was shown to be useful in the BMA context 

by Draper (1995), Raftery et al. (1996) and Volinsky et 

al. (1997). Second, Neuman (2003) proposed evaluating the posterior model probability using an expression 

for p M( k D) derived by expanding the terms in the 

integrand of (4) in a Taylor series about ˆ

θk (Kashyap 

1982). A related approach based on Laplace approximations has been used in the BMA context by Draper 

(1995) and Kass and Raftery (1995). Kashyap’s expression can be written (Ye et al. 2004) as 

( )

( )

( ) 1

1

exp 2

1

exp 2

k k

k K

l l

l

KIC p M

p M

KIC p M =

⎛ ⎞ ⎜ ⎟ − Δ ⎝ ⎠ = ⎛ ⎞ ⎜ ⎟ − Δ ⎝ ⎠ ∑

D (8) 

where 

K k k min Δ IC KIC KIC = − , (9) 

( ) ˆ ln ln , 2 k k k k kk

N KIC NLL N M

π

⎛ ⎞ =+ + ⎜ ⎟ ⎝ ⎠ F D θ (10) 

KICk is the so-called Kashyap information criterion for 

model Mk , KICmin is its minimum value over all candidate models, and 

( ) ( ) ˆ ˆ 2 ln , 2 ln NLL p M p M k kk kk = − − D θ θ (11) 

the negative log likelihood of Mk evaluated at ˆθk . 

Here Nk is the dimension of θ k (number of parameters associated with model Mk ), N is the dimension of 

D (number of discrete data points), and k F is the normalized (by N) observed (as opposed to ensemble 

mean) Fisher information matrix having components 

( ) 2

,

ˆ

1 ln , k k

k ij

i j

k k

p M

N θ θ =

∂ = − ∂ ∂ θ θD θ

F (12) 

In the absence of prior information about the parameters, one simply drops the term ( ) ˆ 2 ln k k − p M θ from 

NLLk . This reflects common practice in model calibration. 
23

Increasing the number of parameters Nk allows 

( ) ˆ ln , k k − p M D θ to decrease and ln N N k to increase. When Nk is large, the rate of decrease does not 

compensate for the rate of increase and KICk grows 

while ( ) k p M D diminishes. This means that a more 

parsimonious model with fewer parameters is ranked 

higher and assigned a higher posterior probability. On 

the other hand, ( ) ˆ ln , k k − p M D θ diminishes with N at 

a rate higher than linear so that as N grows, there may 

be an advantage to a more complex model with larger 

Nk. 

The last term in (10) reflects the information content of 

the available data. Among models having an equal 

number of parameters, which fit a given set of observations equally well, the Fisher information term causes 

KIC to favor models with relatively small information 

content per unit sample or, equivalently, a correspondingly large parameter estimation variance. Looking at 

this from a different but related angle, one expects a 

model having a large specific (per unit sample) information content (and small estimation variance) to exhibit improved performance (better fit for a given complexity) and vice versa. If increasing the specific information content of a model fails to improve its performance relative to another model, then according to 

KIC the former model has a lesser probability of being 

correct than does the model with lesser specific information content; selecting a model with greater information content would not be justified. Among models 

having different numbers of parameters that fit a given 

set of observations equally well, KIC may favor more 

complex models if their specific information content 

per unit sample is comparatively small. 

As shown in Ye et al. (2004), alternative models can 

have different types and numbers of parameters, but the 

latter must be estimated and the models compared considering a single data set D. As additional data become 

available, they can be included in D and the analysis 

updated accordingly. For a comparison of two- and 

three-dimensional models, data distributed in threedimensional space may need to be projected onto a 

two-dimensional plane as done by Ando et al. (2003) 

or averaged in the third dimension as suggested by 

Neuman and Wierenga (2003, Appendix B). 

Previously, KICk has been used (e.g., Carrera and 

Neuman 1986a,b; Samper and Neuman 1989a,b) as an 

optimum decision rule for the ranking of competing 

models. The highest-ranking model is that corresponding to K min IC . Note that KIC has no intrinsic meaning; 

it is only the differences between KIC values that have 

meaning. Thus the use of ΔKIC in (8) reflects the interpretation of p M( k D) as a relative probability suitable for comparing the models within the set M. The 

effect of ΔKIC on the posterior model probability is 

shown in Figure 3-1. Model probability is normalized 

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 1 2 3 4 5 6 7 8 9 10

ΔKICk

p (Mk |D)/max{

p (Mk |D)}

in this figure by the maximum model probability (i.e., 

the probability of the best model). Relatively small KIC

differences result in large differences in model probabilities due to the exponential in (8). Model probability is less than one percent for ΔKIC values greater 

than 10. 

A nonasymptotic version of the familiar Akaike (1974) 

information criterion (AIC) has also been advocated for 

use in computing relative model weights (Burnham and 

Anderson 2002). This version of AIC, derived by Hurvich (1989) can be written as 

2 2

2 ( 1) ( 1)

1 1

kk kk

k kk

k k

NN NN

AICc AIC NLL N

NN NN =+ + + +

= + − − −− (13) 

Model probabilities can be estimated by computing 

A k Δ ICc values as in (9) and substituting these in (8). 

Poeter and Anderson (2005) discuss the use of AICc in 

groundwater applications. Appendix C of this report 

contains an application of MLBMA to recharge model 

uncertainty that includes a comparison of model probabilities computed using KIC and AIC. Ye et al. 

(2007)1

 discuss alternative criteria for computing 

model probabilities and present an analysis that sug-

 

1

 Ye, M., P.D. Meyer, and S.P. Neuman, On model 

selection criteria in multimodel analysis, Water Resour. 

Res., in review, 2007. 

Figure 3-1. The effect of ΔKIC on posterior model 

probability given as a fraction of the 

probability for the best model 
24

gests KIC may be a better model discrimination criterion than AIC or AICc. 

3.1.1.1 Applicability of MLBMA 

Using the maximum likelihood method in BMA has 

several advantages. It can be applied to both complex 

and simplified models. It can be applied to deterministic models as described by Carrera and Neuman 

(1986a,b) and Carrera et al. (1997) and also to stochastic models based on moment equations as demonstrated 

by Hernandez et al. (2002, 2003, 2006). Application of 

maximum likelihood also yields parameter sensitivity 

information. 

Including prior information in the maximum likelihood 

calibration is an option that allows one to condition the 

parameter estimates not only on site monitoring (observational) data but also on site characterization data, 

from which prior parameter estimates are usually derived. When both sets of data are considered to be statistically meaningful, the posterior parameter estimates 

are compatible with a wider array of measurements 

than they would be otherwise and are therefore better 

constrained (potentially rendering the model a better 

predictor). 

Maximum likelihood yields a negative log likelihood 

criterion NLLk (Eq. 11) that includes two weighted 

square residual terms: a generalized sum of squared 

differences between simulated and observed state variables arising from ( ) ˆ 2 ln , k k − p M D θ , and a generalized sum of squared differences between posterior and 

prior parameter estimates arising from 

( ) ˆ 2 ln k k − p M θ . The first is weighted by a matrix 

proportional to the inverse covariance matrix of state 

observation errors. The second is weighted by a matrix 

proportional to the inverse covariance matrix of prior 

parameter estimation errors. Maximum likelihood allows the statistical parameters of the errors to be estimated. When these statistical parameters are known 

(i.e., not estimated), maximum likelihood reduces to 

generalized least squares estimation. In this case, available codes such as PEST (Doherty 2004) and UCODE 

(Poeter et al. 2005) can be applied. 

Maximum likelihood estimation yields an approximate 

covariance matrix for the estimation errors of ˆ

k θ . 

Upon considering the parameter estimation errors of a 

calibrated deterministic model Mk to be Gaussian or 

log Gaussian, one easily determines ( ) ˆ, , k k p M Δ θ D in 

(2) by Monte Carlo simulation of Δ through random 

perturbation of the parameters. The simulation also 

yields corresponding approximations [ ] ˆ, , k k E M Δ θ D

of [ , ] k E M Δ D , and [ ] ˆ, , Var M Δ k k θ D of 

[ , ] k Var M Δ D , in (4) and (5). If Mk is a geostatistical 

model as in the example of Ye et al. 2004 or a stochastic moment model of the kind considered by Hernandez 

et al. (2002, 2003, 2006), it yields [ ] ˆ, , k k E M Δ θ D and 

[ ] ˆ, , k k Var M Δ θ D directly without Monte Carlo simulation. 

One final point regarding the applicability of MLBMA. 

In the most data-limited application, one in which there 

are no system observations with which to calibrate a 

model and the only available parameter information is 

that available from generic databases, (1) reduces to 

( ) ( ) ( ) 1

K

k k

k

p p M pM =Δ= Δ ∑ (14) 

That is, model predictions can still be made using prior 

parameter estimates and model averaging can still be 

carried out, but only with prior model probabilities. 

Since the predictions and model probabilities are not 

conditioned on state variable observations, however, 

the results are expected to be more uncertain and potentially more biased. 

3.2 Incorporation of Scenario 

Uncertainty

MLBMA predictions are computed using the following 

procedure. Each model in M is calibrated in the history-matching period using the dataset D. The calibrated models are then used to simulate the system 

behavior in the predictive period with each model’s 

result weighted by its posterior model probability. For 

calibration, the models must reflect the system makeup, 

processes, and forcing of the history matching period 

and must be capable of producing the quantities in D

(typically head and concentration measurements). For 

prediction, the models must reflect the future scenario(s) and must be able to produce the quantities required to evaluate site safety/performance. This will, in 

general, require that changes be made to the models 

between the history-matching and prediction periods. 

For example, a climate change scenario may require 

modification of the upper boundary condition representing precipitation or recharge. A residential farmer 

scenario may require the inclusion in the models of a 

sink term representing a pumped well. For the purposes 

of the analysis presented in this section, we assume that 

all the models in M retained for prediction (i.e., those 
25

models with non-negligible posterior model probabilities) were constructed such that they can be easily 

modified to simulate any scenario considered. 

3.2.1 Bayesian Model Averaging 

Conditioned on a Specific Scenario 

Formally, scenario uncertainty can be quantitatively 

assessed jointly with model and parameter uncertainties following the methodology described by Draper 

(1995) and applied in a nuclear waste disposal context 

(albeit without the inclusion of model uncertainty) by 

Draper et al. (1999). Consider an uncertain scenario in 

which the uncertainty is modeled discretely as a set of 

alternative scenarios, = ,..., ( ) 1 I S S S . For a given scenario, Si, the posterior distribution of a predicted quantity can thus be interpreted as conditional on that scenario and equation (1) becomes 

( )

( ) ( ) 1

,

,, ,

i

K

ki ki

k

p S

p M S pM S

=

Δ =

∑ Δ

D

D D

. (15) 

Posterior model probability conditional on a given scenario can be expressed similarly by modifying equation 

(3). 

( ) ( ) ( )

( )

( ) ( )

( ) ( ) 1

, | , |

|

|

ki k i

k i

i

k ki

K

l li

l

p M S pM S pM S p S

p M pM S

p M pM S

=

=

=

∑

D

D

D

D

D

 (16) 

The simplifications made in the rightmost equality of 

equation (16) are based on the assumption that the 

dataset, D, is independent of the scenario. That is, the 

occurrence of any particular scenario in the future does 

not affect the probability of observing the data , D, in 

the past. As a result, the model likelihoods, p M (D k ) , 

are not a function of the scenario and do not need to be 

recomputed under each scenario. In contrast, prior 

model probability, ( ) k i p M S , is potentially a function 

of the scenarios. That is, the occurrence of specific 

future hydrologic conditions may have an impact on 

the relative plausibility of the various models. Thus 

posterior model probability is a function of the scenario 

only through the possible dependence of prior model 

probabilities on the scenario. As in equation (7), prior 

model probabilities under a given scenario must sum to 

one. 

( ) 1

| 1

K

k i

k

pM S

=

∑ = . (17) 

Posterior mean and variance of Δ become (see Appendix A) 

( )

( ) ( ) 1

,

,, ,

i

K

ki ki

k

E S

E M S pM S

=

Δ =

∑ Δ

D

D D

 (18) 

( ) ( )

( ) ( )

( )

1

2

2

, ,,

,, ,

,

K

i ki

k

k i ki

i

Var S Var M S

E M S pM S

E S

=

Δ= Δ ⎡

⎣

⎤ + Δ ⎡ ⎤ ⎣ ⎦ ⎥⎦

− Δ ⎡ ⎤ ⎣ ⎦

D D ∑

D D

D

 (19) 

Equation (19) can be rewritten as 

( )

( ) ( )

( ) ( ) ( )

1

2

1

,

,, ,

,, , ,

i

K

ki ki

k

K

k i i ki

k

Var S

Var M S p M S

E M S E S pM S

=

=

Δ =

Δ

+ Δ −Δ ⎡ ⎤ ⎣ ⎦

∑

∑

D

D D

DD D

(20) 

where the two terms on the right hand side represent 

within-model and between-model variance for a given 

scenario. 

3.2.2 Scenario Averaging 

Averaging equation (15) over all scenarios using scenario probabilities pS pS ( i i ) = ( ) D as weights gives 

( ) ( ) ( )

( ) ( ) ( )

1

1 1

,

,, ,

I

i i

i

I K

ki kii

i k

p p S pS

p M S pM S pS

=

= =

Δ= Δ

= Δ

∑

∑∑

D D

D D

 (21) 

where p(Δ D) is implicitly conditional on all scenarios and model structures. Probabilities ( ) , , i k p SM Δ D

and pM S ( k i ,D) can be obtained by Monte Carlo 

simulation and (16), respectively. For the averaging in 

(21) we require that the scenarios given in S

= ( ) 1,..., I S S are mutually exclusive and collectively 

exhaustive. That is, 
26

( ) 1

1

I

i

i

p S =

∑ = . (22) 

The posterior mean of Δ, including the effects of scenario uncertainty, is (see Appendix B) 

( ) ( ) ( )

( ) ( ) ( )

1

1 1

,

,, ,

I

i i

i

I K

ki ki i

i k

E E S pS

E M S pM S pS

=

= =

Δ= Δ

= Δ

∑

∑∑

D D

D D

 (23) 

where ( ) , E S Δ D i is evaluated by (18). The posterior 

variance of Δ is (see Appendix B) 

( ) ( )

( ) ( ) ( )

( )

1 1

2

2

, ,

,, ,

.

I K

k i

i k

k i ki i

Var Var M S

E M S pM S pS

E

= =

Δ= Δ ⎡

⎣

⎤ + Δ ⎡ ⎤ ⎣ ⎦ ⎥⎦

− Δ ⎡ ⎤ ⎣ ⎦

D D ∑∑

D D

D

 (24) 

Equation (24) can be rewritten as (see Appendix B) 

( ) ( ) ( )

( ) ( ) ( )

1

2

1

,

,

I

i i

i

I

i i

i

Var Var S p S

E S E pS

=

=

Δ= Δ

+ Δ −Δ ⎡ ⎤ ⎣ ⎦

∑

∑

D D

D D

 (25) 

where ( ) , E S Δ D i and Var S (Δ D, i ) can be estimated 

by equations (18) and (19) or (20). The first term on 

the right hand side of (25) is the variance within scenarios; the second term is the variance between scenarios. 

By substituting equation (20) into (25), the posterior 

variance can be rewritten in the manner of Draper 

(1995) as 

( )

( ) ( ) ( )

( ) ( )

( ) ( )

( ) ( ) ( )

1 1

2

1 1

2

1

,, , |

,, ,

, |

, |.

I K

ki ki i

i k

I K

ki i

i k

ki i

I

i i

i

Var

Var M S p M S p S

EM SE S

pM S pS

E S E pS

= =

= =

=

Δ =

Δ

+ Δ −Δ ⎡ ⎤ ⎣ ⎦

⋅

+ Δ −Δ ⎡ ⎤ ⎣ ⎦

∑∑

∑∑

∑

D

DDD

D D

D D

D DD

 (26) 

This expression consists of three terms: 

(1) the variance within models and scenarios, 

( )

( ) ( )

1 1

, ,

, |

I K

k i

i k

ki i

Var M S

pM S pS

= =

Δ

⋅

∑∑ D

D D

 (27) 

(2) the variance between models within scenarios, 

( ) ( )

( ) ( )

2

1 1

,, ,

, |

I K

ki i

i k

ki i

EM SE S

pM S pS

= =

⎡ Δ −Δ ⎤ ⎣ ⎦

⋅

∑∑ D D

D D

 (28) 

(3) and the variance between scenarios, 

( ) ( ) ( ) 2

1

, |

I

i i

i

E S E pS =

⎡ ⎤ Δ −Δ ∑⎣ ⎦ D DD . (29) 

The equations provided above can be applied to estimate the individual and collective contribution to 

model predictive uncertainty of parameter, conceptual 

model, and scenario uncertainties. Parameter and conceptual model uncertainty are considered using maximum likelihood Bayesian model averaging (MLBMA) 

in the history-matching period (the period for which 

system state data exist). To incorporate scenario uncertainty, the MLBMA results are repeatedly applied in 

the predictive period under a set of alternative scenarios. Because the scenarios describe future conditions, 

the scenario probabilities represent prior estimates and 

cannot be updated using the (past) system state data. 

Incorporation of scenario uncertainty using the method 

described here thus does not require any additional 

calibration (beyond that conducted in the MLBMA 

analysis), but does require additional probabilistic calculations. For example, solution of equation (21) could 

be accomplished using a Monte Carlo simulation of 

each model within each scenario. This is straightforward, albeit computationally expensive for large or 

complex numerical models. 

As time progresses and additional data are collected, 

these data can be incorporated in an expanded data set, 

D, and the Bayesian analysis updated accordingly. If of 

sufficient duration, the additional data may cause one 

to modify the prior probabilities assigned to the alternative scenarios. 

3.3 Summary of Uncertainty Assessment 

Methodology

To implement MLBMA the following steps are followed. 
27

1. Postulate alternative conceptual-mathematical 

models for a site. Guidance provided in Neuman 

and Wierenga (2003) may be useful in this step. 

2. Assign a prior probability to each model using, for 

example, the method of Ye et al. (2005). 

3. Optionally assign prior probabilities to the parameters of each model, using, for example, guidance 

provided in Meyer and Gee (1999). 

4. Postulate alternative scenarios affecting the future 

hydrologic conditions at the site. 

5. Assign prior probabilities to each scenario using, 

for example, expert elicitation. 

6. Obtain posterior maximum likelihood parameter 

estimates, and estimation covariance, for each 

model by inversion (model calibration). In many 

cases, available codes such as PEST (Doherty 

2004) and UCODE (Poeter et al. 2005) can be applied to this step. 

7. Calculate a posterior probability for each model 

using the model calibration results and the prior 

model probabilities. 

8. Predict quantities of interest using each model. 

9. Assess prediction uncertainty (distribution, variance) for each model using Monte Carlo or stochastic moment methods. 

10. Weight predictions and uncertainties by the corresponding posterior model probabilities and sum 

the results over all models. 

11. Repeat steps 7-10 for each alternative scenario. 

Prior model probabilities may be modified for 

each scenario. 

12. Weight the results of step 10 for each scenario by 

the appropriate scenario probability and sum the 

results over all scenarios. 

A flowchart illustrating the MLBMA approach to combined estimation of conceptual model and parameter 

uncertainty is shown in Figure 3-2. Elements of the 

flowchart related to the analysis of hydrologic scenarios are shown in red. As mentioned above, the methodology can be conducted in an iterative fashion; as additional data become available, they can be included in 

the data set, D, and steps 6-12 repeated. In some cases, 

earlier steps in the methodology may be repeated, such 

as when additional data indicate that other models or 

scenarios should be considered. 
28 

Figure 3-2. Flowchart for the combined estimation of model, parameter, and scenario uncertainties using 

Maximum Likelihood Bayesian Model Averaging. Data boxes contain duplicated 

data/information and are shown separately to clarify the application of the data/information. 

Postulate multiple 

alternative conceptualmathematical models 

1 2 {, ,, } MM M ... K

Available site 

data and related 

information 

Assign a (subjective) prior probability, p M( k ) , to each model 

such that ( )1 k

k

∑ p M =

Calibrate each Mk using 

maximum likelihood 

methods 

Parameter 

Sensitivity 

Analysis 

Flow & transport 

data from site 

monitoring, D

Calculate posterior 

model probabilities 

( ) , k i pM S D

Run each model to estimate 

probability distribution of 

predicted quantity (Δ) 

( ) ˆ, ,, k k i pM S Δ θ D

Find model-averaged prediction weighted by 

model probabilities (MLBMA result) 

( ) ( ) ( )

1

ˆ , , ,, ,

K

i kk i k i

k

p S p M S pM S

=

Δ =Δ D ∑ θ D D

Information to guide 

data collection and 

improve models 

Combined estimate of 

model and parameter 

uncertainty

Assign prior 

parameter 

probability 

distributions

Model 

Information 

Criteria, KICk

Posterior 

Parameter 

Estimates, ˆ

θk

Parameter 

Covariance 

Estimates 

Postulate multiple 

alternative hydrologic 

scenarios 

1 2 {, , , }I SS S ...

Assign a (subjective) prior probability, p S( i ) , to each scenario 

such that ()1 i

i

∑ p S =

Combined estimate of 

model, parameter, and 

scenario uncertainty

Find scenario-averaged 

prediction 

( ) ( ) ( )

1

,

I

i i

i

p p S pS

=

Δ= Δ D D ∑

Information affecting 

future site hydrologic 

conditions 

Once for Each Scenario 
29

4 EXAMPLE APPLICATION

This chapter describes an application of the methods 

presented in the previous chapter to assess the impact 

of conceptual model, parameter, and scenario uncertainty in hydrogeological modeling for dose assessment. The MLBMA methodology for the combined 

estimation of hydrogeological conceptual model and 

parameter uncertainty was applied to the modeling of 

spatial variability in unsaturated, fractured rock by Ye 

et al. (2004). Seven alternative variogram models of air 

permeability were evaluated in that application, with 

the variogram and drift parameters of the models estimated using data from single-hole injection tests. The 

application illustrated the superiority of using the 

model-averaged result from MLBMA over any individual variogram model. The application described 

here advances the uncertainty assessment methodology 

by including the impact of hydrologic scenario uncertainty and by applying the methods to a problem more 

directly relevant to dose assessment: namely, transient 

groundwater flow and transport of a radionuclide from 

an uncontrolled disposal. The presentation in this chapter generally follows the flowchart of Figure 3-2. 

4.1 Available Site Data and Related 

Information

In selecting a site for the application of the uncertainty 

methodology a number of factors were considered. To 

avoid any potential conflict-of-interest issues and limitations on the use of data, NRC-licensed sites were not 

considered. The application site needed to have existing contamination and sufficient historical data to characterize changes in flow and contaminant transport 

over time, to calibrate the models to be developed for 

the site, and to evaluate the methodology. It was desired that the contaminant at the site be a radionuclide 

commonly occurring at NRC decommissioning sites 

and therefore of concern to the NRC staff. In addition, 

the hydrologic characteristics of the site should reflect 

conditions not uncommon at decommissioning sites. 

4.1.1 Application Field Site Background 

The site chosen for the example application was the 

300 Area in the southeastern part of the U.S. Dept. of 

Energy (DOE) Hanford Site in Richland, Washington 

(see Figure 4-1). The 300 Area is an industrial area at 

which uranium fuel was manufactured for use in the 

reactors located in the northern part of the Hanford 

Site. In addition to the former nuclear fuel fabrication 

facilities, the site contains fuels research laboratories, 

several solid waste burial grounds, and liquid effluent 

disposal sites (e.g., process trenches, process ponds) to 

which uranium and other contaminants were discharged during the production process. The primary 

discharges of uranium-laden liquid wastes (in terms of 

liquid volume and uranium activity) occurred at the 

316-2 North and 316-1 South Process Ponds (from 

1944-1975) and at the 316-5 Process Trenches (from 

1975 to 1984). The locations of the 300 Area boundary 

and the waste sites are shown in Figure 4-2. Also indicated on this figure is the boundary of the 300-FF-5 

Operable Unit, a region encompassing potential 

groundwater contamination and defined for regulatory 

purposes. In this report, the term “300 Area” will refer 

to a region that includes that portion of the 300-FF-5 

Operable Unit immediately surrounding the 300 Area 

boundary. 

The Hanford Site is located in the semiarid Pasco Basin 

of the Columbia Plateau in southeastern Washington 

State, within the rain shadow of the Cascade Mountain 

Range. The Hanford Site is characterized as a shrubsteppe ecosystem that is adapted to the region’s midlatitude, semiarid climate (Neitzel 1998). Such ecosystems are typically dominated by a shrub overstory with 

a grass understory. Livestock grazing and agricultural 

production prior to government control of the Hanford 

Site contributed to colonization by non-native vegetation species that currently dominate portions of the 

landscape. In addition, summer range fires have tended 

to eliminate fire-intolerant species and have allowed 

more opportunistic and fire-resistant species a chance 

to become established. The dominant non-native species on the site is cheatgrass. 

Average daily maximum temperature varies from about 

4oC in late December and early January to 33oC in late 

July. On average, there are 48 days during the summer 

months with a maximum temperature greater than or 

equal to 32oC. From late-November through early 

March, minimum temperatures average less than or 

equal to 0oC. The recorded maximum temperature is 

45oC; the recorded minimum is -30oC. 

Precipitation at Richland, Washington (National Climatic Data Center Cooperative Station number 

457015) has averaged 18.1 cm/yr since 1948, with 52 

percent of the annual precipitation occurring from November through February. Days with more than 1.3 cm 

of precipitation occur on average less than once each 

year. Seasonal average snowfall is 22.4 cm with a 

maximum recorded monthly snowfall of 52.1 cm; the 

maximum recorded seasonal snowfall is 99.8 cm. 
30

Figure 4-1. The Hanford Site and its location within Washington State. The 300 Area is in the southeastern 

part of the site adjacent to the Columbia River. 
31

The semiarid climate results in fairly low rates of 

groundwater recharge. Natural recharge rates across the 

Hanford Site are estimated to range from 0 to more 

than 10 cm/yr depending on surface soils, vegetation, 

and topography (Fayer and Walters 1995). Minimal 

recharge rates occur in fine-textured soils where deeprooted plants prevail. Larger recharge rates are likely to 

occur in areas with coarse, gravelly surface sediments 

and little or no vegetation. 

Data from a number of boreholes located in the 300 

Area have been used to develop a geologic conceptual 

model of the site. The location of these boreholes (vertical yellow lines) and the interpreted stratigraphy at 

the 300 Area are shown in Figure 4-3. Units shown in 

Figure 4-3 are 

• u1: Hanford Formation flood deposits consisting 

of high-permeability sand- and gravel-dominated 

facies, 

• u5, u7: Ringold Formation fluvial gravel facies 

of generally lower permeability than the Hanford 

Formation deposits, 

Figure 4-2. Plan view of the 300 Area primary liquid waste discharge sites, boundaries of the 300 Area proper 

and the 300-FF-5 Operable Unit, and the water table elevation in March 2003 (NAVD88, m) with 

inferred groundwater flow directions (from Hartman et al. 2004) 

300 Area Boundary

300-FF-5 Operable 

Unit Boundary 
32

• u6: Ringold Formation overbank deposits consisting of silts and silty sands, 

• u8: Ringold Formation lacustrine deposit consisting of a low-permeability clay, 

• u9: Ringold Formation fluvial deposit of silty 

sand and gravel. 

The low permeability clay unit (u8) is continuous 

across the 300 Area, occurs at an average depth of 

about 30 m, and acts as a confining layer. As occurs 

across the Hanford Site, extensive basalt units underlie 

the 300 Area. East-west cross-sections through this 

geologic model are shown in Figure 4-4; the locations 

of the western 300 Area boundary, the edge of the Columbia River, and the approximate water table in 2001 

are also shown. 

Approximately 30 years of groundwater elevation and 

uranium concentration data exist for wells located in 

the 300 Area. Average depth to groundwater in the 300 

Area is about 12 m with the water table typically occurring in the lower Hanford Formation near the interface with the Ringold Formation. The 300 Area is 

bounded by the Columbia River on the east with 

groundwater flow in the unconfined aquifer generally 

flowing from west to east toward the river, as indicated 

in Figure 4-2. There is some convergence of flow, 

however, with hydraulic head measurements indicating 

a southwesterly flow in the northern portion of the 300 

Area and a northwesterly flow in the southern portion. 

Changes in river stage have been observed to affect 

heads in the unconfined aquifer as much as several 

hundred meters away from the river (Campbell et al. 

Figure 4-3. Stratigraphy of the 300 Area. Red lines indicate the boundary of the 300 Area, the North and 

South Process Ponds, and the 316-5 Process Trenches, vertical yellow lines indicated the location 

of boreholes, u1 indicates the Hanford Formation, and u5-u9 indicate units of the Ringold Formation (vertical exaggeration unknown). 

Columbia River

N
33

Figure 4-4. East-west cross-sections through the 300 Area at (top to bottom) Northing coordinates 

117000, 116500, 116000, 115500, and 115000 m (refer to Figure 4-17 for coordinate locations)
34

Figure 4-5. Average uranium concentrations in groundwater (top of unconfined aquifer) during low river 

stage (December 2002 - top) and during high river stage (June 2003 – bottom) at the Hanford Site 

300 Area (from Hartman et al. 2004) 

Well 399-1-17A
35

1993; Lindberg and Chou 2001). High river stage can 

also impact uranium concentrations a similar distance 

inland from the river (Hartman et al. 2004). Differences in observed uranium concentrations during low 

and high river stages also have been observed, as 

shown in Figure 4-5. 

An extensive characterization of the 300 Area was carried out beginning in the late 1980s as part of a Comprehensive Environmental Response, Compensation, 

and Liability Act (CERCLA) investigation. Data obtained during this investigation illustrate the complex 

interaction between site geochemistry, hydrology, and 

the waste site discharges. After removing contaminated 

soil from the 316-5 Process Trenches (beginning in 

1991), uranium groundwater concentration near the 

southern end of the trenches was significantly reduced. 

Uranium concentration in well 399-1-17A is shown in 

Figure 4-6. This well, whose location is indicated in 

Figure 4-5, is screened near the top of the aquifer. The 

concentration remained low as long as uranium-free 

water continued to be discharged to the trenches. The 

Remedial Investigation/Feasibility Study report completed during this period (DOE 1995) concluded that 

the uranium source was sorption-controlled and that the 

concentration of uranium in the groundwater would fall 

below 20 μg/l throughout the 300 Area by the year 

2003 as a result of dilution and transport/discharge to 

the Columbia River. When all discharges to the process 

trenches ceased (in December 1994), however, the uranium concentration in well 399-1-17A increased rapidly and remains above the 30 μg/l drinking water 

standard. 

Groundwater uranium concentrations continue to exceed the drinking water standard in the 300 Area over 

an area of about 0.4 km2

 (Hartman et al. 2004), indicating that there is a long-term, slow-release source of 

uranium present. Lindberg and Chou’s (2001) conceptual model attributed this source to uranium remaining 

in the vadose zone. During high river stages, river water infiltrates the river banks, raising heads in the aquifer and mobilizing uranium present in the vadose zone 

(see Figure 4-7). Recent geochemical studies of sediments sampled beneath the north and south process 

ponds indicate that uranium desorption in the vadose 

zone is a nonequilibrium process (Zachara et al. 2005). 

0

100

200

300

400

500

600

Dec-86Dec-88Dec-90Dec-92Dec-94Dec-96Dec-98Dec-00Dec-02Dec-04

Date

Uranium (μg/L)

100

101

102

103

104

105

106

107

108

109

Water Level (m NAVD88)

Uranium

Water Level

Figure 4-6. Water level and uranium concentration (top of aquifer) in well 399-1-17A near the southern end 

of the 316-5 Process Trenches. Contaminated soil was removed starting in 1991. Discharge of 

uranium-free water continued until December 1994. 
36

4.1.2 Hanford 300 Area Data Sources 

This section provides a summary of sources of data that 

are available to support modeling of groundwater flow 

and uranium transport in the 300 Area of the Hanford 

Site. Five categories of data are distinguished: 

• surface and borehole geophysics, 

• physical and hydraulic properties, 

• geochemistry, uranium sorption, and transport, 

• source terms and boundary conditions, and 

• hydraulic head and uranium concentration data. 

Each of these data categories is discussed in the following sections. In addition, ongoing research activities 

that are being conducted by other groups are briefly 

described. Data and models developed by these other 

groups could possibly be used to provide additional 

information in support of this modeling effort. 

Most of the available physical and hydraulic property 

and borehole geophysics data for sediments within the 

boundaries of the 300-FF-5 operable unit are contained 

in two reports and/or associated data archives (Schalla 

et al. 1988; Swanson et al. 1992). The report by Schalla 

et al. (1988) is a compendium of technical information 

on the 300 Area, including sediment and groundwater 

chemistry data, and hydrogeologic characterization 

data based on installation of 18 groundwater monitoring wells. Sediment chemistry data from the 300 Area 

Process Trenches and North and South Process Ponds 

are also reported by Schalla et al. (1988; Tables 7.3 and 

7.4). The report by Swanson et al. (1992) summarizes 

characterization data obtained from the installation of 

19 additional groundwater monitoring wells that were 

added to augment the existing 60-well network in the 

300-FF-5 operable unit (which includes the wells installed by Schalla et al. 1998). A general summary of 

these and other available site characterization data is 

provided in the Phase 1 Remedial Investigation Report 

for the 300-FF-5 Operable Unit (DOE 1994). 

More recent sampling has been conducted of nearsurface sediments, and sediments underlying the North 

and South Process Ponds, to characterize geochemical 

conditions and uranium transport behavior in these 

sediments (Serne et al. 2002; Zachara et al. 2005). 

Some of these studies are described in more detail later 

in this section. 

4.1.2.1 Surface and Borehole Geophysics 

Upon completion of each of the 18 groundwater monitoring wells described by Schalla et al. (1988), geophysical logs were taken using natural gamma, neutron, 

and density probes. Plots of these data are provided in 

Appendix A of Schalla et al. (1988), and the original 

logs are maintained in PNNL data archives. 

Figure 4-7. Conceptual model of uranium remobilization in the 300 Area during high river stages (from 

Lindberg and Chou 2001) 
37

Spectral gamma surveys were conducted in 8 of the 

monitoring wells installed by Swanson et al. (1992; 

Table 4). No artificial radionuclides were detected by 

the spectral gamma probe in the wells that were surveyed. Gross gamma surveys were also conducted in 

11 of the monitoring wells installed by Swanson et al. 

(1992). The gross gamma data were said to be marginally useful for correlating thick, fine-grained sequences 

such as the lower Ringold mud unit (Swanson et al. 

1992). 

The results of surface geophysical surveys that were 

conducted in the 300 Area are described by Kunk et al. 

(1993) and Kunk (1993). The objectives of the geophysical surveys were to 

• Evaluate the reflective properties of the Hanford 

and Ringold formations, the water table, and the 

top of the basalt. 

• Determine the existence of a proposed N-S 

trending paleochannel located near the eastern 

boundary of the 300-FF-5 operable unit, parallel 

to the present-day Columbia River. 

• Define the lateral extent of the Ringold lower 

mud unit below the operable unit. 

Four surface geophysical techniques were used: 

• shallow high-resolution seismic reflection, 

• seismic refraction, 

• electromagnetic induction (EMI), and 

• ground-penetrating radar (GPR). 

The shallow high-resolution seismic reflection surveys 

were the first attempt to use this technique at Hanford. 

Kunk et al. (1993) indicated that the contacts between 

Holocene-age eolian soils and Missoula flood deposits, 

the Hanford/Ringold contact, and the lower mud unit of 

the Ringold formation were generally able to be 

mapped using this technique. However, the source frequency and the proximity of the water table to the Hanford/Ringold contact generally made these two features 

appear as one reflector. This was also found to be the 

case for the lower mud unit, which is generally within 

9 m (30 ft) of the basalt bedrock. Kunk et al. (1993) 

suggested that a higher-frequency source might be able 

to distinguish between the water table and the Hanford/Ringold contact, but cautioned that a higherfrequency source would also have lower energy, making it less able to penetrate and resolve features at 

deeper depths. The seismic data did not show the N-S 

trending paleochannel that was proposed by Lindberg 

and Bond (1979). 

The EMI and GPR surveys showed a pervasive reflector between depths of 0 to 4 m (14 ft). This feature was 

interpreted to be a Holocene-age soil horizon that separates the eolian silts and sands from the underlying 

Hanford formation (Kunk et al. 1993). This feature was 

inversely related to the elevation of the ground surface. 

When topographic corrections were made to the data, 

this feature became a relatively flat surface (Kunk et al. 

1993). This interpreted Holocene-age soil horizon was 

absent in several areas, which Kunk et al. (1993) suggested could be due to erosion, human disturbance, or 

that the horizon was never or poorly developed in these 

areas. 

Several locations also showed trough-shaped GPR 

anomalies which coincided with a missing section of 

the interpreted Holocene soil horizon. Kunk et al. 

(1993) suggested that these features could be relatively 

recent paleochannels, but noted that none of these features were of the size, depth, or magnitude suggested 

by Lindberg and Bond (1979). They also noted that 

most of the interpreted paleochannels appear to postdate the Holocene soil horizon. A second reflecting 

horizon was also detected on several profiles, usually 

1.2 to 1.5 m (4 to 5 ft) below the interpreted Holocene 

soil horizon. This horizon was also interpreted to be a 

buried paleosol (Kunk et al. 1993). 

We speculate that the seeps along the banks of the Columbia River adjacent to the 300 Area (Hulstrom 1993) 

occur where one or more of these paleosols intersects 

the shoreline. The capillary barrier effect resulting 

from these finer-grained paleosols overlying coarser 

sand and gravel units should promote lateral flow 

through the finer-grained paleosols. It is also likely that 

a significant fraction of the residual uranium contamination that exists in the vadose zone in the 300 Area 

may reside in these finer-grained sediments, due to 

their greater specific surface area. 

4.1.2.2 Physical and Hydraulic Properties 

All 18 of the wells discussed by Schalla et al. (1988) 

were drilled by the cable-tool method. Sieve data (sand 

fraction only; 0.05 mm < particle diameter ≤ 2 mm) 

were collected for 147 samples taken at approximately 

1.5-m-depth (5 ft.) intervals from five of the wells, at 

depths ranging from 1.2 to 55 m below ground surface. 

Additional particle-size data were collected for selected 

fine-grained sub-samples from some of these wells 

using the hydrometer method (Gee and Bauder 1986). 

Laboratory measurements of vertical saturated hydraulic conductivities were also made on some of these subsamples. All of the physical and hydraulic property 

data reported by Schalla et al. (1988) were available 

only in hard copy. Therefore the sieve data for the 147 

samples (from five wells) reported by Schalla et al. 

(1988) were entered into an Excel® spreadsheet to fa-
38

cilitate later use in estimating hydraulic and transport 

parameters and their spatial distribution. These data 

show gravel (>2 mm particle size) contents ranging 

from 0 to 84%, with an average of 33%. 

Aquifer test analyses were conducted for 13 of the 

wells installed by Schalla et al. (1988). Calculated values of horizontal hydraulic conductivity for wells 

screened in the Hanford formation are up to approximately 15,000 m/d, which reflects the very coarse nature of these sediments. Calculated horizontal hydraulic 

conductivities for the Ringold formation were typically 

lower, ranging from a few to several hundred meters 

per day (Schalla et al. 1988; Table D.2). The pump test 

results were critically evaluated by Spane (1991, unpublished letter report to Craig Swanson, WHC), who 

determined that the results from about half of the pump 

tests were invalid due to the methods used for data 

analysis (semi-log straight-line solutions and log-log 

type curve matching procedures) and recharge boundary effects resulting from the proximity of some of the 

wells to the Columbia River. Some of these pump test 

data have since been re-analyzed.2

The report by Swanson et al. (1992) summarizes data 

obtained during drilling of 19 groundwater monitoring 

wells. Sixteen of the wells were drilled by the cabletool method and 3 were drilled by the sonic-drill 

method. A total of 227 sediment samples were collected during installation of 11 of these wells, at 1.5-mdepth (5 ft.) intervals, using a 10-cm-diameter (4 in.), 

60-cm-long (2 ft.), split-spoon sampler with stainless 

steel or lexan liners. According to DOE (1994, p.2-8), 

the following physical property tests were performed 

on the samples collected by Swanson et al. (1992): 

• sieve analyses, to determine particle-size distributions for the sand to gravel-sized fractions of 

the sediments, 

• hydrometer analyses, to determine particle-size 

distributions for the silt and clay-sized fractions 

of the sediments, 

• permeameter tests, to determine vertical saturated hydraulic conductivities, and 

• moisture content. 

The DOE (1994) report also states that the results of 

these tests are provided in Swanson et al. (1992). Unfortunately, this is not the case. The report by Swanson 

et al. (1992) contains the following data: 

 

2

 Personal communication, Paul Thorne, PNNL, March 

2005. 

• CaCO3 (%), 

• moisture content (%), 

• porosity (%), 

• specific gravity, and 

• bulk density. 

However, no sieve, hydrometer, or permeameter test 

data are contained in the report. Laboratory-measured 

particle-size distribution data for samples from three of 

Swanson’s wells were obtained from the ROCSAN 

database (http://vlprod.rl.gov/vlib/app/). Further inquiries with former staff from the Westinghouse Hanford 

Company Geotechnical Engineering Laboratory 

(GEL), which no longer exists, and subsequent searching of archived data records, led to sieve data for 53 

samples from 10 of Swanson’s wells, and 23 samples 

from various other locations in the 300 Area. All of the 

samples in the GEL records were from depths shallower than 10 m (33 ft.) below ground surface. 

The other missing data that were supposed to be in the 

Swanson et al. (1992) report (hydrometer and permeameter test data) have not been located and may 

never have been reported or published. It should be 

noted, however, that measurements of vertical hydraulic conductivities on core samples of gravel- and cobble-dominated sediments, such as those found in the 

300 Area, are generally considered to be unreliable 

since the measurements would be biased low due to the 

very large particle sizes relative to the size of the core 

barrel. Therefore, even if the permeameter data were 

located, they would probably not be useful. 

The pump test results reported by Schalla et al (1988), 

Swanson (1992), and others were reviewed. The majority of the pump tests were single well tests; storage 

coefficients were not estimated for these tests. Swanson (1992) reports specific yield values of 0.37 and 

0.016 for two multi-well tests conducted near the western 300-FF-5 boundary. Wurstner et al. (1995) and 

Thorne and Newcomer (1992) estimated the specific 

yield of the Hanford Formation to be in the range of 

0.1 to 0.3 and that of the Ringold U5 unit to be in the 

range of 0.05 to 0.2. 

Current estimates of average hydraulic conductivities 

for the different hydrogeologic units underlying the 

300 Area are given in Table 4-1 (column 2). Also 

shown in Table 4-1 are site-wide estimated ranges for 

the hydraulic conductivities of these same units based 

on interpretation of measurements (column 3) and inverse modeling (columns 4 and 5). Hydraulic conductivity values for the 300 Area presented in Table 4-1 

are representative of the horizontal conductivity. Vertical hydraulic conductivities have generally been as-
39

sumed to be 0.01 to 0.1 times the horizontal conductivities in Hanford Site groundwater modeling. 

4.1.2.3 Geochemistry, Uranium Sorption, and 

Transport 

Cantrell et al. (2003) describe a database of adsorption 

data measured on Hanford Site sediment samples for 

contaminants occurring at the site. Measured values for 

Uranium (VI) Kd range from approximately 0 to 1000 

mL/g. For natural groundwater conditions at the Hanford Site, Cantrell et al. (2003) concluded that uranium 

adsorption is relatively low; a range of Kd values from 

0.2 to 4 mL/g being appropriate for these conditions. 

Krupka and Serne (2002) reviewed the geochemical 

factors affecting the behavior of uranium in vadose 

zone sediments. Relevant to the 300 Area uranium 

plume is their conclusion that “near sources of uranium 

release, solubility processes are particularly important 

for those sediments that become partially saturated 

with water or completely dry between periods of recharge, such as the surface soils and vadose zone sediments. Under these conditions, the concentration of 

uranium in the residue pore fluids may exceed the 

solubility limits for U(VI)-containing minerals and/or 

co-precipitates with other minerals, such as iron oxides. 

Characterization studies at DOE sites, such as the Hanford, Fernald, Oak Ridge, and Savannah River sites, 

suggest that sediments and soils contaminated from 

disposal or spills of uranium-containing liquid wastes 

at these sites can contain uranium-containing minerals 

or co-precipitates” (Krupka and Serne 2002, pg. 6.6). 

Such conditions apply to the disposal of uraniumcontaining liquid wastes in the 300 Area and may have 

contributed to the apparent long-term source of uranium in the vadose zone. 

Uranium sorption and transport data from six samples 

of 300 Area sediments, collected in December 2000 

and February 2001, are described by Serne et al. 

(2002). One of the samples, considered to be representative of uncontaminated background sediment, was 

collected from the face of an excavated pit located west 

of the 300 Area. Two samples were collected from the 

North Process Pond; one from an excavated trench 

along the southern border of the pond, and the other 

from the northeast corner of the wall of the pond 

(Figure 4-8). Three additional samples were collected 

near a building in the northern portion of the 300 Area 

proper (the 303-K building). All of the samples were 

sieved in the field to remove gravel particles greater 

than 6.35 mm (1/4 in.). Several types of tests were 

conducted on these samples: column leach tests, batch 

adsorption and leach tests, and column adsorption/desorption tests (Serne et al. 2002). The column 

leach tests were performed on all six samples using deionized water to simulate leaching of residual contaminants by percolating rainwater. The batch leach tests 

were conducted on five sub-samples of the sediments 

Hydrogeologic 

Unit 

Current Average 

Estimates for 

300 Area 

Estimated Range for Hanford Site Based on Pump 

Tests, Slug Tests, and Some 

Lab Tests1

Inverse Model 

Estimates for 

Hanford Site2

Inverse Model 

Estimates for 

Hanford Site3

U1 – Hanford 1500 1 – 1e6 2 – 30,000 192 – 37,100 

U5 – Ringold 

sand/gravel 

150 0.1 – 200 0.1 – 4,000 3 

U6 – Ringold 

overbank 

0.01 0.0003 – 0.09 0.01 – 0.1 

U7 – Ringold 

sand/gravel 

43 0.1 – 200 0.008 – 90 

U8 – Ringold 

lacustrine 

5e-5 0.0003 – 0.09 0.0002 

Basalt 5e-5 

 

1

 Wurstner et al. (1995); Thorne and Newcomer (1992) 

2

 Cole et al. (2001a) 

3

 Vermeul et al. (2003) 

Table 4-1. Estimated hydraulic conductivities (m/d) for the hydrogeologic units underlying the Hanford 300 

Area. 
40

used for the column leach tests, using de-ionized water, 

actual groundwater from the 300 Area, and simulated 

vadose zone pore water (10x ionic strength). Batch 

adsorption tests were performed on one sample using 

uranium-spiked pore-water simulant, and a uraniumspiked pore-water simulant with 10x ionic strength. 

Flow-through column adsorption and desorption tests 

were conducted on one sample using non-spiked and 

uranium-spiked pore-water simulants, and 10x ionic 

strength pore-water simulants. Serne et al. (2002) also 

measured the particle-size distributions and bulk mineralogical characteristics of their samples and estimated Kd values from their column and batch data. 

Uranium leach tests conducted on the 300 Area samples by Serne et al. (2002) using a simulated pore water 

solution resulted in equivalent Kd values between 70 

and 300 mL/g. Based on these results and characterization studies, Serne et al. concluded that the release of 

uranium from the near-surface sediments is dominated 

Figure 4-8. Photos of sediments from the location of samples obtained by Serne et al. (2002) for geochemical 

analyses at the south side (top) and the northeast corner (bottom) of the North Process Pond 
41

by dissolution of discrete uranium minerals or uranium-containing co-precipitates, not by desorption of 

adsorbed uranium. Adsorption tests on uncontaminated 

samples obtained from the 300 Area resulted in Kd values that ranged from 0 mL/g to more than 100 mL/g 

depending on the solution chemistry. Inorganic carbon 

solution concentration had the greatest impact on adsorption. Solution pH was also important, but is expected to be relatively constant in the field due to buffering by the soil. With solution chemistry held constant, a linear adsorption model appears appropriate 

over the range of uranium concentrations observed in 

the 300 Area groundwater. A nonlinear model may be 

appropriate at the higher uranium concentrations expected in the vadose zone. In response to varying solution chemistry (primarily alkalinity), the Kd values are 

expected to vary spatially. Serne et al. (2002) suggested Kd values of 0 to 1 mL/g in the near-surface 

vadose zone, 2 to 4 mL/g in the deeper vadose zone 

and unconfined aquifer, and values at least as high as 7 

mL/g in sediments with solution diluted by Columbia 

River water. 

Additional experimental work on uranium sorption and 

transport in 300 Area sediments is currently being conducted at PNNL and at the U.S. Geological Survey 

(Zachara et al. 2005). The samples used in these experiments were collected from excavations at four locations: two in the North Process Pond, and two in the 

South Process Pond. At each location, samples were 

obtained every 2 ft. to a depth of 4 ft. below the water 

table. (Note that the Ponds had been excavated to remove potentially contaminated materials; as a result, 

the excavations proceeded from locations below grade. 

The water table occurred at 10, 16, 18, and 21 feet below the top of the four excavations.) Figure 4-9 shows 

the face of one of the excavated pits and sediments at 

depths where samples were obtained. Geochemical 

analyses and transport experiments were conducted 

with sub-samples of the bulk sediments consisting of 

only the less than 2 mm particle size fraction. 

Zachara (2004) suggested that the following geochemical reactions should be considered for U(VI) sorption 

and transport in 300 Area sediments: 

Aqueous complexation

Hydrolysis:UO2

2+(aq) + H2O ⇔ UO2OH+

(aq) + H+ 

Carbonate complexation:UO2

2+(aq) + 3CO32- ⇔

UO2(CO3)3

4-(aq) (and many others) 

Adsorption

SOH(s) + UO2OH+

(aq) ⇔ SOUO2OH(s) + H+

2SOH(s) + UO2(CO3)3

4-(aq) + 2H+ ⇔

(SOH2

+

)2UO2(CO3)

2-(s)

where SOH = SiOH, FeOH, AlOH 

Precipitation

UO2

2+(aq) + Na+

(aq) + H4SiO4(aq) + 1.5H2O ⇔

Na[UO2(SiO3OH)](H2O)1.5(s) + 3H+

UO2

2+(aq) + Ca2+(aq) + 2CO32-(aq) + 5H2O ⇔

Ca[UO2(CO3)2](H2O)5(s) 

The significance of U(VI) aqueous complexation reactions is that the net charge of the complexed uranyl ion 

becomes more negative, which could potentially lead 

to enhanced transport (e.g., via anion exclusion effects). However, the complexed aqueous species may 

also undergo surface complexation (or adsorption) reactions, thus rendering the uranyl ion less mobile. 

Various mineral precipitation reactions are also possible. All of these reactions are strongly dependent on 

water chemistry, which is potentially a very important 

consideration for the 300 Area due to the differing river 

and groundwater chemistries. 

The experimental results of Zachara et al. (2005) indicate that the degree of U(VI) sorption in 300 Area 

sediments is dependent on a number of other factors, 

including dissolved inorganic carbon (DIC), silt and 

clay content, extractable Fe(III), and chlinochlore 

(chlorite) content. The calculated Kd values ranged 

from 0.27 to 38.4 mL/g, with higher values generally 

being found for finer-grained samples. In an earlier 

study, Serne et al. (1992) found that U(VI) activity in 

300 Area sediments increased with decreasing particle 

size, presumably due to more reactive surface area, but 

that most of the activity was actually associated with 

the coarser size fractions, simply because the bulk of 

the material in these sediments is very coarse. 

The sorption and desorption data of Zachara et al. 

(2005) reached a common equilibrium point, but significant times were required. Equilibration of samples 

with artificial groundwaters showed rapid desorption 

for the first 24 hours of reaction followed by a slower, 

steady release of uranium for approximately 1 week. It 

is posited that the latter component is a result of the 

slow dissolution of U(VI) in precipitated carbonate 

minerals (Zachara et al. 2005). As a less likely alternative, Zachara et al. (2005) suggest that the slow release 

may also be diffusion of U(VI) from immobile water 

within sediment micro-porosity. 
42

Figure 4-9. Photograph of a pit excavated in the base of the South Process Pond at the Hanford Site 300 Area 

and close-up photos of sediment at discrete depths where samples were taken (from Bjornstad 

2003) 
43

Two models have been developed to represent the 

sorption/desorption data described by Zachara et al. 

(2005). One is a distributed-rate, nonequilibrium reactive transport model (Zachara et al. 2005; see also Liu 

et al. 2004). The other is a semi-empirical surface 

complexation model using the following surface complexation reactions for U(VI) in 300 Area sediments. 

 SOH + UO2

2+ = SOUO2+

 + H+

 SOH + UO2

2+ + H2O = SOUOOH + 2H+

where S refers to surface complexation sites. A total 

site density of 3.84 μmoles/m2

 of BET surface area was 

assumed; the FITEQL model (Westall 1982) was used 

to determine optimal formation constants. 

Davis et al. (2004) also presented “isotherm” data for 

several samples. These data exhibited apparent threshold or maximum adsorption capacities, signifying 

Langmuir-type isotherm behavior. The maximum adsorption capacities appeared to be proportional to 

measured BET surface areas. 

4.1.2.4 Source Terms and Boundary Conditions 

The historical uranium releases and associated artificial 

recharge of water to the 300 Area process ponds and 

trenches are not well documented. Thus uranium transport from the 300 Area has been evaluated within a 

site-wide probabilistic modeling framework (Bryce et 

al. 2003). Records of historical uranium releases and 

artificial recharge of wastewater to the process ponds 

and trenches in the 300 Area have been estimated using 

the Soil Inventory Model (SIM; Simpson et al. 2001). 

Estimated liquid volume and uranium discharges to the 

major facilities at the 300 Area are shown in Figure 

4-10. There are some discrepancies, however, between 

the results from SIM and water discharge estimates to 

the 316-5 facility (North Process Trench) reported by 

Lindberg and Chou (2001). Such differences are attributable to the uncertainties in the actual waste disposal 

history of the 300 Area. 

Because the 300 Area is adjacent to the Columbia 

River, groundwater heads near the river are strongly 

influenced by the river stage, which fluctuates in response to seasonal changes in river flows and operation 

of the hydropower facilities in the Columbia River basin. An hourly record of river stage data at the 300 

Area is available beginning in 1991. A much longer 

record of streamflow data is available from a gage located below Priest Rapids Dam, the nearest upstream 

dam (USGS Gage No. 12472800). Monthly average 

streamflow below Priest Rapids Dam (Figure 4-11) 

illustrates the historical variability of the Columbia 

River flow. The apparent change in variance of the 

discharge in the early 1970’s corresponds with the 

completion of the three Canadian dams constructed as 

part of the Columbia River Treaty. The last of these 

dams, and the largest (Mica Dam), was completed in 

1973. 

Monthly averages underestimate the actual variability 

of the river stage as illustrated in Figure 4-12, which 

shows the monthly and daily averages juxtaposed on 

the hourly data for the river stage at the 300 Area during 1996. Using monthly average river stage as a 

model boundary condition results in an error of greater 

than five feet during peak river stage in 1996 (late February). Daily averages appear to be a much better approximation, although it is evident that there are significant fluctuations on the hourly time scale that 

would not be represented with a boundary condition 

based on daily average river stage. The effect of the 

river stage variability on the groundwater table can be 

seen in Figure 4-13, which shows the water table under 

average (March 2000) and high (June 1997) river stage 

conditions. 

Groundwater velocities shift direction near the river 

during the high stage condition as discharge from the 

aquifer to the river is reduced and recharge to the aquifer from the river increases. Note that the high river 

stage will saturate unsaturated sediments and produce 

three-dimensional flow near the river. Simulations 

conducted by Waichler and Yabusaki (2005) indicate 

that the impact of the river boundary on a tracer extended much farther inland when using hourly river 

data than when using the monthly data. Computational 

requirements of modeling may increase significantly as 

the period of the boundary condition is reduced. 

Waichler et al. (2005) used Columbia River discharge 

data and river stage measurements to calibrate a onedimensional hydrodynamic model of the Hanford 

Reach of the Columbia River for the period of 1940-

2004. This model can be used to generate hourly river 

elevations along the boundary of the 300 Area. 

4.1.2.5 Hydraulic Head and Uranium 

Concentration Data 

Hydraulic head data measured in groundwater wells in 

the 300 Area are available from 1950 to the present 

(see Figure 4-14). The locations of measured heads and 

the measurement intervals have varied over that time 

with significant periods of no data in most wells. Because of the high frequency variation in river levels 

adjacent to the 300 Area and the rapid response of 

groundwater heads to the river level (particularly near 

the river), the frequency of head measurements is often 
44

0.0E+00

5.0E+05

1.0E+06

1.5E+06

2.0E+06

2.5E+06

3.0E+06

3.5E+06

4.0E+06

1945 1955 1965 1975 1985 1995

Year

Liquid Volume (m^3

316-1

316-2

316-3

316-5

0

500

1000

1500

2000

2500

1945 1955 1965 1975 1985 1995

Year

U (kg)

316-1

316-2

316-3

316-5

insufficient to accurately represent actual head variation in time. 

Campbell et al. (1993) instrumented a network of wells 

in the 300 Area with pressure transducers and data loggers. During parts of 1991-1993, hourly head data were 

collected from this well network and were automatically transferred from the field to the lab by radiotelemetry. Hourly head measurements have also recently been resumed in a network of wells. These data 

are being used in conjunction with the pump test results 

from Schalla et al. (1988) and Swanson (1992) to develop and calibrate a three-dimensional flow model in 

support of a review of the Record of Decision for the 

300 Area groundwater (300-FF-5 operable unit). This 

model is similar in spatial extent to the models used 

here, but has differences due to incorporation of the 

most recently obtained data and the specific requirements of the model to support the Record of Decision. 

Figure 4-10. Estimates of liquid volume (top) and U-238 discharged to the 316-1 South and 316-2 North Process Ponds, the 316-5 Process Trenches, and the 316-3 trenches 
45

0

100,000

200,000

300,000

400,000

500,000

600,000

700,000

Jan-17 Jan-27 Jan-37 Jan-47 Jan-57 Jan-67 Jan-77 Jan-87 Jan-97

Date (1917 to 2004)

Discharge (cfs)

103.0

103.5

104.0

104.5

105.0

105.5

106.0

106.5

107.0

107.5

1/1 1/31 3/1 4/1 5/1 6/1 7/1 8/1 8/31 10/1 10/31 11/30 12/31

Date (1996)

River Stage at 300 Area (m)

'Hourly'

'Daily'

'Monthly'

Groundwater samples are collected quarterly or semiannually from selected monitoring wells in the 300 

Area for chemical analyses, including uranium. There 

are scattered measurements of uranium concentrations 

in groundwater in 1959, 1967-68, and the early 1970s, 

but more regular measurements did not begin until the 

late 1970s (see Figure 4-14). Figure 4-15 shows the 

extent of uranium contamination of the groundwater in 

1959 during discharge to the North and South Process 

Ponds. Figure 4-16 shows a time series of uranium 

concentration in groundwater from 1977 to 2004. 

Uranium groundwater concentration data from the 300 

Area have recently been analyzed using geostatistical 

methods to generate estimates of the total mass of uranium in 300 Area groundwater at selected times and 

the uncertainty of these estimates (Murray et al. 2004). 

Figure 4-11. Monthly discharge as measured below Priest Rapids Dam, the nearest dam upriver from the 

Hanford Site 300 Area 

Figure 4-12. Hourly, daily average, and monthly average river stage at the 300 Area in 1996 
46

Figure 4-13. Inferred water table in the 300 Area during (left) average river stage and (right) high river stage. Groundwater movement 

in the uranium source region is indicated by the red arrow. (from Lindberg and Chou 2001). 
47

Chemical analyses were conducted on sediment samples from some of the wells installed by Swanson et 

al. (1992). Sediment concentration data for selected 

radionuclides (including uranium), inorganic metals, 

cyanide, and volatile organic compounds for 51 samples from eight of these wells are reported by DOE 

(1994; Appendix B). These analyses were conducted 

on four to eight samples from each well, with sample 

depths ranging from 8.2 to 54 m (27 to 177 ft) bgs. 

The U-238 data from these samples all have concentrations less than 3.5 μg/g. The concentration of U238 in an “uncontaminated background sample” collected by Serne et al. (2002; sample B11493) was 

about 5 μg/g. Therefore the sediments around the 

process trenches and ponds appear to be at or below 

background concentrations. Serne et al. (2002; p. iv) 

state, however, that “less than 4% of the existing uranium in the contaminated near-surface sediments 

readily leaches into simulated rainwater over a period 

of 6 months.” This statement suggests that it may be 

difficult to accurately determine the amount of uranium that still resides in the unsaturated zone in the 

300 Area. Ongoing data collection efforts are directed at better defining the inventory of contaminants in the unsaturated zone. 

Time-dependent head and uranium concentration data 

were used in model calibrations. There are no timedependent moisture content or pressure head measurements in the 300 Area vadose zone. The only uranium concentration data available in the vadose zone 

are from samples collected for geochemical analysis, 

which represent a single point in time. Total uranium 

concentrations for the sediment samples collected 

from the 300 Area by Serne et al. (2002) ranged from 

5 to 989 μg/g, and for the samples collected from the 

excavations in the North and South Process Ponds by 

Zachara et al. (2005) from 5 to 238 μg/g. Zachara et 

al. (2005) estimated that the fraction of uranium 

available to desorb or dissolve was 4% to 8% for the 

upper samples and 8% to 67% for the deeper vadose 

zone samples. 

102

104

106

108

110

112

1/1/48 12/31/57 1/1/68 12/31/77 1/1/88 12/31/97

Date

Hydraulic Head (m NAVD88)

1

10

100

1000

1/1/59 12/31/68 1/1/79 12/31/88 1/1/99

Date

Uranium (ug/l)

Figure 4-14. Available data from the 300 Area: (top) hydraulic head and (bottom) uranium concentrations 

in groundwater 
48

114500

115000

115500

116000

116500

117000

117500

118000

Northing [m, WCS]

593000 593500 594000 594500 595000

Easting [m, WCS] 500 100 200 

 50 

Columbia River 

Island

Island

Average Uranium

Concentration (μg/l)

2/1/59 - 4/1/59

4.2 Postulate Alternative Conceptual 

Models

As described in Chapter 3, application of the Maximum Likelihood Bayesian Model Averaging method 

for jointly assessing conceptual model and parameter 

uncertainties requires the formulation of alternative 

conceptual models. Based on the characteristics of 

the site and the available data, a number of conceptual model alternatives were considered to potentially 

contribute significant uncertainty. The conceptual 

model elements for which alternative representations 

were considered included the following. 

• Configuration of the geologic units. Units contacts have been interpreted based on available 

borehole data. Uncertainties arise because the 

location of these contacts is not always obvious 

and must be interpolated between boreholes. In 

addition, differences between units u5 and u7 

are small and there is significant overlap in the 

ranges of observed properties for units u1 and 

u5. 

• Heterogeneity within the geologic units. Although it is clear that there is significant spatial 

variability of properties within a geologic unit, 

data is limited to the location of boreholes. As 

a result, it is difficult to specify hydraulic property heterogeneity within geologic units. 

• Spatial and temporal variability of recharge. 

Recharge at the Hanford Site is sensitive to the 

surface material (e.g., building, asphalt, soil), 

Figure 4-15. Uranium concentration (μg/l) contours interpreted from groundwater measurements in 1959 

made at the wells indicated in green
49

1977 Q2

10

30

50

70

90

110

130

150

1978 Q3 1979 Q3

1980 Q3 1981 Q3 1982 Q4

1983 Q4 1984 Q4 1985 Q4

Figure 4-16. Uranium concentration contours derived from measurements in the year and quarter indicated; 

measurement locations shown in green, contours at 10, 30, 50...150 μg/l. (A) 1977-1985 
50

1986 Q4

10

30

50

70

90

110

130

150

1987 Q4 1988 Q4

1989 Q4 1991 Q4 1992 Q4

1993 Q3 1994 Q4 1995 Q4

Figure 4-16 cont. (B) 1986-1995 
51

1996 Q3

10

30

50

70

90

110

130

150

1997 Q3 1998 Q3

1999 Q3 2000 Q3 2001 Q3

2002 Q4 2003 Q4 2004 Q4

Figure 4-16 cont. (C) 1996-2004 
52

the soil texture, and vegetation (if soil is the surface material). Recharge is likely to vary seasonally and from year to year. Nonetheless, the natural recharge flux is small relative to the groundwater flux and the impact on groundwater from 

temporal fluctuations in river stage is likely to be 

much greater than that arising from temporal 

variability in recharge. 

• River boundary transience. River stage is known 

to vary significantly on an hourly basis. The impact on transport of the scale at which this variability is modeled is of potential importance. 

• Uranium release rates. Disposals to the ponds 

have been estimated from available records, but 

significant uncertainties remain. Fluxes of water 

and contaminants to the unsaturated zone and the 

fluxes from the unsaturated zone to groundwater 

are even more uncertain.Spatial and temporal 

variability of uranium adsorption. Reactive 

processes in the unsaturated and saturated zones 

have been quantified recently, but significant uncertainty remains in the appropriate models to 

use. The suitability of applying lab-scale data to 

the field-scale is also uncertain. Geochemical 

modeling uncertainty was not the focus of this 

application, however, so the alternative models 

considered for adsorption were limited. 

• Heads along the lateral boundaries. Boundary 

heads are based on a limited number of boreholes, particularly in the north and northwest part 

of the site. A well was recently added in this 

area. 

• Representation of unsaturated flow and transport. The unsaturated zone at the 300 Area is a 

significant portion of the transport path. There 

are, however, only limited data on the unsaturated hydraulic properties and no in situ record 

of soil water content/pressure data for the 300 

Area unsaturated zone. In addition, there are limited data on uranium concentrations in the unsaturated zone. Some site data indicate that 

transport in the unsaturated zone was primarily 

vertical. At the same time, the extent of the 

groundwater plumes in the past may have been 

increased by lateral transport in the unsaturated 

zone. In any case, it is computationally easier to 

model only saturated flow and transport, thus 

motivating a simplified representation of unsaturated zone flow and transport. 

• Darcy’s Law alternative. As a result of the rapid 

fluctuations in river stage, an alternative form of 

Darcy’s Law that adds a term in the timederivative of flux was considered. This would 

require modification of groundwater flow codes 

and would likely only be applicable very close to 

the river and only during high-frequency river 

stage fluctuations. Waichler and Yabusaki 

(2005) determined that Darcy’s Law was valid at 

the 300 Area. 

To illustrate the application of the uncertainty assessment methodology, the alternative conceptual models 

represented uncertainties in three of these elements: the 

characterization of heterogeneity in hydraulic properties, temporal variability in the Columbia River stage, 

and spatial variability in adsorption. 

4.2.1 Hydraulic Property Heterogeneity 

Two alternatives were postulated that represent a degree of uncertainty in the hydraulic property heterogeneity. One of the alternatives assumed the hydrogeologic units were located as given in the geologic 

model discussed above and that hydraulic properties 

were uniform within each unit. The other alternative 

assumed a homogeneous characterization of hydraulic 

properties over the entire model domain. Justifications 

for this approach include observations that Unit 6 is 

absent over much of the model domain near the river 

(see Figure 4-4), Units 5 and 7 are hydraulically similar, a relatively small portion of the Hanford Formation 

is saturated, and many of the data are from wells 

screened over a length of 4.5 m (15 ft) or more. 

4.2.2 River Boundary Transience 

As discussed above, the Columbia River stage at the 

300 Area fluctuates significantly, sometimes within a 

single day. Representing the transient river behavior 

has an impact on the computational requirements of the 

modeling. As a result, it is not uncommon to average 

river fluctuations over some period of time or to model 

a transient river boundary as a steady-state boundary. 

For this application, two alternatives were considered: 

a steady-state and a transient river boundary condition. 

A monthly averaging period for the transient river alternative was used, based on practical concerns in limiting the computational requirements. 

4.2.3 Adsorption 

Two alternative representations of adsorption were 

included in this application. Both alternatives assumed 

a linear equilibrium adsorption model. One alternative 

assumed a spatially homogeneous Kd throughout the 

model domain. The other alternative used two Kd

zones, one near the river and one for the rest of the 

model domain. This allowed the model to represent, to 

a limited extent, the differences in adsorption due to 

water chemistry differences in the groundwater and 

river water. 
53

4.2.4 Alternative Conceptual Model 

Summary 

The alternative conceptualizations discussed above 

were combined to generate a set of model alternatives 

used in the uncertainty assessment application. The 

alternative models for the 300 Area uranium plume 

were combinations of (1) homogeneous vs. zoned hydraulic conductivity, (2) steady-state vs. transient river 

boundary condition, and (3) homogeneous vs. zoned 

Kd. Eight combinations of these three factors were 

modeled and are listed in Table 4-2. Note that these 

models covered the saturated zone only. 

Preliminary analyses demonstrated that the models 

with a steady-state river boundary (odd numbered 

models in Table 4-2) provided a poor representation of 

observed heads and uranium concentrations and that 

these models would be unable to compete with the 

transient models in terms of posterior model probability. As a result, the steady-state models were dropped 

from the analysis and are not included in the remaining 

discussion of this application. 

4.3 Alternative Model Implementations

The alternative conceptual models described in the 

previous section were implemented using MODFLOW 

(Harbaugh et al. 2000) and MT3DMS (Zheng and 

Wang 1999). This section describes the elements of the 

alternative models. 

The initial models used a simulation period starting in 

1945 and extending through the period of uranium discharges to the waste disposal sites. The recharge representing liquid waste disposal at the process trenches 

and ponds was based on the liquid fluxes used for the 

source terms in the Systems Assessment Capability 

simulations (Bryce et al. 2003). Uranium mass loadings 

for the liquid waste disposal sites were also based on 

information from the Systems Assessment Capability 

project. The calibrated models for the simulations beginning in 1945 were unable to represent the largescale extent of uranium contamination in groundwater 

observed prior to 1976 (e.g., Figure 4-15) and the 

large-scale introduction of significant uranium to the 

groundwater in 1995-97 (Figure 4-16 B and C) either 

through geochemical changes in the aquifer beginning 

in 1995 (with the cessation of discharges to the 316-5 

trench) or the presence of a long-term source of uranium in the vadose zone activated during the flood of 

1997. 

To reduce the uncertainty associated with the uranium 

sources and to simplify the model calibrations, the period of September 1997 to December 2004 was selected as the simulation period for calibration. It was 

assumed that there were no significant sources of uranium to the groundwater during this period. 

4.3.1 Simulation Domain 

The simulation domain for the 300 Area models was 

selected to cover the observed (past and current) and 

anticipated future extent of the uranium plume. A plan 

view of the three-dimensional simulation domain is 

shown in Figure 4-17, including locations of the Columbia River boundary, the boundary of the 300 Area, 

the primary disposal area boundaries, and existing 

Model Hydraulic Property Heterogeneity River Boundary 

Kd Spatial Variability 

1 Homogeneous Ks Steady-State Homogeneous Kd

2 Homogeneous Ks Transient Homogeneous Kd

3 5 Ks Zones Steady-State Homogeneous Kd

4 5 Ks Zones Transient Homogeneous Kd

5 Homogeneous Ks Steady-State 2 Kd Zones 

6 Homogeneous Ks Transient 2 Kd Zones 

7 5 Ks Zones Steady-State 2 Kd Zones 

8 5 Ks Zones Transient 2 Kd Zones 

Table 4-2. Model alternatives considered in the 300 Area application of uncertainty assessment 
54

monitoring well locations. The three-dimensional 

simulation domain was 1386.8 m (dimension of x) by 

2893.1 m (dimension of y) by 47.3 m deep. The rotation angle was 13.6 degrees counterclockwise from the 

east-west direction. 

4.3.2 Discretization 

The simulation domain was discretized into a 

MODFLOW/MT3DMS grid with 15 layers, 54 rows, 

and 32 columns (approximately 26,000 grid cells). Grid 

cell size varied from 34.67 m to 69.34 m in the x direction (perpendicular to the river), from 36.16 m to 

144.66 m in the y direction, and 1.0 m to 5.75 m in the 

z direction. Figure 4-18 illustrates the grid discretization. The grid was dense near the contamination 

sources (the ponds and trenches) with finer discretization also applied to the lateral boundaries. The vertical 

discretization was finer within the depth of the Columbia River than below the river. However, the first layer 

was coarse to incorporate fluctuation of the river stage, 

since the river was treated as a constant head boundary. 

The simulation period for model calibration was from 

September 1997 through December 2004. Because the 

transient river boundary was modeled using monthly 

average river stage, each month constituted a stress 

period for which river stage was constant. Natural recharge was assumed constant in space and time for the 

entire simulation period. 

4.3.3 Hydrogeologic Zonation and Hydraulic 

Parameters 

As discussed earlier, geologic analysis from borehole 

data reveals that there are five major hydrogeologic 

units in the 300 Area among six total units located 

above the underlying basalt. Using unit contact information from the borehole data, a three-dimensional 

model of the spatial distribution of the hydrogeologic 

units was developed (see Figure 4-3 and Figure 4-4). 

This model was used to project the spatial distribution 

of the hydrogeologic units onto the MODFLOW grid 

as illustrated in Figure 4-18 using the MODFLOW 

Hydrogeologic-Unit Flow (HUF) package. Zonation of 

the distribution coefficient was different than the hydrogeologic zonation and will be discussed in Section 

4.3.5. 

Estimated representative (average) values for hydraulic 

conductivity of the 300 Area sediments were given in 

Table 4-1. The parameter estimates were used as initial 

value and their ranges were used to constrain parameter 

adjustment in inverse modeling described in Section 

4.8. 

The upper boundary of the 300 Area flow models was 

a constant flux boundary with an applied flux of 55.4 

mm/yr representing long-term average recharge based 

on the analysis of Fayer and Walters (1995). The bottom boundary of the models was a zero-flux boundary 

representing the top of the confining Lower Mud unit 

(u8). Although flow from the underlying confined basalt aquifer to the unconfined aquifer may occur at locations where the lower mud unit (u8) is incomplete, 

this unit appears to be contiguous throughout the model 

domain. Upward flux from the basalt aquifer was not 

anticipated to affect uranium concentration and was not 

considered in the models. 

The eastern boundary of the model domain was the 

Columbia River. Groundwater is generally discharged 

to the river. During high river stage, there is infiltration 

of river water through the river bank resulting in mixing within the near-river groundwater and a potential 

reversal of groundwater flow (see the discussion in 

Section 4.1.2.4). In the geologic model the river penetrates through the Hanford formation and into the upper 

Ringold (u5) except for a portion of the boundary 

around Northing coordinate 115500 m (see Figure 4-4). 

The shape and depth of the river are irregular, which 

was reflected in both the geologic model and the 300 

Area simulation model grid (Figure 4-18). In the flow 

models, the river was treated as a prescribed head 

boundary with temporally varying head. As discussed 

in Section 4.1.2.4, the river stage fluctuates significantly on an hourly time-scale. For computational reasons, however, the river boundary varied no more frequently than monthly for the simulation models. The 

monthly average river stage at the 300 Area as simulated by Waichler et al. (2005) was taken as the eastern 

head boundary. Figure 4-19 shows the simulated river 

stage over the period of site operation. 

The western boundary of the flow models was treated 

as a prescribed head boundary with the head value 

along the boundary varying in space. The assigned 

head values were based on groundwater heads observed in wells located near the boundary. Observed 

heads supported the assignment of zero flux to the 

southern boundary of the nominal model. The northern 

boundary was also treated as a zero-flux boundary, 

although this assumption has less support from observations since there has only recently been a monitoring 

well in the northwest portion of the model domain. 

The boundaries of the transport model coincided with 

those of the flow model. The upper boundary along 

with the bottom, northern, western, and southern 

boundaries were treated as zero flux transport boundaries. The eastern boundary along the Columbia River 

was treated as an outflow boundary. That is, mass 
55

transport across the boundary occurred by advection 

only and solely in the direction from the groundwater 

to the river. 

4.3.4 Initial Conditions 

The initial condition for the flow models was based on 

average heads across the simulation domain. The initial 

uranium concentration in groundwater was based on 

observed concentrations in August 1997 (Figure 4-20, 

left). Artificial observations were used to provide a 

smooth interpolation of concentration over layer 1 of 

the simulation domain (Figure 4-20, right). These interpolated concentrations were also applied to layers 2-

5 of the model. Fifty percent of the layer 1 concentration was applied to layer 6 and 25% to layer 7 for the 

initial condition. Layers 8-15 had an initial concentration of zero. As discussed in the introduction to Section 

4.3, although groundwater contamination occurred 

much earlier, the simulations for this application were 

begun in September 1997 to reduce the uncertainty 

associated with the uranium sources and to simplify the 

model calibrations. 

4.3.5 Geochemical Modeling 

Desorption/adsorption experiments conducted on 300 

Area soil samples indicate that uranium chemistry at 

the 300 Area may be quite complex (Serne et al. 2002; 

Zachara et al. 2005). Reproducing observed uranium 

transport may thus require a reactive transport model, 

such as the non-equilibrium adsorption model or the 

surface complexation model described in Zachara et al. 

(2005). Application of a reactive transport model was 

beyond the scope of this project, however. 

The alternative models implemented here assumed that 

adsorption at the site can be represented as a timeinvariant, equilibrium process. Serne et al. (2002) indicated that a linear adsorption model is appropriate for 

concentrations currently observed in the groundwater, 

but that a non-linear model may be applicable for 

higher concentrations potentially present in the vadose 

zone. A range of distribution coefficient values 

( 0 100 ≤ ≤ Kd ) have been estimated from laboratory 

experiments on the <2 mm particle size fraction (Serne 

et al. 2002; Zachara et al. 2005). It is generally expected that these experimental Kd values will be reduced in the field due to the presence of coarse (>2 

mm) sediments (Kaplan et al. 2000). Serne et al. (2002) 

concluded that likely ranges for field values of Kd are 

from 0 to 1 mL/g in the near surface vadose zone influenced by evapotranspiration, 2 to 4 mL/g in the unconfined aquifer sediments not influenced by dilution of 

river water, and 7 mL/g or higher when groundwater is 

diluted by river water. Their proposed spatial variability in Kd values was primarily due to differences in 

dissolved inorganic carbon. 

The effect of river water on uranium adsorption was 

approximated in the model alternatives implemented 

here by using a spatially variable, zoned Kd with different underlying values for the groundwater within a 

mixing zone adjacent to the Columbia River. For the 

alternative models with heterogeneous Kd, the model 

domain was separated into two parts by the black lines 

shown in Figure 4-18. This zonation was chosen based 

on an approximate zone of mixing as reported by 

Waichler and Yabusaki (2005) using observed nitrate 

concentrations. Prior values for Kd are discussed in 

Section 4.5. 

4.4 Prior Model Probability

For the 300 Area application, equal prior model probabilities were assumed to simplify the analysis. This 

reflects the a priori belief that the alternative models 

are all equally plausible. For many applications, nonuniform prior model probabilities may be appropriate. 

In these cases, expert judgment combined with the 

method of Ye et al. (2005) can be used to assign prior 

model probabilities. Appendix C contains an application of MLBMA to alternative recharge models in 

which a panel of experts was used to assign prior 

model probabilities. 

Ye et al. (2005) proposed to maximize a measure of 

entropy composed of prior model probabilities subject 

to constraints representing prior expert knowledge of 

the relative model probabilities. Specifically, they proposed to assign prior model probabilities by solving the 

following nonlinear, constrained optimization problem. 

1

max log k

K

k k p k

H p p

=

= −∑ (30) 

subject to 

1

0 1,...

0 1,...

1

i

j

K

k

k

g i I

h jJ

p

=

≤ =

= =

∑ =

 (31) 

where H is the entropy measure, p pM k k ≡ ( ) , the 

prior model probability, and i g and j h are specified 

relationships between the various k p values. For example, the constraint 2 1 2 0 p p − ≤ expresses the prior 

judgment that Model 1 is at least twice as plausible as 
56

114500

115000

115500

116000

116500

117000

117500

118000

Northing [m, WCS]

593000 593500 594000 594500 595000

Easting [m, WCS]

1-1

1-2 1-3/7/8

1-4

1-5

1-6

1-10

1-11

1-13 1-12

1-14

1-15

1-17A/B 1-16A/B

1-18A/B

1-19

2-1

2-2

2-3

3-1

3-2 3-3

3-6 3-7

3-8 3-9

3-10

3-11

3-12

4-1

4-5

4-7

4-8

4-9

4-10

4-11

4-12

5-1

6-1

8-1 8-2

8-3

S19-E13

S27-E14

S29-E12

S28-E12

Columbia River 

Island

Island

 Model Boundary

 Columbia River Boundary

 300 Area Boundary

 Waste Discharge Site

 Well Location

Figure 4-17. Simulation domain and locations of major features of the 300 Area simulation model 
57

Y

Z

X

Figure 4-18. Discretization of three-dimensional simulation domain. Cells are inactive within the Columbia 

River (shown in blue). Hydrogeologic units are also shown in the figure as follows: u1 (teal), u5 

(green), u6 (grey), u7 (yellow), u8 (red). The black lines indicate an adsorption zone near the 

river.
58

101

102

103

104

105

106

107

108

109

110

111

Monthly Average River Stage at 300 Are

Jan-45 Jan-55 Jan-65 Jan-75 Jan-85 Jan-95 Jan-05

a

(m NAVD88)

114500

115000

115500

116000

116500

117000

117500

118000

Northing [m, WCS]

593000 593500 594000 594500 595000

Easting [m, WCS]

 250

 200

 150

 130

 110

 90

 70

 50

 30

 10

Uranium (μg/l)

114500

115000

115500

116000

116500

117000

117500

118000

Northing [m, WCS]

593000 593500 594000 594500 595000

Easting [m, WCS]

Figure 4-19. Monthly average river stage at the 300 Area simulated by Waichler et al. (2005) and used as a 

boundary condition in the groundwater flow models 

Figure 4-20. Contours of (left) observed uranium concentration in groundwater in August 1997 and (right) 

modified values used as the initial condition in layer 1 of the simulation models 
59

Model 2. If the constraints are logical in that they conform to Ockham’s razor and to behavior expected on 

theoretical and/or empirical grounds (i.e., if one excludes arbitrary solutions that are not based on the 

principle of parsimony coupled with sound expert 

knowledge), then the corresponding informed solution 

is preferred over the non-informative neutral choice 

(i.e., 1/ k p K = ). 

If an expert, or a group of experts, is unable to select 

one set of constraints among several alternative sets, 

Ye et al. (2005) recommended choosing among alternative expert opinions a posteriori, that is, on the basis 

of posterior measures of model quality. If sufficient 

data are available to conduct a cross-validation, Ye et 

al. (2005) recommended choosing the set of prior probabilities that yields optimum predictive performance as 

measured by an objective criterion (such as mean 

squared prediction error). If there are not enough data 

to conduct a meaningful cross-validation Ye et al. 

(2005) suggested selecting the set of prior probabilities 

that maximizes the likelihood of the set of model alternatives in light of the data. That is, they suggested 

choosing the set of prior probabilities that maximizes 

( ) ( ) ( )

( )

1

1

1

exp 2

K

k k

k

K

k k

k

p p M pM

KIC p M

=

=

=

⎛ ⎞ ≈ −⎜ ⎟ ⎝ ⎠

∑

∑

D D M

 (32) 

Once KICk values, which are independent of prior 

model probabilities, have been computed, then Eq. (32) 

can be easily calculated for any admissible (i.e., arrived 

at via entropy maximization on the basis of prior expert 

judgment) prior probability set. 

4.5 Prior Parameter Probability

The methodology described in Chapter 3 allows for the 

application of prior parameter probabilities as an optional component of the maximum likelihood objective 

function used in the parameter estimation (Eq. 11 and 

Step 3 of Section 3.3) and as an uncalibrated, but uncertain parameter input to the computation of prediction uncertainty in Step 9 of Section 3.3. The assessment of prior parameter uncertainty was discussed in 

general terms in Section 2.1. This section describes the 

use of generic and site-specific data to estimate uncertainty in the average linear adsorption coefficient at the 

Hanford Site 300 Area and application of this data to 

specify prior values of Kd in the model calibrations. 

4.5.1 Modeling Adsorption in Contaminant 

Transport Models 

Adsorption is one of the most important geochemical 

mechanisms for slowing or preventing the migration of 

contaminants through the vadose zone and in groundwater systems. The most common method used to describe contaminant adsorption on complicated matrices 

such as soils and sediments is the distribution coefficient or Kd model: 

Kd = S/Caq (33)

where S is the concentration of the contaminant on the 

solid and Caq is the concentration in the aqueous phase. 

Implicit assumptions typically made when using this 

model in groundwater transport codes are that the Kd is 

constant (conditions that affect the Kd do not change), 

adsorption reaches equilibrium quickly relative to the 

groundwater flow rate, and the adsorption is reversible. 

These are not necessarily realistic assumptions. The 

popularity of the distribution coefficient approach 

among modelers stems from its simplicity and the fact 

that it can be applied to complex matrices and solutions 

for which it would be difficult or impossible to obtain 

all the required thermodynamic and mechanistic surface adsorption data. This advantage is also the primary 

drawback of this approach. Because the model is empirical, it should be strictly applied only to the same 

geochemical conditions under which the Kd was measured. This condition can be relaxed if a variable is 

known to have no or minimal influence on adsorption 

of the contaminant of interest. When geochemical conditions that affect adsorption vary significantly within 

the system being modeled, then the constant Kd model 

can produce erroneous results. This problem can be 

avoided by varying the Kd spatially and temporally as 

needed due to changing geochemical conditions. In 

addition, if mechanisms other than adsorption are important for immobilizing contaminants of interest (such 

as precipitation), the Kd model may not be appropriate. 

Surface complexation modeling is a mechanistic-based 

approach for describing adsorption that can account for 

variable geochemical conditions. Most commonly used 

groundwater transport codes do not have the capability 

to model sophisticated geochemical interactions. Reactive transport codes that integrate thermodynamics to 

model complex geochemical interactions are computationally expensive and typically have intensive input 

data requirements. As a result, these more sophisticated 

approaches to contaminant transport modeling are not 

generally applied, but are applied at sites where resources permit more in-depth research. 
60

An alternative to using the more geochemically comprehensive reactive transport codes, is to use surface 

complexation data and geochemical equilibrium codes 

to calculate Kd values that can be used in groundwater 

transport codes that rely on Kd values to describe adsorption. In this approach Kd values that vary spatially 

and/or temporally can be used as input data to the 

groundwater transport code. 

Keeping in mind the caveats discussed above, adopting 

the use of Eq. (33) requires estimating an appropriate 

value of Kd and the uncertainty in that value. A variety 

of data sources are available to provide a basis for this 

estimation. Table 4-3 lists a possible hierarchy of this 

data, ranging from generic literature values to detailed, 

site-specific geochemical measurements. The availability of data determines the appropriate methods used to 

estimate Kd values. It is assumed in Table 4-3 that as 

the quantity and quality of site-specific characterization 

data become available, the quality of adsorption data 

that can be determined or measured will increase and 

the uncertainty will decrease. The challenge is to converge on an appropriate balance between minimizing 

Kd uncertainty and minimizing the resources needed to 

reduce that uncertainty. 

4.5.2 Hanford Site 300 Area Uranium 

Example 

Application of the concepts embodied in Table 4-3 are 

illustrated here using as an example uranium adsorption at the Hanford Site 300 Area. 

4.5.2.1 No Site-Specific Data Case 

A number of literature compilations of Kd values are 

available. Sheppard and Thibault (1990) provide a 

compilation of Kd values for four major soil types 

(sand, loam, clay, and organic). Across all soil types, 

the overall range for the uranium Kd is from 0.03 to 

4.0x105

 L/kg (mL/g). A look-up table for estimating Kd

values as a function of pH (for pH values ranging from 

3 to 10) is provided in EPA (1999). The overall range 

of Kd values across the entire pH range of 3 to 10 was 

<1 to 1.0x106

 L/kg. It is clear from these data sources, 

that the range of Kd values for a wide range of possible 

conditions is very large for uranium. From a transport 

modeling perspective, this indicates that uranium mobility can range from being essentially completely mobile (moves at the same rate as water) to essentially 

completely immobile. One could define a Kd probability distribution to reflect this large range of possible 

values, but the resulting transport uncertainty would 

likely be so large as to be unworkable. 

4.5.2.2 Known Soil Texture and pH Case 

When the soil texture and pH are known and can be 

used to represent average site conditions, this information can be used to narrow the range of Kd values used 

for transport modeling and hydrologic uncertainty assessments. As indicated previously, Sheppard and Thibault (1990) provide a compilation of Kd values for 

four major soil types (sand, loam, clay, and organic). In 

this compilation, mineral soils were categorized by 

texture into sand, clay, or loam. Soils that contained ≥

70% sand-sized particles were classified as sand soils. 

Available Site-Specific Input Data Methodology Used to Determine 

Best Estimates and Range of Kd

Values 

Resulting Uncertainty in Adsorption Modeling Results 

No Site-Specific Data Available Literature Compilations of 

Generic Kd Values

Soil Texture and pH Refinement of Kd Values Using 

Available Literature Compilations of 

Generic Kd ValuesBased on Limited 

Site-Specific Data

Site-Specific Batch Kd Measurements Better Defined Kd Values and 

Ranges Based on Site-Specific Empirical Measurements 

Detailed Geochemical Information, 

Site-Specific Adsorption Measurements and Model Development 

Use of More Scientifically Defensible Adsorption Models and SiteSpecific Data to Calculate Kd Values 

for Variable Conditions 

Decreasing Uncertainty, 

Increasing Resource Requirements 

Table 4-3. Hierarchical application of data to determine best estimates and uncertainty in Kd values 
61

Mineral soils that contained ≥ 35% clay-sized particles 

were classified as clay soils. Loam soils had an even 

distribution of sand-, clay-, and silt-sized particles or 

consisted of up to 80% silt-sized particles. Soils containing > 30% organic matter were classified as organic 

soils. 

Previous work conducted on 300 Area sediments can 

be used as a source for soil texture and pH data 

(Zachara et al. 2005). Table 4-4 contains major size 

fraction data for a sediment sample collected at 16 feet 

below ground surface (bgs) from the 300 Area North 

Process Pond (NPP) along with the pH value measured 

on a 1:2 water extract. Included are percentages for the 

whole sample and the sample fraction less than 2 mm. 

Because the 300 Area sediments are composed of predominantly cobbles, they don’t fit into the classification scheme of Sheppard and Thibault (1990). To handle this problem, the Sheppard and Thibault method for 

estimating Kd values will be used on the less than 2 mm 

size fraction and a gravel correction will be handled 

separately. For the NPP-1 (16 ft bgs) sample, the less 

than 2 mm size fraction contains 70% sand. As a result, 

this sediment is classified as a sand using the Sheppard 

and Thibault (1990) scheme. From Table 3 in Sheppard 

and Thibault (1990), the mean uranium Kd value for 

sand is 35 L/kg, with a range of 0.03 to 2200 L/kg. 

A methodology for adjusting Kd values to account for 

the gravel content of soils and sediments is provided in 

Cantrell et al. (2003). This report indicates that for high 

Kd contaminants (Cs, Sr, and Pu), the following equation is recommended (see Appendix A, Kaplan and 

Serne 2000). 

Kdgc = (1-f) Kd<2mm + (f)0.23 Kd<2mm (34) 

where Kdgc is the gravel-corrected Kd value, f is the 

weight fraction gravel, and Kd<2mm is the Kd value determined using the less than 2 mm material. For low Kd

contaminants, the following equation is recommended. 

Kdgc = (1-f) Kd<2mm (35) 

Sediment ID Entire Sample Fraction <2mm 

Cobbles (%) 91.7 - 

Sand (%) 6.5 70 

Silt (%) 0.77 13 

Clay (%) 1.01 17 

pH 7.98 7.98 

The potential range of uranium Kd spans values generally considered to be both low and high. Because the 

mean Kd value for uranium is 35 L/kg, Eq. (34) was 

used for the gravel corrections. 

Using a value of 0.917 for f in Eq. (34), the gravelcorrected mean uranium Kd value for 300 Area sediments was calculated to be 10 L/kg with a range of 

0.009-650 L/kg, based on Table 3 in Sheppard and 

Thibault (1990). 

This Kd value estimation methodology has reduced the 

uncertainty for the Kd values by over two orders of 

magnitude and has provided a mean estimate as well. A 

probability distribution consistent with the mean and 

range could be assigned to represent Kd uncertainty. 

This approach still produces a relatively large uncertainty for uranium Kd values. 

4.5.2.3 Available Regional Batch Kd Values 

Kd values measured on Hanford Site sediments and 

Hanford groundwater range from 0.16 to 104 L/kg (on 

< 2 mm size materials) (Cantrell et al. 2003). None of 

these measurements were on 300 Area sediments. As 

part of the composite analysis being conducted for the 

Hanford Site, best estimate Kd values were selected for 

various waste chemistry/source categories and impact 

categories (Last 2004). Estimates of minimum and 

maximum values were also provided. These values 

were determined based upon a thorough review of the 

data compiled in Cantrell et al. (2003). For natural 

Hanford groundwater conditions, the best estimate Kd

value for uranium was taken to be 0.8 L/kg, with a 

minimum of 0.2 L/kg and a maximum of 4 L/kg. Applying the gravel correction appropriate for the 300 

Area (Eq. 35 with f = 0.917), a best estimate of 0.07 

L/kg was determined, and a minimum and maximum of 

0.02 and 0.33 L/kg were determined. Even though 

these Hanford Site values could be considered site specific, they are not specific to the 300 Area. 

4.5.2.4 Available Site-Specific Batch Kd Values 

Leaching and adsorption characteristics of uranium in 

near-surface sediments collected from the 300 Area of 

the Hanford Site were studied by Serne et al. (2002). 

Adsorption Kd values were measured over a range of 

solution variables. These included two artificial 

groundwater compositions; one referred to as high 

ionic strength solution and one referred to as low ionic 

strength. The low ionic strength solution was a 10:1 

dilution of the high ionic strength solution. Kd measurements with these solutions were conducted over a 

range of uranium concentrations that varied from 50 to 

5000 ppb U. Other experiments were conducted at 

Table 4-4. Textural characteristics and pH for 

sediment sample NPP-1 obtained 16 ft. 

below ground surface from the North 

Process Pond (Zachara et al. 2005) 
62

variable pH values and variable bicarbonate concentrations. The Kd values measured under conditions that 

most closely resemble those of 300 Area groundwater 

were those measured using the low ionic strength solution. Only these Kd values will be reviewed here. 

Ten Kd measurements were made using the low ionic 

strength solution. As indicated, the U concentration 

was varied from 50 to 5000 ppb. Results from both the 

low ionic strength and high ionic strength solutions 

indicated that the adsorption was linear over this range 

of U concentration (i.e., the Kd model was appropriate). 

The average and standard deviation for the ten Kd values measured at low ionic strength were 3.77 and 0.71 

L/kg, respectively, with a minimum of 2.99 and a 

maximum of 5.04. Applying the same gravel correction 

used previously, a corrected average and standard deviation of 0.31 and 0.06 L/kg, respectively, were calculated. The geometric mean was 0.31 L/kg also. Assuming Kd follows a lognormal distribution, the mean plus 

and minus three standard deviations yields an approximate range of values from 0.18 to 0.53 L/kg. 

4.5.2.5 Available Site-Specific, MechanisticallyBased, Adsorption Model Case 

To account for the variation in the observed Kd values, 

a new geochemical conceptual model was developed to 

describe U(VI) adsorption on 300 Area sediments 

(Zachara et al. 2005). This conceptual model included 

a surface complexation model (SCM) based on the 

generalized composite approach (Davis et al. 2004, 

2002; Davis and Curtis 2003). A preliminary SCM 

based on this work was used here (Davis, J.A., personal 

communication, 2005). U(VI) adsorption to 300 Area 

sediments was measured over a range of uranium concentrations and alkalinity that was significantly greater 

than those observed at the 300 Area. This ensured that 

the SCM was applicable to all groundwater conditions 

that could be expected at the 300 Area. 

Two equations were found to provide the best fit to all 

the experimental adsorption data. The two equations 

are given below, including the log of the equilibrium 

constant and its standard deviation: 

SOH + UO2

2+ + H2O = SOUO2OH + 2H+

 Log K = -4.476 ±0.048 

SOH + UO2

2+ + 2H2CO3 = SOUO2(HCO3)2-

 + 3H+

 Log K = -4.081 ±0.036 

These equations were used along with major ion composition data available from the HEIS (1994) database 

for 300 Area wells to calculate Kd values. The equilibrium distribution between the solid and aqueous phase 

was calculated using the SpecE8 module of Geochemists Workbench® (Bethke 2005). The thermodynamic 

database thermo.com.v8.r6+.dat was modified to include the same formation constants for U(VI) solution 

species used in the preliminary SCM. 

Kd values were calculated for groundwater compositions determined from 19 wells in the 300 Area. Only 

wells that had a complete analysis set were used. This 

included pH, major cations and anions, and alkalinity. 

Wells that were near the river were avoided because 

these wells are seasonally influenced by the Columbia 

River. Only one analysis was used for each well to 

calculate the Kd values. In general, the latest available 

analysis that included the required suite of analytes was 

used. Table 4-5 shows the groundwater compositions 

used to calculate the Kd values, along with the calculated Kd values. A gravel fraction of 0.917 in Eq. (35) 

was used to correct the derived Kd values. 

The average and standard deviation for the nineteen Kd

values that were calculated was 3.18 and 2.64 L/kg, 

respectively, with a geometric mean of 2.14 L/kg. The 

minimum and maximum values calculated were 0.25 

and 9.96 L/kg. Assuming Kd follows a lognormal distribution, the mean plus and minus three standard deviations yield an approximate range of values from 

0.10 to 44 L/kg. This large variation in calculated Kd

values illustrates that relatively minor changes in solution chemistry can have relatively large impacts on 

how strongly U(VI) is adsorbed to Hanford sediments. 

The variation in Kd values calculated using the SCM 

approach was significantly greater than that determined 

from both the 300 Area site-specific Kd measurements 

and Kd measurements made using sediments collected 

throughout the Hanford Site. In the case, of the 300 

Area site-specific measurements, the limited variation 

of the Kd measurements was due to the fact that the 

background solution composition was identical for 

each measurement. For the Hanford-wide Kd measurements, natural Hanford groundwater samples (uninfluenced by waste site releases) were used as the background solution. In the case of the Kd values calculated 

for the 300 Area using the SCM, variable solution 

compositions measured from wells throughout the 300 

Area were used. In some cases, relatively low pH values were observed in wells near the process trenches 

and ponds. Natural Hanford groundwater pH values 

typically range from about 7.8 to 8.2. At the 300 Area 

some pH values as low as 7.35 were used to calculate 

the Kd values. This explains the reason for the relatively large variation in Kd values calculated using the 

SCM. 
63

Well ID Sample Date Alkalinity Ca Cl Mg Nitrate pH K Na Sulfate Kd

399-1-12 6/6/1996 120000 44000 16300 9500 22150 7.47 2600 20000 51300 6.52

399-1-13A 7/17/1992 120000 48500 16014 9900 25650 7.86 4500 22000 40720 1.26

399-1-14A 6/4/1996 140000 47000 16500 11000 23300 7.46 5500 22000 53200 4.84

399-1-17A 12/16/2004 140000 56800 18400 12800 23000 7.46 5040 27700 53800 3.46

399-1-21A 4/30/1992 121000 47200 20200 9505 17300 7.40 4710 18250 46000 7.42

399-1-5 5/20/1992 111000 47950 18500 9710 21200 7.35 2550 17250 40720 9.96

399-3-11 6/11/1992 123000 46150 17500 9255 15100 7.55 5195 20250 39000 4.05

399-3-12 4/23/1992 129000 45550 18200 9350 15100 7.80 5105 20650 41000 1.51

399-3-2 12/9/1991 116000 41650 16014 8705 19000 7.98 5105 16700 27000 1.15

399-3-3 5/13/1992 98000 31550 8900 6420 7970 7.79 3660 12350 23000 4.78

399-4-1 12/10/1991 120000 41000 16014 8050 25650 7.82 4800 18000 40720 1.95

399-4-11 12/11/1991 120000 42000 16014 8100 25650 7.61 4600 18000 40720 3.99

399-5-1 4/21/1992 158000 63850 17300 12800 66400 7.84 6605 24250 54000 0.54

399-6-1 4/21/1992 134000 50700 20700 11150 31400 8.30 5985 22150 40000 0.25

399-8-1 7/15/1992 110000 43000 14200 8750 15300 8.16 5250 14000 36200 0.73

399-8-4 1/8/1992 106000 40750 16014 8865 15900 7.87 3420 17550 25000 2.04

399-8-5A 4/16/1996 110000 43000 14000 8900 24000 7.80 5200 14000 32000 2.21

699-S27-E14 11/29/1990 151000 51200 11600 10600 24680 7.47 6250 21000 35300 3.28

699-S28-E12 4/21/1992 146000 64450 11900 13400 68600 7.90 6835 24700 54000 0.48

Average 124895 47174 16014 9829 25650 7.73 4890 19516 40720 3.18

Standard Deviation 16141 7922 2931 1778 15702 0.26 1184 3910 9713 2.64

Coeff. of Variation (%) 12.9 16.8 18.3 18.1 61.2 3.4 24.2 20.0 23.9 83.0

In summary, a hierarchy of data was applied to specifying the uncertainty in Kd values at the Hanford Site 300 

Area. Uncertainty in this case was represented as a 

range of possible values, which was generally expected 

to be reduced as more site-specific data was applied. 

Figure 4-21 displays the resulting ranges of Kd values 

derived for the five levels of data. If an average value 

was available that is shown also. (As noted above, 

when the Kd estimates are supported by limited data, 

the resulting uncertainty may be unworkably large. 

When batch Kd measurements are the only source of 

data, they may underestimate the uncertainty by failing 

to account for potential variation in water chemistry. 

The use of site-specific adsorption measurements and 

an SCM, when applied to observed water chemistry 

data resulted in a realistic range of Kd values. Since 

these values were computed using a common gravel 

fraction, the derived Kd uncertainty would be somewhat 

larger if the variation in sediment particle size distribution were included. 

4.5.3 Prior Values of Kd for 300 Area Model 

Calibration 

The distribution of Kd based on the 300 Area surface 

complexation model (Section 4.5.2.5) could be used in 

model calibration by specifying a prior estimate for the 

value of Kd as the mean of the distribution and deriving 

a weight for the prior estimate from the variance of the 

distribution.. Because two of the models (Models 6 and 

8) have two distinct Kd zones (and thus two Kd parameters for which prior information could be supplied), the 

data described in Section 4.5.2.5 were supplemented 

with a set of seven Kd estimates derived from the surface complexation model using water chemistry data 

from seven wells near the river (Table 4-6). The mean 

Kd for the near-river wells was 20.1 L/kg with a standard deviation of 17.5 L/kg. For the entire set of 26 Kd

estimates, the mean was 7.75 L/kg with a standard deviation of 11.7 L/kg. 

The mean values for the inland and near-river Kd estimates were used as prior estimates for the two Kd parameters in Models 6 and 8. For Models 2 and 4 in 

which the Kd was uniform, the mean value for the enTable 4-5. 300 Area groundwater compositions (μg/L) used to calculate Kd (L/kg) values using the SCM, 

along with the calculated Kd values. Units for alkalinity are in μg/L as CaCO3. Included at the bottom are the average, standard deviation, and coefficient of variation (COV) for each of the chemical constituents and Kd values for all the wells. 
64

tire set of Kd estimates was used as the prior estimate. 

For all models, the weight applied to the prior estimates was taken as the inverse of the standard deviation of the entire set of Kd estimates multiplied by the 

number of estimates (26 in this case). Because the log 

of Kd was estimated in the calibration, the prior values 

and the weight were computed on the log Kd estimates. 

Prior estimates and the weight for the log(Kd) parameters are given in Table 4-7 for each of the model alternatives. 

Well ID Kd (L/kg) 

399-1-10A 7.3 

399-1-16A 5.5 

399-3-10 31.7 

399-3-9 51.1 

399-4-10 18.9 

399-4-7 1.5 

399-4-9 25.0 

10-2 10-1 100 101 102 103 104 105

Kd (L/kg)

No site-specific data

Soil texture data

Regional batch Kd data

Site-specific batch Kd data

Site-specific data, SCM

Parameter Model 

Applicability 

Prior 

Estimate 

Weight 

Kd 2 and 4, 

full domain 

-2.46443 45.51 

Kd1 6 and 8, 

inland 

-2.67047 45.51 

Kd2 6 and 8, 

near-river 

-1.905192 45.51 

4.6 Alternative Hydrologic Scenarios

The baseline hydrologic scenario for the uncertainty 

assessment application to uranium transport at the 300 

Area was based on a continuation of current conditions 

at the site. That is, it was assumed that the current hydrologic conditions at the site will continue in the future. This was implemented by simulating a 20-year 

period from January 2005 through December 2024 

with the river boundary condition taken to be the river 

stage during the period 1975 to 1994. This boundary 

condition was selected for convenience. In a more sophisticated analysis, the river stage could be modeled 

as a random variable (correlated in time) sampled from 

a distribution consistent with the statistics of the observed river stage. 

When considering alternative hydrologic scenarios at 

the site, several changes to the future hydrologic conditions can be reasonably proposed. (1) Future land uses 

in the surrounding area may change as restoration of 

the Hanford Site progresses. One such change considered for a performance assessment of waste disposal at 

the Hanford Site is agricultural development of land 

after completion of Site restoration (anticipated to be 

mid-21st century). Evans et al. (2000) examined the 

potential for agricultural development at the Hanford 

Table 4-6. Kd values estimated for near-river wells 

using a surface complexation model 

Figure 4-21. Ranges of Kd values for the Hanford Site 300 Area derived using a hierarchy of data 

Table 4-7. Prior estimates and calibration weight 

for the log of distribution coefficient parameters 
65

Site and estimated recharge rates resulting for the anticipated irrigation. Mixed use development is also 

being considered for the site. (2) The operation of dams 

on the Columbia River may change as a result of 

changes to dam structures, dam aging, and water use 

priorities. These changes will have an impact on the 

magnitude and variability of river stage at the 300 

Area. (3) Climate change is anticipated to impact the 

hydrology within the Columbia River basin in a number of ways (Leung et al. 2004). Changes in precipitation and temperature may directly impact the recharge 

rate at the 300 Area and may indirectly impact the river 

stage through resulting changes in dam operations. 

One alternative hydrologic scenario was considered in 

this application. This scenario assumed that changes to 

dam operation results in Columbia River flows that are 

closer to its free-flowing state. This was implemented 

by simulating a 20-year period from January 2005 

through December 2024 with the river boundary condition taken to be the river stage during the period 1955 

to 1974. The river stage during this time period reflects 

the much higher peak spring/summer flows that existed 

prior to the completion of the Canadian dams on the 

Columbia River (Figure 4-19). While this scenario is 

not plausible in the near-term, it was used here to demonstrate the uncertainty methodology in this application using a scenario that has a significant impact on 

flow and transport. 

4.7 Prior Scenario Probability

As discussed in Section 2.3.1, since scenarios describe 

future conditions, prior scenario probabilities must be 

assigned to each scenario based on a subjective understanding and judgment of the plausibility of the scenario. While formal methods could be used to estimate 

scenario probabilities [e.g., expert judgment and the 

approach of Ye et al. (2005) could be applied], for this 

application, arbitrary prior scenario probabilities were 

used to illustrate the uncertainty assessment methodology. A prior probability of 0.7 was assigned to the 

baseline hydrologic scenario and a probability of 0.3 

was assigned to the alternative scenario. 

4.8 Model Calibration

Model calibration (inverse modeling) was conducted to 

obtain maximum likelihood parameter estimates so that 

the NLL (Eq. 11) and Fisher information matrix (Eq. 

12) could be calculated to compute KIC (Eq. 10) and 

posterior model probability (Eq. 8). The calibration 

was implemented using PEST, whose utilities for 

MODFLOW and MT3DMS parameter estimation were 

used to link MODFLOW/MT3DMS/PEST. The utilities contain a series of subroutines to extract simulated 

heads and concentrations from MODFLOW and 

MT3DMS output files corresponding to measured 

heads and concentrations selected for model calibration. Residuals between the simulated and measured 

heads and concentrations form an objective function, 

which was minimized iteratively to find optimum 

model parameters. In this application, prior information 

on the Kd parameter(s) was used and included in the 

objective function. 

4.8.1 Selection of Calibration Data 

The available head and concentration observations in 

the 300 Area for the period of September 1997 to December 2004 were reviewed for use in the calibration 

of the alternative models. Observations from 21 wells 

were selected to provide spatially and temporally representative coverage (Figure 4-22). Three wells were 

screened in the lower portion of the aquifer (well numbers are indicated in Figure 4-22) while the remaining 

18 wells were screened near the top of the aquifer. A 

subset of data from each of the 21 wells was used in 

the calibration with the general goal of two observations per year at each well if available. The distribution 

in time of the selected calibration data is shown in 

Figure 4-23. In total there were 430 observations used 

in the calibration, of which 222 were head measurements and 208 were concentration measurements. 

4.8.2 Weights Associated with Calibration 

Data 

The weights for head measurements were proportional 

to the inverse of estimated observation errors and included components related to well-altitude error, welllocation error, nonsimulated transient error, and measurement error. More discussion of these individual error components can be found in Belcher et al. (2004). 

At the Hanford Site, the well-location error was small 

enough to be negligible (Vermeul et al. 2003). The 

well-altitude error was less than 0.01 m, given a maximum head gradient of about 0.25%. The measurement 

error was about 0.03 m corresponding to a depth of 

observation of 30 m. To incorporate nonsimulated transient error, final head observation error was considered 

as 0.06 m. 

According to Hill (1998), given a measurement accuracy, h a , so that *

h h ha h ha − ≤ ≤+ ( * h and h being 

“true” and measured head values), the standard deviation of observation error is /1.96 h h σ = a . This assumes that head observation error follows a normal 

distribution with zero mean, and 1.96 is determined 

from the 95% confidence interval. Therefore, the 
66

weights associated with head observations can be estimated via 

1 1 1.96 * = /1.96 h hh

hh h

w ha h ha σ a a = = − ≤ ≤+ . (36) 

With a head observation error of ah = 0.06 m , this 

gives a uniform weight of 32.667 for each head observation used in the calibration. 

The weights associated with concentration observations 

were initially estimated using lab-reported analytical 

errors for uranium concentration measurements at well 

399-1-17A, a well near the southern boundary of the 

316-5 Process Trenches with a long history of detectable uranium concentrations. The total analytical error 

reported is shown in Figure 4-24 and can be seen to be 

an approximately constant proportion of the reported 

uranium concentration. Assuming normally distributed 

measurement errors and a 95% probability that the true 

uranium concentration is within the range defined by 

114500

115000

115500

116000

116500

117000

117500

118000

Northing [m, WCS]

593000 593500 594000 594500 595000

Easting [m, WCS]

1-1

1-6

1-7

1-10A/B

1-12

1-14A

1-15

1-17A/B1-16A/B

2-1

2-2

3-6

3-10 3-11

3-12

4-1 4-7

4-11

Columbia River 

Island

Island

 Model Boundary

 Columbia River Boundary

 300 Area Boundary

 Waste Discharge Site

 Wells Selected for Calibration

Figure 4-22. Distribution of observation wells from which calibration data were selected 
67

103.0

103.5

104.0

104.5

105.0

105.5

106.0

106.5

107.0

107.5

9/1/1997 9/1/1998 9/1/1999 8/31/2000 9/1/2001 9/1/2002 9/1/2003 8/31/2004

Hydraulic Head (

m

0.00

50.00

100.00

150.00

200.00

250.00

300.00

350.00

9/1/1997 9/1/1998 9/1/1999 8/31/2000 9/1/2001 9/1/2002 9/1/2003 8/31/2004

Uranium (ug/l)

the measured concentration plus or minus the reported 

analytical error, the standard deviation of the measurement errors was estimated as the reported analytical 

error divided by 1.96.The resulting concentration 

weights decrease as the observed concentrations increase. The calibration thus emphasized those observations where the concentrations were relatively small. 

The resulting weighted concentration residuals were 

not independent of the weighted simulated concentrations as shown in the upper plot of Figure 4-25. Observations with very low values of concentration (such as 

those from the deeper wells) also contributed an inordinate amount to the calibration objective function. 

To improve the calibration results, equal weights were 

applied to the concentration observations. The weight 

selected, 0.349, was the median of the weights computed as inversely proportional to the observed concentrations. This corresponds to a standard deviation of 

observation error of 2.87 μg/l, or a uniform measurement error of 5.62 μg/l. The weighted concentration 

residuals from a calibration with equal concentration 

observation weights were independent of the weighted 

simulated concentration as shown in the lower plot of 

Figure 4-25. Weighted concentration residuals increased with simulated concentrations, suggesting that 

the weights on the lowest concentration observations 

may be too small. 

4.8.3 Joint Model Calibration and 

Calibrated Model Parameters 

Joint calibration of the flow and transport models was 

necessary, since the Fisher information matrix (Eq. 12) 

of MLBMA must be obtained simultaneously by 

jointly calibrating the flow and transport processes. 

This results in joint calibration of maximum likelihood 

estimates for flow and transport parameters. In addition, Sun and Yeh (1990) have shown that using concentration observations in a joint calibration can improve the calibration of flow model parameters. 

Figure 4-23. Distribution of calibration data in time for (top) head and (bottom) concentration observations
68

0

50

100

150

200

250

0 100 200 300 400 500 600

Measured Uranium Concentration (μg/L)

Total Analytical Error Reported (g/L) 

Well 399-1-17A

-40

-30

-20

-10

0

10

20

0 10 20 30 40 50

Weighted Simulated Concentration

Weighted Conc Residuals

-40

-20

0

20

40

60

80

0 10 20 30 40 50 60 70

Weighted Simulated Concentration

Weighted Conc Residuals

Figure 4-24. Plot of total analytical error with observation of uranium concentration at well 399-1-17A 

Figure 4-25. Weighted concentration residuals versus weighted simulated concentrations for Model 4 using 

(top) concentration weights inversely proportional to observed concentrations and (bottom) 

equal concentration weights 
69

Following Medina and Carrera (1996), NLL (Eq. 11) of 

MLBMA was defined as 

*1 * ln(2 ) ln ( ) ( ) T NLL N π − = + +− − C dd C dd (37) 

where N are the number of head and concentration observations plus the number of prior parameter estimates, * d is a vector of observations and prior parameter estimates (length N), d is a vector of computed 

head, concentration, and parameter values (length N), 

and C is the combined covariance matrix of the observations and prior parameter estimates. The covariance 

matrix was taken to be diagonal with elements equal to 

the inverse of the weights determined in Section 4.8.2 

for the head and concentration observations and Section 0 for the prior parameter estimates. As the first two 

terms on the right hand side of (Eq. 38) were constant, 

the PEST implementation used the objective function 

*1 * ( )( ) T S − =− − dd C dd (38) 

which was minimized iteratively using the GaussLevenberg-Marquardt method (Doherty 2004). 

Preliminary parameter sensitivity analysis examined 

the impact on the flow and transport model results of 

horizontal hydraulic conductivity (e.g., Kh_1 for unit 

1), the ratio of horizontal to vertical hydraulic conductivity (VANI), the specific yield of unit 1 (Sy_1), porosity, longitudinal dispersivity (alphaL), the ratio of 

vertical to longitudinal dispersivity (R-alphaV), and the 

linear adsorption distribution coefficient (Kd). The 

method of Morris (1991) was used to estimate the distributions of elementary effects for these parameters. 

The mean of each distribution represents the first-order 

effect of each parameter. The standard deviation of 

each distribution represents the effect of parameter 

interactions (without information about which parameters are interacting). Results for Model 4 are presented 

in Figure 4-26. The distribution of values represented 

by the box plots in this figure results from computing 

the sensitivities for each of the model outputs used in 

the calibration (all head and concentration observations). The most important parameters to include in the 

calibration, as suggested by the sensitivity analysis, 

were the horizontal hydraulic conductivity of units 1 

and 5, the linear adsorption distribution coefficient, 

longitudinal dispersivity, and the ratio of vertical to 

longitudinal dispersivity. Model results were relatively 

insensitive to the remainder of the considered parameters. 

Preliminary calibrations confirmed the sensitivity results. Calibration results were more stable when the 

insensitive parameters were not included in the estimation process. The calibration results were also improved by fitting the log of the hydraulic conductivity 

and distribution coefficient parameters. In addition, 

because of a high correlation between alphaL and RalphaV, the final calibrations did not fit R-alphaV. The 

values of all non-fitted parameters were fixed during 

calibration at estimates based on literature values and 

professional judgment. 

Table 4-8 lists initial values of the calibrated parameters and the ranges used to constrain parameter variation within PEST for the four alternative models. Parameter ranges were set large enough to ensure that the 

optimal parameter values fell between the limits. In this 

table, the homogeneous horizontal hydraulic conductivity of models 2 and 6 was denoted by Kh. The heterogeneous hydraulic conductivity of models 4 and 8 

was denoted by Kh_1 and Kh_5 for units 1 and 5. The 

homogeneous distribution coefficient of models 2 and 

4 was denoted as Kd while the heterogeneous Kd for 

zones 1 and 2 of models 6 and 8 were denoted as Kd1 

and Kd2, respectively. The initial parameter values 

were based on preliminary calibration results for Kh_1, 

Table 4-1 for Kh_5, mean values from the surface 

complexation model for the Kd parameters, and judgment for alphaL. Parameters that were fixed in the 

calibrations are listed in Table 4-9. In most cases, values of the fixed parameters were based on previous 

model applications at the Hanford Site or the literature. 

4.9 Model Calibration Results

Each of the four alternative models was calibrated over 

the period of September 1997 to December 2004 using 

the same set of head and uranium concentration observations. As stated previously, the flow and transport 

models were calibrated jointly. Final calibrated parameter values for each model are listed in Table 4-10 

and presented graphically in Figure 4-27. Of the parameters estimated, sensitivities to Kh_5 and alpha 

were the lowest. This is reflected in the large differences between models for the calibrated values of these 

parameters and in the relatively large confidence limits 

for Kh_5. 

Calibrated results for the hydraulic conductivity parameters differ from the current average estimates for 

the 300 Area given in Table 4-1, but are consistent with 

the ranges of values based on the field tests and previous inverse modeling for the Hanford Site (Table 4-1). 

Calibrated hydraulic conductivity for the homogeneous 

models was significantly less than the unit 1 hydraulic 

conductivities from the heterogeneous models. 
70

200

150

100

50

0

Mean Elementary Effect (

μ)

Kh_1Kh_5Kh_6Kh_7

Sy_1

VANI

alphaL

R-alphaV

Kd

porosity

200

150

100

50

0

Standard Deviation of Elementary Effect (

σ)

Kh_1Kh_5Kh_6Kh_7

Sy_1

VANI

alphaL

R-alphaV

Kd

porosity

Calibrated values of the dispersivity were small for the 

homogeneous models – much smaller than the grid cell 

size in the primary flow direction (35 m) and very 

close to the initial value. For the heterogeneous models, the calibrated values of the dispersivity were much 

larger, on the order of the grid cell size. 

Calibrated values for the Kd parameters were greater 

than the mean values computed using the surface complexation model results (Sections 4.5.2.5 and 0) but 

were within the estimated limits derived from the geochemical modeling. Calibration of Model 6 resulted in 

nearly equal values for Kd1 and Kd2 while the Model 

8 calibration resulted in a value of Kd1 greater than 

Kd2. Each of these results is contrary to the postulated 

geochemical conceptualization and indicates that a 

more sophisticated geochemical model than used here 

may be necessary. 

Figure 4-26. Estimated means (top) and standard deviations (bottom) of the elementary effects of Model 4 

computed using the method of Morris (1991). Boxplots represent the distribution over all head 

and concentration measurements used in the calibration. Median, 10, 25, 75, and 90 percentiles 

shown; outliers are outside 5 and 95 percentiles.
71

Kh 

(m/d) 

Kh_1 

(m/d) 

Kh_5 

(m/d) 

alphaL

(m) 

Kd

(L/kg) 

Kd1

(L/kg) 

Kd2

(L/kg) 

Model 2 15000 

[1,2x105

] 3.0 [0.1,100] 

8.0 

[0.1,500] 

Model 4 15000 

[1,2x105

] 

150 

[0.1,5000] 

3.0 

[0.1,100] 

8.0 

[0.1,500] 

Model 6 15000 

[1,2x105

] 3.0 [0.1,100] 

3.0 

[0.1,500] 

20.0 

[0.1,500] 

Model 8 15000 

[1,2x105

] 

150 

[0.1,5000] 

3.0 

[0.1,100] 

3.0 

[0.1,500] 

20.0 

[0.1,500] 

Parameter Value 

Kh_6 0.01 m/d 

Kh_7 43 m/d 

Kh_8 5.0x10-5 m/d 

Sy 0.2 

VANI 10 

Ratio of transverse to 

longitudinal dispersivity 

1.0 

R-alphaV 0.1 

Bulk Density 2100 kg/m3

Molecular Diffusion 

Coefficient 

8.64x10-6 m2

/d 

The initial and final calibration objective function components are listed in Table 4-11 for each alternative 

model. The total sum of squared weighted residuals 

(SSWR) is the sum of the SSWR due to the head and 

concentration observations as well as the prior estimates of the Kd parameters (Eq. 39). The total objective 

function favors Model 4 over Model 8 by a small margin and favors the models with heterogeneous hydraulic conductivity (Models 4 and 8) by a wide margin 

over the homogeneous models (Models 2 and 6). 

The better fit of Models 4 and 8 is due primarily to the 

improved fit to the head observations and primarily to 

the improved head fit at wells 3-6, 4-1, and 4-11 (see 

Figure 4-22 for the locations of these wells). The 

SSWR for the head observations actually increased 

during the calibration for Models 2 and 6. This was 

apparently necessary to decrease the SSWR for the 

concentration observations from their large initial values. The initial concentration SSWR for Models 2 and 

6 was significantly larger than for Models 4 and 8. 

Kh 

(m/d) 

Kh_1 

(m/d) 

Kh_5 

(m/d) 

alphaL

(m) Kd (L/kg) Kd1

(L/kg) 

Kd2

(L/kg) 

Model 2 8097 

[3385,19371] 4.9 [3.2,6.6] 

26.3 

[10.8,64.2] 

Model 4 23,508 

[9733,56775] 

11 

[0.4,263] 

49.6 

[22.8,76.3] 

18.0 

[7.6,42.2] 

Model 6 6237 

[3261,11926] 4.6 [3.3,5.9] 

19.0 

[9.9,36.2] 

20.5 

[10.5,40.1] 

Model 8 26,093 

[13079,52055] 

780 

[179,3410] 

25.0 

[6.9,43.1] 

30.2 

[14.8,61.4] 

14.3 

[7.0,29.1] 

Table 4-8. Initial values and upper and lower limits (in brackets) for parameters estimated in the calibration. 

Empty spaces indicate that the parameter was not present in a model. Log-transformed parameters were calibrated for all parameters except alphaL. 

Table 4-9. Values of parameters that were fixed 

in the calibrations 

Table 4-10. Calibrated parameter values and 95% upper and lower linear confidence limits (in brackets)
72

0

10000

20000

30000

40000

50000

60000

Model 2 Model 4

Kh_1

Model 4

Kh_5

Model 6 Model 8

Kh_1

Model 8

Kh_5

K (m/d)

0.0

10.0

20.0

30.0

40.0

50.0

60.0

70.0

Model 2 Model 4 Model 6

Kd1

Model 6

Kd2

Model 8

Kd1

Model 8

Kd2

Kd (L/kg)

0

10

20

30

40

50

60

70

80

Model 2 Model 4 Model 6 Model 8

alpha (m)

There were relatively small differences between models in the final fit to the concentration observations. 

Model 8 with its two Kd zones fits both the head and 

concentration observations slightly better than Model 

4, but the contribution to the objective function from 

the Kd prior component is significantly larger for 

Model 8. 

Plots of simulated versus observed heads and uranium 

concentrations are shown in Figure 4-28 for Model 4. 

Results for the other models appear similar and are not 

shown here. Observed heads are underestimated for the 

highest heads and overestimated for the lowest heads, 

primarily as a result of the fixed head on the western 

boundary of the model. The pattern is similar for the 

uranium concentrations with the largest concentrations 

being somewhat severely underestimated. These results 

suggest that there is room for improvement in the models. 

A comparison between observed and simulated values 

at selected wells is shown in Figure 4-29 for two wells 

located between the major uranium sources. Well 399-

2-2 is close to the river while well 399-1-17A is farther 

inland. (See Figure 4-22 for well locations). Calibration 

data indicate observations that were included in the 

calibration. Measured data are observations that were 

not included in the calibration. There are only minor 

differences between heads simulated by the four model 

alternatives at these wells. For concentrations, Models 

2 and 6 produce similar results, but there are differences in the results of the other two models. In general, 

the calibrated models appear to fit the observed heads 

better than the observed concentrations. Figure 4-30 

presents calibration results for two wells located south 

of the major uranium sources. Well 399-3-10 is located 

near the river while well 399-4-11 is much farther 

inland. Models 2 and 6 produce very similar results at 

these wells for both heads and concentrations. Unlike 

the other wells, at well 399-4-11 there are noticeable 

differences between the heads simulated by the models. 

Simulated uranium concentrations at the end of December 2004 are compared in Figure 4-31 to the observed values from the fourth quarter of 2004. Results 

are shown for Model 4. Simulated values are presented 

as banded color contours where the color interfaces 

represent simulated concentration values of 10, 30, 50, 

70, 90, and 110 μg/l. The observed values are represented by labeled contours at 10, 30, 50, and 70 μg/l. In 

general, simulated concentrations appear to be somewhat lower than the observed concentrations except 

near the river where there are two regions of elevated 

concentration in the simulated results. Overall, the 

simulated results reflect the general character of the 

observed concentrations at the end of the calibration 

period. 

Model: 2 4 6 8 

Initial 32405 29949 32405 29949 

SSWR_head Final 32824 28595 33104 28170 

Initial 93077 44495 96113 48351 

SSWR_conc Final 30219 30404 30474 29869 

Initial 280 280 133 133 

SSWR_prior Final 1619 1070 1961 2747 

Initial 125761 74724 128651 78433 

SSWR Final 64662 60069 65539 60786 

Figure 4-27. Comparison of final calibrated parameter values: (top) hydraulic conductivity parameters, (middle) adsorption parameters, and (bottom) dispersivity 

Table 4-11. Calibration objective function components for each alternative model 
73

103

103.5

104

104.5

105

105.5

106

106.5

107

107.5

108

103 104 105 106 107 108

Observed Head (m)

Simulated Head (m)

0

50

100

150

200

250

300

350

0 50 100 150 200 250 300 350

Observed U Conc (ug/l)

Simulated U Conc (ug/l)

4.10 Posterior Model Probabilities

The results of the model calibrations were used to 

compute posterior model probabilities as described in 

Section 3.1.1. Equation (8) provides the expression for 

posterior model probabilities and contains two needed 

quantities: the prior model probabilities, which were 

assumed to be an equal 25% for the four models of this 

application, and the ΔKIC value for each model. The 

expression for KIC (Eq. 10) contains three terms. The 

negative log likelihood term was computed here as 

2 NLL N = ln( ) σ (39) 

Where σ2

 is the maximum likelihood estimate of the 

error variance 

2 SSWR

N

σ = (40) 

NLL represents the fit to the observations used in the 

calibration. The smaller this value, the better the fit. As 

can be seen in Table 4-12, Model 4 has the smallest 

value of NLL followed by Model 8, Model 2, and 

Model 6. 

Figure 4-28. Simulated versus observed (top) heads and (bottom) uranium concentrations for Model 4 
74

103

104

105

106

107

108

9/1/97 9/1/98 9/1/99 8/31/00 9/1/01 9/1/02 9/1/03 8/31/04

Date

Water Level (m NAVD88)

Measured Calibration Data Model 2

Model 4 Model 6 Model 8

0

50

100

150

200

250

300

350

9/1/97 9/1/98 9/1/99 8/31/00 9/1/01 9/1/02 9/1/03 8/31/04

Date

Uranium (μg/L)

399-2-2

104

104.5

105

105.5

106

106.5

107

107.5

108

9/1/97 9/1/98 9/1/99 8/31/00 9/1/01 9/1/02 9/1/03 8/31/04

Date

Water Level (m NAVD88)

Measured Calibration Data Model 2

Model 4 Model 6 Model 8

0

50

100

150

200

250

300

9/1/97 9/1/98 9/1/99 8/31/00 9/1/01 9/1/02 9/1/03 8/31/04

Date

Uranium (μg/L)

399-1-17A

Figure 4-29. Calibration results for (top two plots) well 399-2-2 and (bottom two plots) well 399-1-17A
75

103

104

105

106

107

108

9/1/97 9/1/98 9/1/99 8/31/00 9/1/01 9/1/02 9/1/03 8/31/04

Date

Water Level (m NAVD88)

Measured Calibration Data Model 2

Model 4 Model 6 Model 8

0

50

100

150

200

9/1/97 9/1/98 9/1/99 8/31/00 9/1/01 9/1/02 9/1/03 8/31/04

Date

Uranium (μg/L)

399-3-10

104

104.5

105

105.5

106

106.5

107

107.5

108

9/1/97 9/1/98 9/1/99 8/31/00 9/1/01 9/1/02 9/1/03 8/31/04

Date

Water Level (m NAVD88)

Measured Calibration Data Model 2

Model 4 Model 6 Model 8

0

10

20

30

40

50

9/1/97 9/1/98 9/1/99 8/31/00 9/1/01 9/1/02 9/1/03 8/31/04

Date

Uranium (μg/L)

399-4-11

Figure 4-30. Calibration results for (top two plots) well 399-3-10 and (bottom two plots) well 399-4-11
76

 70 

 50 

 50 

 30 

 10 

2004 Q4

 30 

The second term in KIC, Nk ln(N/2π), is a function of 

the number of calibrated parameters, Nk, which varies 

between models, and the number of observations, 

which is constant for all models. This term is sometimes interpreted as a representation of model complexity. In this sense, KIC favors models that are less complex. Values of this term are given in Table 4-12 and 

can be seen to favor Model 2 over all other models and 

Models 4 and 6 over Model 8. 

The third term in the computation of KIC is the normalized observed Fisher information matrix (Eq. 12). 

This term can be expressed as 

1 ln ln ln( ) ln ln Nk N NN k N

− FI I I = = =− + (41) 

where I is the observed Fisher information matrix, 

computed here as 

1 ln ln ln − I = Σ =− Σ (42) 

and Σ is the maximum likelihood estimate of the parameter covariance matrix. If the parameter covariance 

matrix is reported by the parameter estimation software, the determinant can be computed directly with 

care taken to observe two requirements. 

Figure 4-31. Model 4 simulated uranium concentrations in December 2004 (color) overlain by contours based 

on observed uranium concentration (μg/l) in the fourth quarter of 2004. Color levels indicate the 

10, 30, 50, 70, 90, and 110 μg/l locations. Dark blue indicates the Columbia River. 
77

Model: 2 4 6 8 

SSWR 64662 60069 65539 60786 

Nk 3 4 4 5 

σ2

=SSWR/N 150.38 139.70 152.42 141.36 

NLL=Nln(σ2

) 2155.65 2123.97 2161.44 2129.07 

Nk ln(N/2π) 12.68 16.90 16.90 21.13 

ln | | F -20.20 -38.69 -13.48 -39.71 

KIC 2148.13 2102.18 2164.87 2110.49 

Rank 3 1 4 2 

ΔKIC 45.95 0.00 62.69 8.32 

p(Mk|D) 0.00% 98.46% 0.00% 1.54% 

For the first requirement, if the parameter covariance is 

reported for transformed parameters, as is done in 

PEST, the covariance must be recomputed for the untransformed parameters. For PEST this can be accomplished by setting the parameter NOPTMAX=-1 in the 

PEST control file and rerunning PEST without parameter transformations and with initial parameter values set 

to the optimal values. 

For the second requirement, if the parameter estimation 

software uses least squares (as does PEST) instead of 

maximum likelihood, the parameter covariance reported is the least squares estimate and must be converted to the maximum likelihood estimate as follows. 

LS

N Nk

N

− Σ= Σ (43) 

where ΣLS is the least squares estimate of the parameter 

covariance matrix. 

PEST reports the eigenvalues of the (least squares) 

parameter covariance matrix, which were used to 

evaluate Eq. (42), 

LS

1

ln ln ln ln

ln ln

k

k

N

k k

LS

N

k

k i

i

NN NN

N N

N N N

N =

− − ⎛ ⎞ Σ= Σ = + Σ ⎜ ⎟ ⎝ ⎠

⎛ ⎞ − = + ⎜ ⎟ ⎝ ⎠ ∑ λ

 (44) 

where λi are the eigenvalues. The third term of KIC

was thus computed using the PEST output as 

1

ln ln ln ln

Nk

k

kk i

i

N N N NN

N =

⎛ ⎞ − =− − − ⎜ ⎟ ⎝ ⎠ F ∑ λ (45) 

Eigenvalues were evaluated by running PEST with the 

control parameter NOPTMAX=-1 and with all parameters untransformed and set at their optimal values. 

As discussed in Section 3.1.1, the Fisher information 

term causes KIC to favor models with relatively small 

information content per unit sample or, equivalently, a 

correspondingly large parameter estimation variance. 

Values of ln | | F for the four alternative models are 

given in Table 4-12. It can be seen that this term favors 

Model 8 over Model 4 (slightly) with Model 2 and 

Model 6 having much larger values. 

Final KIC values are given in Table 4-12 with the result that Model 4 is preferred over Models 8, 2, and 6 

in that order. The ΔKIC value for Model 8 is 8.32 with 

Models 2 and 6 having much higher values. With equal 

prior probabilities, the posterior probabilities were 

computed using Eq. (8) and are also given in Table 

4-12. The result is that Model 4 is preferred to the near 

exclusion of the other models. Referring to Figure 3-1, 

it can be seen that the sensitivity of model probability 

to ΔKIC means that the small differences in the three 

terms of KIC can produce a large difference in posterior model probabilities. 

4.11 Prediction

The fact that Model 4 dominates the other alternatives 

in terms of posterior model probabilities means that 

only Model 4 would need to be applied in the predictive period. To better illustrate the full application of 

the uncertainty methodology, however, we used all 

four models for prediction under the baseline scenario. 

For the alternative scenario, only Model 4 was applied. 

There is an independent reason to apply even lowprobability models in the predictive period. If the conditions of the predictive period are significantly different than the calibration period, differences between 

models that were not apparent in the calibration may 

appear during prediction. Applying even lowprobability models in the predictive period may bring 

these differences to light. At that point, the conditions 

of the predictive and calibration periods and the data 

used in the calibration can all be reviewed to determine 

whether the model probabilities should be re-evaluated. 

4.11.1 Predictive Simulation Description 

The predictive period for the 300 Area application was 

from January 2005 to January 2025. The simulation 

domain and model discretization were unchanged for 

the predictive simulations. Groundwater heads and 

uranium concentrations from the end of the calibration 

period were used as the initial conditions for the preTable 4-12. Calibration results and computation of 

model probabilities using KIC
78

dictive modeling. Boundary conditions used in the predictive simulations were unchanged except for the head 

boundary representing the Columbia River stage. For 

the baseline scenario, the river stage simulated by 

Waichler et al. (2005) for the period 1975 to 1994 was 

used in the predictive period (see Figure 4-19). For the 

alternative scenario, the river stage simulated by 

Waichler et al. (2005) for the period 1955 to 1974 was 

used. In both cases, monthly average river stage was 

used with one-month stress periods in the flow models. 

4.11.1.1 Modeled Parameter Uncertainty 

The calibration results were used to simulate parameter 

uncertainty in the predictive period by assuming that 

the calibrated parameters (or the log of the parameters 

in the case of the hydraulic conductivity and distribution coefficient parameters) followed a multivariate 

Gaussian distribution with mean values given by the 

optimal parameter values and a covariance given by the 

maximum likelihood estimated parameter covariance 

(Eq. 43). 

The Latin hypercube sampling code of Iman and 

Shortencarier (1984) was used to generate 200 realizations of the calibrated parameters for each of the four 

model alternatives. The resulting parameter distributions and correlations accurately reflected the input 

distributions determined from the calibration results. 

Sample distributions are shown in Figure 4-32 for 

Kh_1 and alphaL of Model 4. 

As mentioned in Section 2.3.1, characteristics of the 

scenarios could have been modeled as random variables. Specifically, the future river stage is clearly uncertain – modeling the river stage as a temporally correlated stochastic process would have been justified. 

The simulated results of Waichler et al. (2005) could 

be used to derive appropriate statistics of the stochastic 

process for the two scenarios. Each Monte Carlo realization would then include not only the randomly sampled parameters described above, but also a randomly 

sampled river stage time series. Scenario uncertainty 

would thus include not only the effect of the two scenarios (baseline and the alternative), but also the random variation in the river stage time series. 

4.11.2 Baseline Scenario 

The parameter realizations discussed in the previous 

section were used in a Monte Carlo simulation to compute flow and uranium transport over the predictive 

period. The resulting head and concentration distributions included the impact of parameter uncertainty. A 

Monte Carlo simulation was carried out for each of the 

four model alternatives to incorporate the impact of 

model uncertainty. 

There are various types of prediction results that may 

be of interest. For example, the mean and variance of 

concentration can be viewed as a function of time to 

examine the evolution of the plume over the predictive 

period and where the greatest concentration uncertainty 

exists (that portion due to parameter uncertainty). 

Figure 4-33 shows the concentration mean and variance in January 2010. This is a view looking northwest 

with the river grid cells removed. The variance is generally largest along the river and in the areas of high 

mean concentration. 

It may also be of interest to examine the predicted output at a specific location. Figure 4-34 shows the uranium concentration at well 399-1-1 near the river as 

predicted by Model 4. Concentrations computed for all 

200 realizations are shown as a function of time along 

with the average concentration. The concentration variance grows over time at this location. In addition, the 

concentration at specific times appears to be somewhat 

skewed. 

For regulatory or management applications, the predicted value of interest is likely to be a single quantity 

that can be derived from the spatial and temporal distributions of concentration. Examples include the 

maximum concentration along a compliance boundary 

during the predictive period, the length of time for the 

maximum concentration to fall below a limit, and the 

peak dose over the predictive period as computed from 

predicted concentrations and a specified exposure scenario. A probability distribution for each of these quantities could be estimated from the Monte Carlo simulation results. These distributions, one for each alternative model, are the (| ,) k p M Δ D of Eq. (1). To illustrate such results for the 300 Area application, the distributions of predicted uranium concentration at well 

399-1-1 on 1/1/2025 are shown in Figure 4-35 for each 

of the four model alternatives and the model average 

using the posterior model probabilities from Table 

4-12. Empirical probability density functions based on 

the 200 realizations are shown in the upper plot while 

the lower plot shows the empirical cumulative distribution functions. As determined by the posterior model 

probabilities, the model-average distribution closely 

follows the distribution from Model 4. 

The distributions of concentration for Models 2 and 6 

were nearly identical at this time and location - the 

heterogeneity of Kd (Model 6) had little impact on the 

predicted concentration when the hydraulic conductivity was homogenous throughout the domain. This was 

not unexpected as the calibrated values of Kd1 and 
79

12

10

8

6

4

2

0

Number

6x104 1 2 3 4 5

Kh_1 (m/d)

N = 200

Min = 8294.6

Max = 66623

Mean = 25890

StDev = 11770

Median = 23483

10

8

6

4

2

0

Number

20 30 40 50 60 70 80

alphaL (m)

N = 200

Min = 17.945

Max = 81.172

Mean = 49.556

StDev = 13.37

Median = 49.578

1.0

0.8

0.6

0.4

0.2

0

Empirical cdf

6x104 1 2 3 4 5

Kh_1 (m/d)

1.0

0.8

0.6

0.4

0.2

0

Empirical cdf

20 30 40 50 60 70 80

alphaL (m)

60x103

50

40

30

20

10

Kh_1 (m/d)

20 30 40 50 60 70 80

alphaL (m)

Kd2 for Model 6 were nearly the same (Table 4-10) 

with a correlation coefficient of 0.86. In addition, the 

calibrated value of the homogeneous Kd for Model 2 

was similar to the calibrated values of Kd1 and Kd2 for 

Model 6. Models 4 and 8 with heterogeneous hydraulic 

conductivity had much larger mean uranium concentrations at well 399-1-1 on 1/1/2025 than the homogenous 

models. The variance of predicted concentration was 

also much larger for the heterogeneous models. The 

heterogeneity of Kd in Model 8 had a noticeable effect 

on the predicted uranium concentration distribution, 

producing a smaller mean and somewhat smaller variance than Model 4 with homogeneous Kd. 

Figure 4-32. Sample histograms, cumulative distribution functions (mean +/- one standard deviation shown), 

and scatterplot for Model 4 parameters Kh_1 and alphaL resulting from 200 Latin hypercube 

realizations (correlation = 0.27) 
80

ConcVar

10

9

8

7

6

5

4

3

2

1

0

ConcAvg

105

95

85

75

65

55

45

35

25

15

5

80

60

40

20

0

Uranium Concentration (ug/l) at Well 399-1-1

1/1/2005 1/1/2010 1/1/2015 1/1/2020 1/1/2025

Date

Model 4, Baseline Scenario

(200 Realizations)

 Monte Carlo Realization Result

 Average

Figure 4-33. Monte Carlo simulation results for Model 4 in January 2010: (left) concentration mean (μg/l) 

and (right) concentration variance 

Figure 4-34. Monte Carlo simulation results for Model 4 under the baseline scenario: uranium concentration 

(μg/l) at well 399-1-1 for 200 realizations and average concentration 
81

The model-averaged concentration mean and variance 

can be computed from the individual model results 

using Eqs. (18) and (19) or (20). Because of the dominant probability for Model 4 in this application, the 

model-averaged mean and variance were essentially 

the same as the mean and variance of Model 4. The 

model-averaged distribution shown in Figure 4-35 can 

be used to estimate risk-related quantities. For example, the probability that the uranium concentration at 

well 399-1-1 exceeds 30 μg/l on 1/1/2025 is estimated 

to be about 1%. 

4.11.3 Alternative Scenario 

As discussed previously, the alternative scenario considered in this application assumed variability in Columbia River stage at the 300 Area reflective of conditions prior to 1975. This was modeled by simulating 

uranium transport from January 2005 through December 2024 using a head boundary condition along the 

river equal to the estimated river stage from 1955 to 

1974. Compared to the baseline scenario, this boundary 

condition was more variable with significantly higher 

late spring/early summer peaks (Figure 4-19). For the 

alternative scenario, the random realizations of parameter values used in the Monte Carlo simulation were 

0.4

0.3

0.2

0.1

0.0

Empirical pdf

5 10 15 20 25 30

Predicted Uranium Concentration (ug/l) at Well 399-1-1 on 1/1/2025

 Model 2

 Model 4

 Model 6

 Model 8

 Model Average

1.0

0.8

0.6

0.4

0.2

0

Empirical cdf

5 10 15 20 25 30

Predicted Uranium Concentration (ug/l) at Well 399-1-1 on 1/1/2025

Mean +/- 1 standard deviation shown

 Model 2

 Model 4

 Model 6

 Model 8

 Model Average

Figure 4-35. Probability distributions for the predicted uranium concentration at well 399-1-1 on 1/1/2025: 

(top) empirical probability density functions for the four model alternatives and the model average and (bottom) empirical cumulative distribution functions 
82

identical to the values used in the baseline scenario. 

200 realizations of groundwater flow and uranium 

transport were simulated. 

Uranium concentration at well 399-1-1 over the entire 

predictive period is shown for all 200 realizations in 

Figure 4-36. These results can be compared to the 

baseline scenario results shown in Figure 4-34. For the 

alternative scenario, the mean concentration is much 

more variable and falls from its peak value more 

quickly, particularly in the first few years of the simulation. The concentration variance appears to grow 

more rapidly in the early part of the simulation and 

appears be greater throughout the predictive period 

than under the baseline scenario. 

Model-averaged results could be computed for the alternative scenario in a manner identical to that discussed above for the baseline scenario and shown in 

Figure 4-35 for the predicted concentration at well 399-

1-1 on 1/1/2025. Corollary results for the alternative 

scenario are not shown here because the modelaveraged results are nearly identical to the results from 

Model 4 (discussed in the following section) due to the 

dominant probability of Model 4. 

4.11.3.1 Scenario-Averaged Prediction 

Probability distributions for predicted quantities that 

include parameter, model, and scenario uncertainty can 

be computed using Eq. (21). The model-averaged 

probability distributions for each scenario were averaged over the alternative scenarios using the scenario 

probabilities as weights. For this application, the baseline scenario had a probability of 0.7 and the alternative scenario had a probability of 0.3. Model-averaged 

probability density functions for the uranium concentration at well 399-1-1 on 1/1/2025 are shown in the 

top plot of Figure 4-37 for the two scenarios. The scenario-averaged density function was computed as the 

weighted average of these two densities. The corresponding cumulative distribution functions are shown 

in the lower plot of Figure 4-37. The mean concentration is smaller for the alternative scenario while the 

concentration variance is larger. The distribution for 

the alternative scenario also appears to be more highly 

skewed. 

Risk-related quantities can be estimated directly from 

the scenario-averaged distribution. For example, the 

probability that the uranium concentration at well 399-

1-1 exceeds 25 μg/l on 1/1/2025 is estimated to be 

about 30 percent. This estimate includes the impact of 

parameter uncertainty, conceptual model uncertainty, 

and scenario uncertainty. At the same time, the contribution to the total uncertainty from each of these components is made clear in the figures (Figure 4-35 and 

Figure 4-37). These contributions can also be computed using the equations presented in Section 3. The 

mean and variance (Eqs. 23 and 24) of the predicted 

80

60

40

20

0

Uranium Concentration (ug/l) at Well 399-1-1

1/1/2005 1/1/2010 1/1/2015 1/1/2020 1/1/2025

Date

Model 4, Alternative Scenario

(200 Realizations)

 Monte Carlo Realization Result

 Average

Figure 4-36. Monte Carlo simulation results for Model 4 under the alternative scenario: uranium concentration (μg/l) at well 399-1-1 for 200 realizations and average concentration 
83

concentration at well 399-1-1 on 1/1/2025 are given in 

Table 4-13. The variance is broken down into the 

within-scenario and between-scenario components as 

computed from Eq. (25). 

The impact of changes in the model or scenario probabilities can be readily computed and viewed graphically. In addition, differences in the distributions of 

predicted values due to the model or scenario alternatives are readily apparent. Thus the impact on riskrelated quantities can be easily seen. For example, the 

probability that the uranium concentration at well 399-

1-1 exceeds 25 μg/l on 1/1/2025 is estimated to be 

about 45 percent under the baseline scenario and just 5 

percent under the alternative scenario. Such information may be valuable in justifying the use of specific 

scenarios and supporting decisions based in part on the 

simulation results. 

 Baseline 

Scenario 

Alternative 

Scenario 

Scenario 

Average 

Probability 0.7 0.3 

Mean 24.40 18.30 22.57 

Variance 8.70 24.77 21.34 

Std Dev 2.95 4.98 4.62 

Within-Scenario Variance 13.52 

Between-Scenario Variance 7.82 

Table 4-13. Mean, variance, and standard deviation 

of individual scenarios and the scenario 

average for the predicted concentration 

at well 399-1-1 on 1/1/2025 
84

0.20

0.15

0.10

0.05

0

Empirical pdf

10 15 20 25 30

Predicted Uranium Concentration (ug/l) at Well 399-1-1 on 1/1/2025

 Baseline Scenario - Model Average

 Alternative Scenario - Model Average

 Scenario Average

1.0

0.8

0.6

0.4

0.2

0

Empirical cdf

10 15 20 25 30

Predicted Uranium Concentration (ug/l) at Well 399-1-1 on 1/1/2025

 Baseline Scenario - Model Average

 Alternative Scenario - Model Average

 Scenario Average

Mean +/- 1 standard deviation shown

Figure 4-37. Probability distributions for the predicted uranium concentration at well 399-1-1 on 1/1/2025: 

(top) empirical probability density functions for Model 4 under the two scenarios and the scenario average and (bottom) empirical cumulative distribution functions 
85

5 CONCLUSIONS

This report has described a methodology to systematically and quantitatively assess predictive uncertainty in 

groundwater flow and transport modeling with consideration of the combined impact of hydrogeologic uncertainties associated with the conceptual-mathematical 

basis of a model, model parameters, and the scenarios 

to which the model is applied. The application of this 

methodology to realistic, field-scale groundwater flow 

and transport modeling has been demonstrated using 

uranium contamination at the 300 Area of the U.S. 

Department of Energy Hanford Site as an example. 

A review of published results from hydrogeologic 

model post-audits and other model applications demonstrated that the primary errors in groundwater modeling predictions can often not be assigned to errors in 

parameter values. In 9 of the 16 applications reviewed, 

conceptual model errors were most significant. Model 

scenario errors were the most significant in 4 of the 16 

applications. Parameter errors were most significant in 

three of the applications. This (limited) review suggests 

that a comprehensive approach to uncertainty assessment in hydrogeologic modeling should not be limited 

to parameter uncertainties, but must also consider the 

potential for significant conceptual model and scenario 

uncertainties if a realistic estimate of predictive uncertainty is desired. 

In a quantitative uncertainty analysis, parameter uncertainty is typically characterized using continuous probability distributions. When characterizing conceptual 

uncertainty in hydrogeologic modeling, specifying a 

continuum of conceptual model possibilities is likely to 

be infeasible. Instead, it is generally more appropriate 

to postulate a discrete set of alternative conceptual 

models. Given a set of alternatives, one might suggest 

that model uncertainty be resolved by simply comparing predictions from the alternative models. This approach has a serious drawback, however. Without a 

quantitative measure of the degree of plausibility of 

model alternatives, it is impossible to determine the 

risk of a decision based on the model predictions. Similarly, justifying the adoption of the most conservative 

model for use in prediction will be easier if the plausibility of the conservative model can be compared to the 

alternatives. 

Maximum Likelihood Bayesian Model Averaging was 

used here to estimate model probabilities, which are a 

measure of the relative degree of plausibility of a set of 

alternative models. MLBMA is an optimal and computationally feasible way of combining the predictions of 

several competing models and assessing their joint 

predictive uncertainty. It is a general method, applying 

to deterministic and stochastic models, and to complex 

and simplified models. 

The model-averaging approach has been criticized on 

the basis that there is only one model corresponding to 

the physical reality and therefore an average over several models has no physical interpretation. Although 

the models considered here are physically-based, the 

inherent complexity of the hydrogeologic environment 

means that all models considered will be an approximation to physical reality. When appropriately formulated, each model alternative will have some merit in 

reproducing aspects of the physical system, this merit 

being quantified by each model’s probability. Since we 

interpret model probability as a relative measure with 

respect to the other model alternatives considered, a 

model-averaged consequence has an intuitive and consistent meaning. Note that in an application of 

MLBMA the predicted consequences of the physics as 

rendered by the models are averaged, not the underlying physics. 

Model averaging has also been criticized for masking 

information essential to the decision maker. It is demonstrated in this report, both in the abstract and in the 

concrete application to uranium transport at the 300 

Area, that the MLBMA results can be presented so as 

to fully inform a decision, clearly illuminating the effect of model uncertainty and the importance of a consistent and defensible means to estimate model probabilities. Model averaging reduces the risk of relying 

on an overly conservative model and provides a consistent and quantitative way to address model uncertainty 

in the context of a regulatory decision. 

In the framework adopted in this report, estimated parameter values and model probabilities are based on 

information and data available for the model calibration, referred to here as the history-matching period. 

That is, parameter values and model probabilities are 

estimated as part of the process of model development 

and evaluation. The characterizations of parameter and 

model uncertainty resulting from this process can be 

projected into the period over which model predictions 

are needed, referred to here as the predictive period. As 

defined here, a scenario is a description of the future 

conditions under which a model is applied, conditions 

that are inherently uncertain and apply only in the predictive period. A comprehensive assessment of predictive uncertainty thus requires that scenario uncertainty 

be explicitly considered. However, in the limited definition used here, a scenario can be reduced to a set of 
86

conditions described by the three elements of any 

simulation model: geometry and parameterization, 

structure of governing equations, and forcing. Since we 

presently know, albeit imperfectly, how to deal with 

parameter uncertainty, uncertainty in the model structure, and uncertainty in the forcing terms, we know 

mathematically how to deal with scenario uncertainty. 

Most of the time a scenario will impact mainly forcing 

terms, which are often easier to deal with. 

MLBMA was extended to include scenario uncertainty 

in a joint estimation of predictive uncertainty. This 

requires postulating a set of alternative scenarios and 

the probability of their occurrence. As with model 

probabilities, the scenario probabilities are interpreted 

as relative measures of the degree of plausibility of the 

scenarios. Unlike model probabilities, at the time of 

prediction there is no opportunity to condition the scenario probabilities on observation. Scenario probabilities are thus prior probabilities and are appropriately 

interpreted as subjective measures based on current 

knowledge, experience, and judgment. The MLBMA 

methodology requires that alternative scenarios be mutually exclusive, a condition that can be enforced by 

defining the scenarios as possible combinations of alternative events that need not be mutually exclusive. 

Scenario probabilities can be determined from estimates of the marginal and conditional probabilities of 

these events. 

In deriving MLBMA with scenario averaging, the assumption was made that the dataset used for model 

calibration is independent of the scenario. That is, the 

occurrence of any particular scenario in the future does 

not affect the probability of observing the data in the 

past. As a result, the model calibrations are not a function of the scenario and do not need to be recomputed 

under each scenario. This significantly reduces the 

computational requirements of the methodology. To 

incorporate scenario uncertainty, predictive simulations 

are conducted with the alternative models under each 

alternative scenario. While this is computationally expensive, it is straightforward and would be carried out 

in any case unless scenario uncertainty is completely 

ignored. Although not considered in the application 

presented here, it is possible to include additional parameter uncertainty specific to a scenario (i.e., not present in the history-matching period) in the predictive 

simulations. 

A previous application of MLBMA illustrated the superiority of using the model-averaged result from 

MLBMA over any individual model. The application 

to the Hanford Site 300 Area uranium plume advances 

the uncertainty assessment methodology by including 

the impact of hydrologic scenario uncertainty and by 

applying the methods to a more complex problem and 

relevant system: transient groundwater flow and transport of a radionuclide from an uncontrolled disposal. 

The application should be viewed as an example, however, since significant uncertainties were ignored (primarily uncertainty in uranium sources) and models 

were simplified in critical ways (e.g., the geochemistry, 

saturated flow only). The application illustrates the 

very real tradeoffs that must be made when modeling 

complex systems between incorporating complexity in 

the model and the computational demands of model 

calibration and uncertainty assessment. Application of 

the methodology to a similarly-scaled, threedimensional, variably-saturated model incorporating 

reactive geochemistry is currently likely to be computationally infeasible. In spite of these limitations, the 

application illustrates all aspects of the methodology 

and can serve as a template for future applications. 

As described in the report, there exists at the 300 Area 

a significant amount of data regarding aquifer heads, 

uranium concentrations, and site geochemistry. Despite 

this, there are significant uncertainties that remain, 

including the hydraulic properties of the unsaturated 

zone, the extent and magnitude of the current uranium 

contamination in the unsaturated zone, and the historical flux of uranium to groundwater. As is the case at 

the 300 Area, the historical data is not always optimal 

for use in model calibration. Where models are to be 

used in site management, data collection needs to be 

targeted for use in model development and evaluation, 

including calibration and uncertainty assessment. 

Data continue to be collected at the 300 Area site and 

the regulatory models continue to be refined. Four 

large-diameter boreholes were completed recently with 

continuous split-spoon samples collected to better define the distribution of uranium. Physical and geochemical properties have been measured on more than 

100 of these samples. Slug tests and pump tests were 

performed on the completed wells to provide hydraulic 

parameter estimates. In addition, a large-scale tracer 

test was recently conducted using one of the wells as 

the injection well. These recent data were not used in 

the application presented here. Due to the iterative nature of the methodology, however, these data could be 

incorporated in an updated analysis in a very straightforward manner. 

For this application, prior parameter information was 

valuable in producing reasonable estimates of the uranium adsorption coefficients. Without the prior information, adsorption coefficients were unreasonably 

large. Since the prior information exists, including it in 

the calibration was justified. The poor calibration results obtained without the prior information suggest, 
87

however, that the geochemical conceptualization of the 

models needs to be improved. This is also suggested by 

the failure of the calibrations to produce the expected 

relationship between adsorption coefficients in the two 

zones used in Models 6 and 8, and by the inability to 

reproduce large observed concentrations. 

Selection of weights for the calibration data had a significant effect on the calibration results. Actual laboratory measurement error for the uranium concentration 

data is likely to be proportional to the concentration. 

Using weights based on estimated laboratory measurement errors produced inferior calibrations to those 

obtained using equal weights for the concentration observations. The use of equal weights implies that a 

concentration-independent sampling error dominates 

the uranium concentration measurement error. 

Model probabilities are very sensitive to the magnitudes of the KIC differences. Thus changes in model 

assumptions and inputs, such as fixed parameter values 

and uranium sources, could alter the model probabilities. Changes in calibration inputs, such as the selection 

of data, may have the same effect. The sensitivity of 

model probability to factors influencing KIC differences (as well as other model discrimination criteria 

such as AIC) warrants further investigation. 

Although the methodology is formulated using maximum likelihood parameter estimation, results from a 

least squares parameter estimation can be used as well. 

The 300 Area application used the least squares code 

PEST to complete the model calibrations. Computation 

of the model probabilities from the PEST results were 

completed in a spreadsheet. 

For the 300 Area application, one model dominated the 

other alternatives. This result implies that only the 

dominant model needed to be used in predictive simulations. For illustrative purposes, however, all four 

models were used. In some cases, it may be reasonable 

to apply low-probability models to the prediction. If the 

conditions of the predictive period are significantly 

different than the calibration period, differences between models that were not apparent in the calibration 

may appear during prediction. Applying even lowprobability models in the predictive period may bring 

these differences to light. At that point, the conditions 

of the predictive and calibration periods and the data 

used in the calibration can all be reviewed to determine 

whether the model probabilities should be reevaluated. 

Predictive simulations were carried out for the 300 

Area application over the period 2005-2025 using all 

four models and under two alternative scenarios. 

Monte Carlo simulation was used to simulate parameter uncertainty. Selected uranium concentration results 

were shown to illustrate the model and scenario averaging. These figures and the accompanying tables illustrate the individual contributions to predictive uncertainty of the model, parameter, and scenario uncertainties. In this case, model uncertainty was negligible due 

to the dominance of a single model. Parameter and 

scenario uncertainties were significant. 

In a regulatory application of the methodology, probability distributions of concentration would also likely 

be computed. The regulatory quantity of interest may 

be a quantity derived from predicted concentration, 

however, such as peak dose over a given time period or 

a mortality risk to a specific population. In these cases, 

the execution of the methodology would be identical 

except that the desired regulatory quantity would be 

computed for each Monte Carlo realization and its distribution would be averaged over the alternative models and scenarios. Figures illustrating the results would 

be for the regulatory quantity instead of for contaminant concentration. 

Application of the uncertainty methodology requires 

the calibration and probabilistic simulation of multiple 

models and is therefore computationally demanding. 

For the 300 Area application, however, computations 

were carried out on inexpensive desktop computers. In 

addition, the application was completed using primarily 

public-domain software tools. The 300 Area application thus demonstrates the practicability of applying a 

comprehensive uncertainty assessment to large-scale, 

detailed groundwater flow and transport modeling. 
89

6 REFERENCES

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 A.1 

Appendix A 

Derivation of Posterior Mean and Variance Considering Model 

Structure and Parameter Uncertainty for a Given Scenario 

Posterior mean ( ) , E S Δ i D can be evaluated via 

ES pS d ( ) Δ i i , , =Δ Δ Δ ( ) D D ∫ (A1) 

Substituting the expression of ( ) ,i p S Δ D (8) into (A1) gives 

( ) ( ) ( )

1

, ,, ,

K

i k i ki

k

E S p M S pM S d =Δ =Δ Δ Δ D DD ∫ ∑ (A2) 

which can be rewritten as 

( ) ( ) ( )

1

, ,, ,

K

i ki ki

k

E S p M S d pM S =Δ = ΔΔ Δ D DD ∑∫ (A3) 

Considering that E ( ) Δ =Δ Δ Δ MS p MS d ki ki ,, ,, ( ) D D ∫ , (A3) leads to (18) directly. 

Posterior variance ( ) , Var S Δ i D can be evaluated via 

( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

2 2

2 2

, , ,,

, 2, , , ,

i i ii

i ii i i

Var S E E S E S p S d

p S d ES pS d ES pS d

⎛ ⎞ Δ = Δ− Δ = Δ− Δ Δ Δ ⎡ ⎤⎡ ⎤ ⎜ ⎟ ⎣ ⎦⎣ ⎦ ⎝ ⎠

= Δ Δ Δ− Δ Δ Δ Δ+ Δ Δ Δ ⎡ ⎤ ⎣ ⎦

∫

∫∫ ∫

D DD D D

D DD D D

 (A4) 

Recalling (A1) and ( ) , 1 i pS d Δ Δ= ∫ D , the last two terms of (A4) are 

( ) ( ) ( ) ( ) ( ) 2 2

2, , , , , ES pS d ES pS d ES ii i i i − Δ Δ Δ Δ+ Δ Δ Δ=− Δ ⎡ ⎤ ⎡⎤ ∫ ∫ ⎣ ⎦ ⎣⎦ DD D D D (A5) 

and (A4) thus becomes 

( ) ( ) ( ) 2 2 ,, , Var S p S d E S ii i Δ = Δ Δ Δ− Δ ⎡ ⎤ ∫ ⎣ ⎦ DD D (A6) 

Substituting the expression of ( ) ,i p S Δ D (15) into the first term at the right hand side of (A6) gives 
 A.2 

( ) ( ) ( )

( ) ( )

( ) ( )

2 2

1

2

1

2

1

, ,, ,

,, ,

,, ,

K

i ki ki

k

K

ki ki

k

K

k i ki

k

p S d p M S pM S d

p M S d pM S

E M S pM S

=

=

=

Δ Δ Δ= Δ Δ Δ

= ΔΔ Δ

= Δ

∫ ∫ ∑

∑∫

∑

D DD

D D

D D

 (A7) 

Substituting the expression of variance 

( ) ( ) ( ) 2 2 ,, ,, ,, E M S Var M S E M S ki ki ki Δ = Δ +Δ ⎡ ⎤ ⎣ ⎦ D DD (A8) 

into (A7) gives 

( ) ( ) ( ) ( ) 2 2

1

, ,, ,, ,

K

i ki k i ki

k

p S d Var M S E M S p M S =

⎡ ⎤ Δ Δ Δ= Δ + Δ ⎡ ⎤ ⎢ ⎥ ⎣ ⎦ ∑⎣ ⎦ ∫ D D DD (A9) 

Substituting (A9) into (A6) yields (19) directly. 

Equation (19) can be written as 

( )

( ) ( ) ( ) ( ) ( ) 2 2

1 1

,

,, , ,, , ,

i

K K

ki ki k i ki i

k k

Var S

Var M S p M S E M S p M S E S

= =

Δ =

Δ + Δ −Δ ⎡ ⎤ ⎡⎤ ∑ ∑⎣ ⎦ ⎣⎦

D

DD D D D

(A10) 

Consider the expression 

( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

2 2

1 1

2

1 1

,, , , ,, ,

2 , ,, , , ,

K K

k i i ki k i ki

k k

K K

i k i ki i ki

k k

E M S E S pM S E M S pM S

E S E M S pM S E S pM S

= =

= =

⎡ ⎤ ⎡⎤ Δ −Δ = Δ ⎣ ⎦ ⎣⎦

−Δ Δ +Δ ⎡ ⎤ ⎣ ⎦

∑ ∑

∑ ∑

DD D D D

D DD D D

 (A11) 

Given (17) and (18), (A11) becomes 

( ) ( ) ( )

( ) ( ) ( )

2

1

2 2

1

,, , ,

,, , ,

K

k i i ki

k

K

k i ki i

k

E M S E S pM S

E M S pM S E S

=

=

⎡ ⎤ Δ −Δ = ⎣ ⎦

⎡ ⎤ ⎡⎤ Δ −Δ ⎣ ⎦ ⎣⎦

∑

∑

DD D

D DD

 (A12) 

Substituting (A12) into (A10) leads to (20) directly.
 B.1 

Appendix B 

Derivation of Posterior Mean and Variance Considering Model 

Structure, Parameter, and Scenario Uncertainties 

Posterior mean E( ) Δ D can be evaluated via 

E pd (Δ ) =Δ Δ Δ ( ) D D ∫ (B1) 

Substituting the expression of p( ) Δ D (21) into (B1) gives 

( ) ( ) ( ) ( )

1 1

,, ,

I K

ki ki i

i k

E p M S pM S pSd = =Δ =Δ Δ Δ D DD ∫ ∑∑ (B2) 

which can be rewritten as 

( ) ( ) ( ) ( )

1 1

,, ,

I K

ki ki i

i k

E p M S d pM S pS = =Δ = ΔΔ Δ D DD ∑∑∫ (B3) 

Considering that E ( ) Δ =Δ Δ Δ MS p MS d ki ki ,, ,, ( ) D D ∫ , (B3) leads to (23) directly. 

Posterior variance Var( ) Δ D can be evaluated via 

( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

2 2

2 2 2

Var E E E p d

p dE p dE p d

⎛ ⎞ Δ = Δ− Δ = Δ− Δ Δ Δ ⎡ ⎤⎡ ⎤ ⎜ ⎟ ⎣ ⎦⎣ ⎦ ⎝ ⎠

= Δ Δ Δ− Δ Δ Δ Δ+ Δ Δ Δ ⎡ ⎤ ⎣ ⎦

∫

∫∫ ∫

D DD D D

D DD D D

 (B4) 

Recalling (B1) and p d ( ) Δ Δ=1 ∫ D , the last two terms of (B4) are 

( ) ( ) ( ) ( ) ( ) 2 2

− Δ Δ Δ Δ+ Δ Δ Δ=− Δ 2E p dE p d E ⎡ ⎤ ⎡⎤ ∫ ∫ ⎣ ⎦ ⎣⎦ DD D D D (B5) 

and (B4) thus becomes 

( ) ( ) ( ) 2 2 Var p d E Δ = Δ Δ Δ− Δ ⎡ ⎤ ∫ ⎣ ⎦ DD D (B6) 

Substituting the expression of p( ) Δ D (21) into the first term on the right hand side of (B6) gives 
 B.2 

( ) ( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )

2 2

1 1

2

1 1

2

1 1

,, ,

,, ,

,, ,

I K

ki ki i

i k

I K

k i ki i

i k

I K

k i ki i

i k

p d p M S pM S pSd

p M S d pM S pS

E M S pM S pS

= =

= =

= =

Δ Δ Δ= Δ Δ Δ

= ΔΔ Δ

= Δ

∫ ∫ ∑∑

∑∑∫

∑∑

D DD

D D

D D

 (B7) 

Substituting the expression of variance 

( ) ( ) ( ) 2 2 ,, ,, ,, E M S Var M S E M S ki ki ki Δ = Δ +Δ ⎡ ⎤ ⎣ ⎦ D DD (B8) 

into (B7) gives 

( )

( ) ( ) ( ) ( )

2

2

1 1

,, ,, ,

I K

k i k i ki i

i k

p d

Var M S E M S p M S p S = =

Δ Δ Δ=

⎡ ⎤ Δ +Δ ⎡ ⎤ ⎢ ⎥ ⎣ ⎦ ⎣ ⎦

∫

∑∑

D

D DD

 (B9) 

Substituting (B9) into (B6) yields (24) directly. 

Considering (19), equation (24) can be written as 

( ) ( ( ) ( ) ) ( ) ( )

( ) ( ) ( ) ( ) ( )

2 2

1

2 2

1 1

, ,

, ,

I

i ii

i

I I

ii i i

i i

Var Var S E S p S E

Var S p S E S p S E

=

= =

Δ= Δ + Δ − Δ ⎡ ⎤ ⎡⎤ ⎣ ⎦ ⎣⎦

= Δ + Δ −Δ ⎡ ⎤ ⎡⎤ ⎣ ⎦ ⎣⎦

∑

∑ ∑

D DD D

D DD

 (B10) 

Consider the expression below 

( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( ) ( )

2

1

2 2

1 11

,

,2 ,

I

i i

i

I II

i i ii i

i ii

E S E pS

E S pS E E S pS E pS

=

= ==

⎡ ⎤ Δ −Δ ⎣ ⎦

= Δ −Δ Δ +Δ ⎡ ⎤ ⎡⎤ ⎣ ⎦ ⎣⎦

∑

∑ ∑∑

D D

D DD D

 (B11) 

which, using (22), becomes 

( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( )

2

1

2 2

1 1

,

,2 ,

I

i i

i

I I

i i ii

i i

E S E pS

E S pS E E S pS E

=

= =

⎡ ⎤ Δ −Δ ⎣ ⎦

= Δ −Δ Δ +Δ ⎡ ⎤ ⎡⎤ ⎣ ⎦ ⎣⎦

∑

∑ ∑

D D

D DD D

 (B12) 

Using (19), (B12) can be written as 
 B.3 

( ) ( ) ( )

( ) ( ) ( ) ( )

( ) ( ) ( )

2

1

2 22

1

2 2

1

,

, 2

,

I

i i

i

I

i i

i

I

i i

i

E S E pS

E S pS E E

E S pS E

=

=

=

⎡ ⎤ Δ −Δ ⎣ ⎦

= Δ − Δ +Δ ⎡ ⎤ ⎡ ⎤⎡ ⎤ ⎣ ⎦ ⎣ ⎦⎣ ⎦

= Δ −Δ ⎡ ⎤ ⎡⎤ ⎣ ⎦ ⎣⎦

∑

∑

∑

D D

D DD

D D

 (B13) 

Considering (22), (B13) leads to 

( ) ( ) ( ) ( ) ( ) ( ) 22 2

1 1

, ,

I I

ii i i

i i

E S pS E E S E pS = =

⎡ ⎤ ⎡ ⎤⎡ ⎤ Δ − Δ = Δ −Δ ∑ ∑ ⎣ ⎦ ⎣ ⎦⎣ ⎦ D D DD (B14) 

Substituting (B14) into (B10) gives (25) directly. 
 C.1 

Appendix C 

On Evaluation of Recharge Model Uncertainty: 

A Priori and A Posteriori 

M. Ye, K. Pohlmann, J. Chapman, and D. Shafer 

Previously published in Proceedings of the International High-Level Radioactive Waste Management 

Conference 

April 30 – May 4, 2006, Las Vegas, Nevada 

Abstract 

Hydrologic environments are open and complex, rendering them prone to multiple interpretations and 

mathematical descriptions. Hydrologic analyses typically rely on a single conceptual-mathematical 

model, which ignores conceptual model uncertainty and may result in bias in predictions and underestimation of predictive uncertainty. This study is to assess conceptual model uncertainty residing in five 

recharge models developed to date by different researchers based on different theories for Nevada and 

Death Valley area, CA. A recently developed statistical method, Maximum Likelihood Bayesian Model 

Averaging (MLBMA), is utilized for this analysis. In a Bayesian framework, the recharge model 

uncertainty is assessed, a priori, using expert judgments collected through an expert elicitation in the form 

of prior probabilities of the models. The uncertainty is then evaluated, a posteriori, by updating the prior 

probabilities to estimate posterior model probability. The updating is conducted through maximum 

likelihood inverse modeling by calibrating the Death Valley Regional Flow System (DVRFS) model 

corresponding to each recharge model against observations of head and flow. Calibration results of 

DVRFS for the five recharge models are used to estimate three information criteria (AIC, BIC, and KIC) 

used to rank and discriminate these models. Posterior probabilities of the five recharge models, evaluated 

using KIC, are used as weights to average head predictions, which gives posterior mean and variance. The 

posterior quantities incorporate both parametric and conceptual model uncertainties. 

1 INTRODUCTION 

Hydrologic analyses are commonly based on a single conceptual-mathematical model. Yet hydrologic 

environments are open and complex, rendering them prone to multiple interpretations and mathematical 

descriptions. This is true regardless of the quantity and quality of available hydrologic information and 

data. Focusing on only one conceptual-mathematical model may lead to a Type I model error, which 

arises when one rejects (by omission) valid alternative models. It may also result in a Type II model error, 

which arises when one adopts (fails to reject) an invalid conceptual-mathematical model. Indeed, critiques 

of hydrologic analyses, and legal challenges to them, typically focus on the validity of the underlying 

conceptual (and by implication mathematical) model. If a proposed model is found to be severely 

deficient, hydrologic analysis based on the single model may damage professional credibility of the work; 
C.2 

result in the loss of a legal contest; and lead to adverse environmental, economic and political impacts ([1-

2]). 

The need to properly assess conceptual model uncertainty has motivated the recent development of a 

Maximum Likelihood Bayesian Model Averaging method (MLBMA) [3-4]. MLBMA is being applied in 

our study to assess conceptual model uncertainty in the Death Valley Regional Flow System (DVRFS) 

model, developed by the U.S. Geological Survey [5] to simulate the regional flow system in southwest 

Nevada and southeast California. This area includes the U.S. Department of Energy proposed Yucca 

Mountain nuclear repository, the nation’s first long-term permanent geologic repository of spent nuclear 

fuel and high-level radioactive waste. 

Our study is focused on assessing conceptual model uncertainty due to five alternative recharge models 

listed in Table C-1: (1) the Maxey-Eakin (ME) model [6], (2) two distributed parameter watershed (DPW) 

models, one with and one without a runon-runoff component [7], and (3) two chloride mass balance 

(CMB) models, each with different zero-recharge masks, one for alluvium and one for both alluvium and 

elevation [8]. These five models are based on different methodologies for estimating recharge and have 

different levels of complexity, and they all have been used for groundwater modeling in Nevada. 

Recharge estimates of the five models are plotted in Figure C-1, and they are significantly different. A 

large amount of conceptual model uncertainty exits in the recharge models [9]. Since recharge 

significantly affects modeled groundwater flow paths and travel times, it is important to evaluate the 

recharge model uncertainty and quantify its propagation through the groundwater modeling process. 

Using MLBMA, we assess the recharge model uncertainty a priori and a posteriori. The terms “a priori” 

and “a posteriori” refer primarily to how or on what basis our assessment is conducted. An assessment is 

conducted a priori if it is based on prior information without calibrating the regional flow model (of 

which the recharge model is a component) against site observations (e.g., hydraulic head and groundwater 

flux). The prior information includes assessment of model uncertainty from a similar site and/or expert 

judgments based on one’s professional experience. An assessment is conducted a posteriori when the 

regional flow model (of which the recharge model is a component) is calibrated against site observations. 

In MLBMA, assessments a priori and a posteriori are quantified by prior and posterior model 

probabilities, as discussed below. 
C.3 

Figure C-1. Illustration of the five recharge models: (a) ME, (b) DPW1, (c) DPW2, (d) CMB1, and (e) 

CMB2. 

 Table C-1. Abbreviation and Description of the Five Recharge Models. 

Models Model Description 

ME Maxey-Eakin model 

DPW1 Distributed parameter watershed model with runon-runoff component 

DPW2 Distributed parameter watershed model without runon-runoff component 

CMB1 Chloride mass balance model with fluvial mask 

CMB2 Chloride mass balance model with fluvial and elevation masks 

2 MAXIMUM LIKELIHOOD BAYESIAN MODEL AVERAGING (MLBMA) 

To render our paper complete and self-contained, we start with a brief description of MLBMA; for 

additional details the reader is referred to [3-4]. If ∆ is the desired 

predicted quantity given a set of K alternative models, then its posterior distribution, given a discrete set 

D of site data, is 

( ) ( ) ( ) 1

,

K

k k

k

p p M pM =Δ= Δ D DD ∑ (C1) 

where p(∆|Mk,D) is the posterior distribution of ∆ under model Mk and p(Mk|D) is posterior probability of 

Mk. With consideration of parametric and conceptual model uncertainty, mean and variance of ∆ are 

( ) 1

,

K

k k

k

E E M pM =⎡⎤ ⎡ ⎤ Δ= Δ ⎣⎦ ⎣ ⎦ DD D ∑ (C2) 

( ) ( ) ( ) 2

1 1

, ,

K K

kk k k

k k

Var Var M p M E M E p M = =⎡⎤ ⎡ ⎤ ⎡ ⎤⎡⎤ Δ = Δ + Δ −Δ ⎣⎦ ⎣ ⎦ ⎣ ⎦⎣⎦ D D D D DD ∑ ∑ (C3) 
C.4 

where , E Mk ⎡ ⎤ Δ⎣ ⎦ D and , Var Mk ⎡ ⎤ Δ⎣ ⎦ D are mean and variance of ∆ under model Mk due to uncertainty 

of parameters associated with Mk. The weight p(Mk|D) used to average model predictions and 

corresponding predictive variance is posterior model probability of model Mk, evaluated using Bayes’ rule 

( ) ( ) ( )

( ) ( ) 1

k k

k K

l l

l

p M pM p M

p M pM =

=

∑

D

D

D

 (C4) 

where p(D|Mk) is likelihood of model Mk (a measure of consistency between model predictions and site 

observations D) and p(Mk) is prior probability of Mk. Estimating prior model probability will be discussed 

in detail in Section III.The posterior model probability is conditioned on site observations explicitly and 

prior information implicitly. According to [4], equation (C4) can be approximated as 

( )

( )

( ) 1

1

exp 2

1

exp 2

k k

k K

l l

l

KIC p M

p M

KIC p M =

⎛ ⎞ ⎜ ⎟ −

⎝ ⎠ ≈ ⎛ ⎞ ⎜ ⎟ −

⎝ ⎠ ∑

D (C5) 

where KIC is Kashyap information criterion defined as [10] 2 ( )ln ln 2 ln | | T KIC N N N k k k k kk =− − + σ π X ωX (C6) 

where N is number of calibration data D, Nk is number of parameters θk associated with model Mk, e is 

the natural number, ω (the same for all models) is weight matrix associated with calibration data D, and X

is sensitivity matrix with element ' X D k ij k i k j , ,, =∂ ∂θ evaluated at maximum likelihood parameter 

estimates, , ˆ

θk j ( ' Dk i, being predictions at locations of Di by model Mk). , ˆθk j can be estimated using 

maximum likelihood (or, equivalently, generalized least square) methods, which also gives the calculated 

error variance, 2 σ k , 

2

T

kk k

k

WSSR

N N = = e ωe

σ (C7) 

where ' e DD = − is residual and WSSRk is weighted sum of squared residual of model Mk. All the 

quantities above can be estimated based on results of model calibration using common software such as 

MODFLOW2000 [11]. 

3. EVALUATE RECHARGE MODEL UNCERTAINTY: A PRIORI

Conceptual uncertainty of the five recharge models is first evaluated, a priori, using prior probabilities of 

the models. Prior model probability is interpreted by [4] as subjective values reflecting the analyst’s (or a 

group of analysts’) belief about the relative plausibility of each model (or a group of models) based on its 

apparent (qualitative, a priori) consistency with available knowledge and data. The analyst’s perception, 

degree of reasonable belief [12], or confidence [13] in a model is ideally based on expert judgment, which 

is considered by [14] as the basis of conceptual model development. Hence we view integrating expert 

judgment in MLBMA (by specifying subjective prior probabilities) to be a strength rather than a 

weakness. According to this view, the models included in the model set must be those (and only those) 

that experts consider being of potential relevance to the problem at hand. Given a set of alternative 

models, their prior probabilities sum up to one, 
C.5 

( ) 1

1

K

k

k

p M

=

∑ = (C8) 

This implies that all possible models of relevance are included in the model set (collective 

exhaustiveness), and that all models in the set differ from each other sufficiently to be considered 

mutually exclusive (the joint probability of two or more models being zero). 

Following the process suggested by [15], an expert elicitation is conducted to elicit professional 

judgments from seven experts on uncertainty of the five recharge models. Elicited prior probabilities of 

the five models are plotted in Figure C-2, which shows that the maximum and minimum prior 

probabilities are 45% and 5%, respectively. Models ME, DPW1, and CMB2 are the three most plausible 

models, and do not receive the minimum prior probability from any expert. Although the experts evaluate 

the models from various aspects (e.g., model assumptions and sensitivity of model predictions to mode 

parameters), no experts place more than 50% prior probability on any model. 

The prior model probabilities are aggregated using simple averaging, i.e., 

1

1 NE

k ik

i

P P

NE =

= ∑ (C9) 

where NE=7 is number of experts and Pik is the prior probability expert i assigns to alternative model Mk,. 

The aggregated prior model probabilities are plotted in Figure C-3. Models DPW1 and DPW2 have the 

largest and smallest probability, respectively, since the experts regard that including the runon-runoff 

component is more realistic. Probability of the model CMB2 is larger than that of CMB1, since experts 

regard that elevation mask is an important feature in recharge estimation. Model ME is ranked as the 

second most plausible model and has prior model probability of 25%, although this model is the simplest 

one and its recharge estimation is significantly different from that of other four models. Although prior 

probabilities given by each expert are significantly different (Figure C-2), the aggregated probabilities are 

more or less uniform, considering that the equally likely prior probability is 20%. The largest deviation 

from the equally likely prior probability is only 10% for model DPW1. This manifests the inherent 

uncertainty in the recharge models, since they are developed independently based on solid physical 

principles and assumptions, calibrated with site measurements, and have all been applied to water 

resource management in Nevada. Since none of the models dominates over other models and all models 

have prior model probabilities larger than 5%, there is no justification to select one model and discard 

others, a priori.

0.00

5.00

10.00

15.00

20.00

25.00

30.00

35.00

40.00

45.00

50.00

ME DPW1 DPW2 CMB1 CMB2

Prior Model Probability (%)
C.6 

Figure C-2. Column chart of prior probabilities of the five models given by seven experts. Columns of 

each model represent elicited prior model probability from one expert. Model names are explained in 

Table C-1.

CMB1

13%

DPW2

11%

CMB2

20% ME

25%

DPW1

31%

Figure C-3. Prior probabilities of the five recharge models obtained through an expert elicitation. Model 

names are explained in Table C-1. 

4. EVALUATE RECHARGE MODEL UNCERTAINTY: A POSTERIORI 

Recharge model uncertainty is assessed, a posteriori, by maximum likelihood model calibration against 

site observations. Results of model calibration are used to estimate model likelihood p(D|Mk), which, in 

turn, is used to evaluate posterior model probability p(Mk|D) in (C4). Whereas prior model probabilities 

must in our view remain subjective, the posterior model probabilities are modifications of these subjective 

values based on an objective evaluation of each model’s consistency with available data. 

4.1 Model Calibration Using MODFLOW2000 

Plausibility and uncertainty of each of the five recharge models is evaluated by calibrating the Death 

Valley Regional Flow System (DVRFS) model, of which the recharge model is a component. DVRFS 

was modeled by [5] using MODFLOW2000, and a three-dimensional hydrogeologic framework based on 

characterization of regional geology, hydrology, and hydrogeology. The recharge model used in DVRFS 

is DPW1 developed by [7]. Our study is to assess recharge model uncertainty in the modeling framework 

of DVRFS, without modifying its other components. DVRFS was calibrated using MODFLOW2000 

against a total of 4,963 observations of head (2,227), head change (2,672), discharge (49), and constanthead flow (15). These observations are also used in our calibration. 

Our calibration process, however, is different from that of DVRFS, which calibrated 55 model 

parameters, 23 in the steady-state model and 32 in the transient model. Our model calibration is based on 

the transient model only, since there is insufficient information to identify how the 23 parameters are 

calibrated in the steady-state model. In addition, only some of the 55 parameters are calibrated in our 

study, due to different purposes of our study. Specifically, 32 of the 55 parameters are calibrated for 

DPW1 and DPW2. The two models estimate precipitation (not recharge), which is converted to recharge 

within DVRFS by dividing the top model layer into five recharge zones. Recharge coefficients in two 

zones are calibrated against site observations. Since the other three models estimate recharge directly, 
C.7 

recharge coefficients are not used and therefore only 30 parameters are calibrated. All other calibration 

parameters are the same as those used in DVRFS. Although MLBMA allows different models having 

different numbers of calibrated parameters, we intend to calibrate the same model parameters for all the 

recharge models so that model ranking and uncertainty analysis are on the same basis. In the same line, 

model calibration is conducted in the same manner for all the recharge models. Specifically, all the model 

calibrations use identical initial parameter values, convergence criterion, and other calibration variables 

such as parameter log transform and damping factors. 

Model calibration results corresponding to the five recharge models are summarized in Table C-2 and 

Figure C-4. Table C-2 lists WSSR (weighted sum of squared residuals) of the four kinds of observations, 

respectively, and total WSSR. WSSR of DVRFS is also listed for comparison. The table shows that, 

except for recharge model ME, the values of WSSR of the models are close and are lower than that of 

DVRFS. This is not surprising since our calibration is based on model calibration of DVRFS to a certain 

extent, and can be regarded as further calibration of DVRFS. The largest relative differences of WSSR 

occur for the observations of discharge and constant-head flow. Figure C-4 plots some of the calibrated 

parameters whose values are noticeably different between the recharge models (values of other calibrated 

parameter are close). The values of calibrated parameters in DVRFS are also plotted for comparison. 

Although WSSR corresponding to the recharge models are similar, some parameter values are different, 

indicating different responses of the regional flow system simulation to the recharge models, which 

provides a basis for model discrimination. The largest difference of parameter values occurs for hydraulic 

conductivity of volcanic rock units (K3) such as K3BRU123 and K3CTM. While most of the parameters 

are within the parameter ranges given in [5], several parameter values exceed the ranges. This, however, 

is not surprising, since the ranges are based on limited information of site measurements. All values of the 

calibrated parameters are considered reasonable. 

TABLE C-2. Weighted Sum of Squared Residuals (WSSR) for all Kinds of Observations Corresponding 

to the Five Recharge Models and DVRFS. 

Type of 

observation 

Observaion 

Number 

DVRFS ME DPW1 DPW2 CMB1 CMB2 

Hydraulic head 2227 23083.22 26321.55 20030.92 20296.37 20215.87 19803.57 

Head changes 2672 13348.08 11805.63 12599.57 12752.66 12372.07 12057.11 

Discharge 49 637.64 2078.36 674.43 611.12 1001.34 1062.06 

Constant-head 

flow 

15 438.15 1520.28 296.94 350.56 863.24 641.49 

Total 4963 37507.10 41725.82 33601.86 34010.71 34452.52 33564.23 
C.8 

1.00E-02

1.00E-01

1.00E+00

1.00E+01

1.00E+02

1.00E+03

K3211TMVA 

K4_VF_AQ 

RCH_2 

K241LCA_T1

K3_VOLVANI

RCH_35 

K3215BCU34

K4_VF_CU 

K3BRU123 K243_LCA 

K242A_LCA K42UP_VSU 

K3C_TM 

K321521_PP

DVRFS

ME

USGS1

USGS2

CMB1

CMB2

Figure C-4. Comparison of values of some calibrated parameters corresponding to the five recharge 

models. 

4.2 Posterior Model Probability 

Posterior model probabilities of the recharge models are calculated using equation (C6) based on the 

model calibration results listed in Table C-3, which also lists three information criteria (AIC, BIC, and 

KIC) commonly used to rank alternative models. AIC and BIC are evaluated via 

( ) 2 2ln 2 ln A k k k kk IC p M N N N =− + = + D σ (C10) 

( ) 2 2ln ln ln ln B k k k kk IC p M N N N N N =− + = + D σ (C11) 

These information criteria rank alternative models not only based on their goodness-of-fit (as measured 

by WSSR) but also on the principle of parsimony, which states that a simple model (with lower number 

of parameters) is considered more plausible than a complex model if their predictions fit observations 

equally well. The three information criteria rank the five recharge models at almost the same order, with 

models ME, DPW2, and CMB1 ranked as the least plausible. In addition, the information criteria show 

that DPW1 and CMB2 are more plausible than DPW2 and CMB1, respectively, which is consistent with 

the results of the expert elicitation (Figure C-3). Nevertheless, model ME is ranked as least plausible after 

model calibration. AIC and BIC rank CMB2 as the best model, while KIC ranks DPW1 as the best one. 

Inconsistency of model ranking given by different information criteria is not uncommon. Among the three 

criteria, KIC is favored since it incorporates quality of data used for model calibration [3] and can yield 

more reliable model rankings in various circumstances [e.g., 4]. 

TABLE C-3. Quality Criteria, Ranking, And Prior/Posterior Probabilities Associated With The Five 

Recharge Models. 

ME DPW1 DPW2 CMB1 CMB2 

Nk 30 32 32 30 30 

WSSR 41726 33602 34011 34453 33564 

ln|F| 360 346 344 349 346 

AIC 10627 9556 9616 9676 9547 

Rank 5 2 3 4 1 

BIC 10822 9764 9824 9871 9742 

Rank 5 2 3 4 1 
C.9 

KIC 10808 9718 9775 9852 9720 

Rank 5 1 3 4 2 

p(Mk) 25% 30% 11% 13% 20% 

p(Mk|D) 0 83.16% 0 0 16.84%

p(Mk) 20% 20% 20% 20% 20% 

p(Mk|D) 0 76.70% 0 0 23.30%

Posterior model probabilities are evaluated using (C5) for two sets of prior model probabilities. One set 

has informative priors obtained from the expert elicitation and the other one treats the five models equally 

likely. Regardless of prior probabilities, posterior probabilities of models ME, DPW2, and CMB2 are 

zero, indicating that they are implausible given the calibration data. This is so even though model ME 

received a relatively large prior probability from the experts. The effects of prior on posterior model 

probability is observed for models DPW1 and CMB2. Posterior probability of DPW1 decreases 6.43% 

when its prior probability decreases 10%. This results in a concomitant increase of 6.43% in the posterior 

probability of model CMB2, even though its prior probability does not change. Although it is expected 

that sensitivity of posterior to prior model probability diminishes as the amount conditioning (calibration) 

data increases, this study shows that, even with 4,963 observations, sensitivity to prior probability does 

not disappear. In this case, using informative prior model probability (obtained from expert elicitation in 

this study) may increase accuracy of model uncertainty assessment, as suggested in [16]. Note that just 

like prior probabilities, posterior probabilities are valid only in a comparative, not in an absolute, sense. 

They are conditional on the choice of models, calibration data, and prior information used to estimate 

prior model probabilities. 

5. BAYESIAN MODEL AVERAGING 

Based on equations (C2) and (C3), Bayesian model averaging is used to yield the posterior mean and 

variance to incorporate both parametric and conceptual model uncertainty. The posterior mean represents 

the optimum prediction and the posterior variance measures the associated predictive uncertainty. Monte 

Carlo simulation is used to assess parametric uncertainty and estimate , E Mk ⎡Δ ⎤ ⎣ D ⎦ and , Var Mk ⎡ ⎤ Δ⎣ ⎦ D

for model Mk. Multivariate normal distributions are used to generate 200 parameter realizations of the 

calibrated parameters. The mean of the normal distribution is the maximum likelihood parameter estimate 

ˆ

θk of model Mk and the covariance matrix is 2 1 ( ) T

k k

− σ X ωX [3, 11]. Estimation of posterior mean and 

variance using equations (C2) and (C3) is straightforward. Figure C-5 plots mean head predictions 

corresponding to the five recharge models (Figures C-5b – C-5f) and the MLBMA (Figure C-5a) 

posterior mean head in the first (top) model layer at stress period 87 (1998, the last year of the transient 

model). The MLBMA mean head (Figure C-5a) is an average of the mean heads for DPW1 and CMB2, 

since the other three models have zero posterior probabilities. Mean head contours of DPW1 and DPW2 

are similar to each other, as are the contours of CMB1 and CMB2, owing to the similarity of the two pairs 

of recharge models. Since these four recharge models are different from ME, the contour of ME is 

different from the four contours in Figures C-5c – C-5f. Figure C-6 plots the cumulative distribution 

function (CDF) of the mean head and standard deviation of head predicted by models DPW1 and CMB2 

and MLBMA for the entire simulation domain at stress period 87. For mean head predictions, the CDFs 

of DPW1, CMB2, and MLBMA are almost identical, due to the similarity of mean head predictions of 

DPW1 and CMB2 (shown in Figures C-5b and C-5f). Nevertheless, the standard deviation of head 
C.10 

prediction of MLBMA is larger than that of models DPW1 and CMB2, since MLBMA considers both 

parametric and conceptual model uncertainty, while the two single models address only parametric 

uncertainty. 

6. CONCLUSIONS 

This study assesses conceptual model uncertainty of five recharge models within the modeling framework 

of DVRFS, of which each recharge model is a component. Conceptual model uncertainty is first assessed, 

a priori, using expert judgment gathered from an expert elicitation. The experts placed higher 

probabilities on DPW1and CMB2 than DPW2 and CMB1, respectively. However, since the recharge 

models are developed by different researchers based on different theories, prior model probabilities 

elicited from the experts are around the average value of 20%. This indicates that one cannot select one 

model for predictions and discard all others a priori. Since prior information cannot fully assess 

conceptual model uncertainty, model calibration is needed to assess conceptual model uncertainty a 

posteriori based on observations of head and flow. DVRFS is used as the framework for numerical 

modeling and only its recharge component varies for different recharge models (other components remain 

the same). Based on model calibration results using MODFLOW2000, three information criteria (AIC, 

BIC, and KIC) are evaluated to rank the models. Model ranking of AIC and BIC are the same, but 

different from that of KIC. Consistent with results of expert elicitation, DPW1 and CMB2 are ranked 

more plausible than DPW2 and CMB1, respectively. However, as opposed to the results of expert 

elicitation, model ME is ranked as least plausible. This suggests the importance of uncertainty assessment 

a posteriori. Posterior model probabilities are evaluated using KIC, which is considered superior to AIC 

and BIC. Models ME, DPW2, and CMB1 have zero posterior probabilities. Sensitivity of posterior to 

prior probabilities for models DPW1 and CMb2 does not disappear, although 4,963 observations are used 

for model calibration. Note that posterior probabilities are valid only in a comparative, not in an absolute, 

sense. They are conditional on the choice of models, calibration data, and prior information used to 

estimate prior probabilities. Bayesian model averaging is conducted to estimate posterior mean and 

variance of head and flux. Posterior variance of MLBMA is larger than the variance of any single model, 

since conceptual model uncertainty is also addressed. Our research results can be extended to incorporate 

conceptual model uncertainty in flow path delineation, which can in turn be used to design networks for 

detection and monitoring of potential radionuclide transport in the saturated zone of the Death Valley 

Regional Flow System, where Yucca Mountain is located. 
C.11 

Figure C-5. Mean head predicted by (a) MLBMA, (b) ME, (c) DPW1, (d) DPW2, (e) CMB1, and (f) 

CMB2 in the first (top) layer at stress period 87 (1998). 

Figure C-6. Cumulative distribution function (CDF) of mean head and standard deviation of head 

prediction over the whole simulation domain at stress period 87 (1998). 

ACKNOWLEDGMENTS 

Research is funded through contract DE-AC52-00NV13609 with the U.S. Department of Energy, 

National Nuclear Security Administration Nevada Site Office (NNSA/NSO). We appreciate Dr. Glendon 

Gee, Dr. Greg Pohll, Mr. Chuck Russell, Dr. Joe Hevesi, and Mr. Randy Laczniak for their participation 

in the expert elicitation. 

REFERENCES 

[1] NATIONAL RESEARCH COUNCIL, “Conceptual Models of Flow and Transport in the Fractured 

Vadose Zone”, National Academy Press, Washington, DC (2001). 
C.12 

[2] S.P. NEUMAN and P.J. WIERENGA, “A comprehensive strategy of hydrogeologic modeling and 

uncertainty analysis for nuclear facilities and sites”, NUREG/CR-6805, U.S. Nuclear Regulatory 

Commission, Washington, D.C. (2003). 

[3] S.P. NEUMAN, “Maximum likelihood Bayesian averaging of alternative conceptual-mathematical 

models, Stochastic Environmental Research and Risk Assessment”, 17(5), 291-305, DOI: 

10.1007/s00477-003-0151-7 (2003). 

[4] M. YE, S.P. NEUMAN, and P.D. MEYER, “Maximum likelihood Bayesian averaging of spatial 

variability models in unsaturated fractured tuff”, Water Resour. Res., 40, W05113, 

doi:10.1029/2003WR002557 (2004). 

[5] W.R. BELCHER (ed). “Death Valley regional ground-water flow system, Nevada and California – 

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Investigation Report 2004-5205 (2004). 

[6] G.B. MAXEY and T.E. EAKIN, “Ground water in White River Valley, White Pine, Nye, and Lincoln 

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[8] C.E. RUSSELL, and T. MINOR, “Reconnaissance Estimates of Recharge Based on an Elevationdependent Chloride Mass-balance Approach”, DOE/NV/11508-37. Prepared for the U.S. Department of 

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[12] H. JEFFREYS, “Scientific Inference” (2nd ed.), Cambridge, U.K., Cambridge University Press, 

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[13] E. ZIO, and G.E. APOSTOLAKIS, “Two methods for the structured assessment of model 

uncertainty by experts in performance assessments of radioactive waste repositories”, Reliability 

Engineering and System Safety, 54, 225-241 (1996). 

[14] J. BREDEHOEFT,” The conceptualization model problem-surprise”, Hydrogeol. J., 13, 37-46 

(2005). 

[15] R.L. KEENEY AND D. VON WINTERFELDT, “Eliciting probabilities from experts in complex 

technical problems”, IEEE Transactions On Engineering Management, 38 (3): 191-201 (1991). 

[16] M. YE, S. P. NEUMAN, P. D. MEYER, and K. POHLMANN (2005), “Sensitivity analysis and 

assessment of prior model probabilities in MLBMA with application to unsaturated fractured tuff”, Water 

Resour. Res., 41, W12429, doi:10.1029/2005WR004260. 
BIBLIOGRAPHIC DATA SHEET

(See instructions on the reverse)

NRC FORM 335

(9-2004)

NRCMD 3.7

U.S. NUCLEAR REGULATORY COMMISSION 1. REPORT NUMBER

 (Assigned by NRC, Add Vol., Supp., Rev.,

 and Addendum Numbers, if any.)

NUREG/CR-6940

PNNL-16396

3. DATE REPORT PUBLISHED

MONTH

July

YEAR

2007

4. FIN OR GRANT NUMBER

Y6465

2. TITLE AND SUBTITLE

Combined Estimation of Hydrogeologic Conceptual Model, Parameter, and Scenario Uncertainty

with Application to Uranium Transport at the Hanford Site 300 Area

5. AUTHOR(S)

P.D. Meyer (PNNL) , M.Ye (DRI), M.L. Rockhold (PNNL), S.P. Neuman (UA), and K.J. Cantrell

(PNNL)

(PNNL - Pacific Northwest National Laboratory, DRI - Desert Research Institute, UA - University

of Arizona)

6. TYPE OF REPORT

Technical

7. PERIOD COVERED (Inclusive Dates)

August 2002 - February 2007

8. PERFORMING ORGANIZATION - NAME AND ADDRESS (If NRC, provide Division, Office or Region, U.S. Nuclear Regulatory Commission, and mailing address; if contractor,

Pacific Northwest National Laboratory Desert Research Institute University of Arizona

9. SPONSORING ORGANIZATION - NAME AND ADDRESS (If NRC, type "Same as above"; if contractor, provide NRC Division, Office or Region, U.S. Nuclear Regulatory Commission, 

Division of Fuel, Engineering and Radiological Research

provide name and mailing address.)

and mailing address.)

10. SUPPLEMENTARY NOTES

T.J. Nicholson, NRC Project Manager

11. ABSTRACT (200 words or less)

A methodology was developed and applied to systematically and quantitatively assess predictive uncertainty in ground-water

flow and transport modeling. The methodology considers the combined impact of hydrogeologic uncertainties associated with the

conceptual-mathematical basis of a model, model parameters, and the scenario to which the model is applied. The methodology

is based on an extension of a Maximum Likelihood implementation of Bayesian Model Averaging. Model uncertainty is

represented by postulating a discrete set of alternative conceptual models for a site with associated prior model probabilities.

The prior model probabilities reflect a subjective belief about the relative plausibility of each model based on its apparent

consistency with available knowledge and data. Posterior model probabilities are computed and parameter uncertainty is

estimated by calibrating each model to observed system behavior. Scenario uncertainty is represented as a discrete set of

alternative future conditions affecting boundary conditions, source/sink terms, or other aspects of the models. A joint

assessment of uncertainty results from combining model predictions computed under each scenario using as weights, the

posterior model and prior scenario probabilities. The uncertainty methodology was applied to modeling of ground-water flow and

uranium transport at the Hanford Site 300-Area. Results demonstrate the feasibility of applying a comprehensive uncertainty

assessment to large-scale, detailed ground-water flow and transport modeling. Results also illustrate the ability of the

methodology to provide better estimates of predictive uncertainty, quantitative results for use in assessing risk, and an improved

understanding of the system behavior and the limitations of the models.

12. KEY WORDS/DESCRIPTORS (List words or phrases that will assist researchers in locating the report.)

Bayesian Model Averaging

dose assessment

ground-water modeling

hydrogeologic conceptual model

model calibration

model uncertainty

parameter uncertainty

scenario uncertainty

uncertainty

14. SECURITY CLASSIFICATION

13. AVAILABILITY STATEMENT

unlimited

(This Page)

unclassified

(This Report)

unclassified

15. NUMBER OF PAGES

16. PRICE

NRC FORM 335 (9-2004) PRINTED ON RECYCLED PAPER

P.O. Box 999 Las Vegas, NV 89119 Tucson, AZ 85721

Richland, WA 99352

Office of Nuclear Regulatory Research

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