MODFLOW and More 2008: Ground Water and Public Policy - Conference Proceedings, Poeter, Hill, & Zheng - www.mines.edu/igwmc/
Parameter Estimation in MODFLOW Using a Bayesian Inverse Approach 1
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Michael Fienen , Tom Clemo , Randall J. Hunt U.S. Geological Survey,
[email protected],
[email protected], Middleton, Wisconsin, USA 2 CGISS, Boise State University,
[email protected], Boise, Idaho, USA ABSTRACT
Bayesian geostatistical inverse modeling is rooted in probability theory and provides the most probable, minimum variance parameter set in addition to the best fit parameter set, given a set of field observations and a conceptual model. Posterior covariance, a measure of uncertainty in the estimated parameter field, can also be calculated. Tractability is obtained in a probabilistic framework through imposition of general prior knowledge rather than the reduction of the domain into homogeneous zones. As a result, many parameter values must be estimated. Major roadblocks limiting the application of Bayesian parameter estimation in MODFLOW include the computational expense of calculating sensitivities for many parameters and lack of a general purpose tool. Adjoint-state versions of MODFLOW-2005 and MT3DMS, allowing efficient calculation of parameter sensitivity values, remove the first roadblock. This work addresses the second roadblock by presenting an integration of the Bayesian inverse approach into the parameter estimation software PEST. This implementation will use the JUPITER application programming interface (API). The implementation is described, and the method is illustrated using field data collected by the USGS Wisconsin Water Science Center and the UW-Madison Center for Limnology in the Trout Lake basin of northern Wisconsin. OVERVIEW OF THE BAYESIAN GEOSTATISTICAL INVERSE METHOD The core of the Bayesian geostatistical inverse method is Bayes’ theorem where
is the prior probability density function (pdf) of the parameters s (what we think the
parameters should be prior to the experiment), tell us), and
is the likelihood function (what the observations
is the posterior pdf of s given the observations y (what we believe the parameters
should be after considering the observations). In groundwater modeling, y is typically a vector of head or concentration data (or both), and s is a vector of hydraulic conductivity, porosity, storage, or dispersivity values. This is a maximum likelihood method meaning that the parameter set that maximizes the posterior pdf is the best estimate; the most probable answer given the data and conceptual model. The posterior covariance can be calculated and provides confidence intervals about the best estimate. Equally likely realizations may be generated that honor the statistical properties of . These distributions are multi-Gaussian, characterized through their first and second moments; mean and covariance. The prior pdf,
incorporates knowledge about the parameters that exists prior to the collection of
measurements y. In the geostatistical approach, a variogram model is chosen that imparts an appropriate characteristic such as smoothness or continuity on the distributed parameters. Intrinsic variability of the medium, an irreducible source of uncertainty, is accounted for in the covariance model for the prior pdf. A prior estimate of the mean and uncertainty of that estimate can also be included but is not required. The likelihood function,
, expresses data misfit. This is the degree to which model results, based
on a specific estimate of s, agree with observations. We do not expect y to be exactly correct because of epistemic uncertainty which accounts for measurement error, measurement paucity, conceptual model
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MODFLOW and More 2008: Ground Water and Public Policy - Conference Proceedings, Poeter, Hill, & Zheng - www.mines.edu/igwmc/
error, round-off error, and other sources of uncertainty related to the framing of the problem. These errors are reducible to a limited extent by addressing the specific sources of uncertainty, but it is never possible eliminate them so an epistemic error term is incorporated into the covariance of the likelihood function. In groundwater applications, the solution for the best estimate, , is nonlinear so must be found through iterative co-Kriging equations, solved using Gauss-Newton, as discussed in Kitanidis (1995). The variogram parameters and the epistemic error term are structural parameters. Optimal structural parameters are found by restricted maximum likelihood (Kitanidis and Vomvoris, 1983) and they determine the balance between best fit of data reproduction and smoothness based on prior information. COMPUTATIONAL CONSIDERATIONS A critical element of parameter estimation with all gradient-based methods, including the Gauss-Newton solution used for this work, is the calculation of the Jacobian or sensitivity matrix. Determining the Jacobian matrix is the most computationally expensive part of most parameter estimation problems. Often, this matrix is calculated using either the sensitivity equation (Harbaugh et al., 2000), or finite differences, requiring either m+1 or 2(m+1) model runs where m is the number of parameters. This computational expense has impeded widespread adoption of Bayesian methods which assume a distributed approach with many parameters. Adjoint-state methods reformulate the sensitivity calculation problem such that n+1 model runs are required to calculate the Jacobian where n is the number of observations. In the Bayesian approach, usually m>>n so adjoint-state is vital. To this end, we employ adjoint-state versions of both MODFLOW-2005 (Harbaugh, 2005) and MT3DMS (Zheng and Wang, 1999) to efficiently calculate sensitivities for flow and transport problems (Clemo, 2007, Clemo in prep.). The availability of adjoint-state model removes a major roadblock that has discouraged widespread application of Bayesian techniques for groundwater parameter estimation using popular modeling programs. In the Gauss-Newton solution, a Levenberg-Marquardt formulation can provide a balance between the Gauss-Newton direction and steepest descent. This is useful to stabilize nonlinearity in the inverse problem. We adopt a modified Levenberg-Marquardt formulation (Nowak and Cirpka, 2004) which splits the problem into two steps; innovation and projection. Each step has its own Marquardt parameter (λ) that controls overshooting in the case of innovation, and solution deterioration in the case of projection. 2
As the number of parameters (m) grows, the size of the prior covariance matrix grows at the rate of m . Another major computational consideration is therefore the need for the efficient storage and manipulation of very large matrices. Methods based on Fourier and Karhunen-Loeve transformations have been derived for uniform grids (Nowak et al., 2003) and unstructured grids (Li and Cirpka, 2006), respectively. IMPLEMENTATION IN PEST Another roadblock that has limited widespread adoption of Bayesian inversion for groundwater problems is the lack of a general purpose tool. Many research codes have been developed for specific problems, but this work is the first attempt to create a Bayesian geostatistical inverse module in a widely available parameter estimation software package. PEST is a popular, model-independent inverse modeling code including regularization and other techniques for underdetermined problems (Doherty, 2007). The Bayesian geostatistical inverse method is a natural extension of PEST. Many practitioners are familiar with PEST which motivates incorporation into this existing code. The main inverse engine, while similar to the Levenberg-Marquardt inversion in PEST, can stand alone; the connection to PEST is largely through input-output conventions and tools, and other processing utilities. The module implementing the Bayesian inverse method into PEST has an interface compatible with JUPITER API protocols (Banta et al., 2006). The JUPITER protocols serve two purposes. First, they are compatible with other inverse codes such as UCODE_2005 (Poeter, et al., 2005) so many elements of input will be transferable among various inverse approaches. A second purpose is to allow for a hierarchy in parameter input that simplifies input and reduces input errors. PEST requires several variables specific to the inversion algorithm. Many users are not familiar with the details of these variables and default
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MODFLOW and More 2008: Ground Water and Public Policy - Conference Proceedings, Poeter, Hill, & Zheng - www.mines.edu/igwmc/
values are sufficient for the majority of cases. The Bayesian geostatistical module adds more variables where once again default values are usually appropriate. The JUPITER protocol enables the user to define only the variables for which non-default values are to be used. The protocol is flexible such that, after scanning the input for variable values, for all variables not indicated by the user, default values may be used. This allows users to focus on the problem-specific data and parameter information and to only consider adjusting the default variables in special cases. Sensitivity matrices in PEST are typically calculated using finite differences, although accommodation is made for sensitivities to be provided from outside of PEST. For MODFLOW and MT3D projects, the adjoint-state implementation discussed above will be crucial to success of the highly parameterized inverse models. For other models to which the Bayesian module is applied, the user must provide a method of calculating sensitivities for efficiency. In the case of short run times, or large parallel computing arrays, the existing PEST finite difference calculation of sensitivities can still be used. Input requirements are similar to typical PEST projects with some additional parameters required by the Bayesian module. Output options include the best estimate of the parameters, posterior covariance, and conditional realizations. The realizations, however, come at significant computational cost and are optional. A key feature of PEST is the ability to define interaction between model input and output and parameter estimation input and output through template and instruction files (see Doherty (2007) for details). This functionality is used without modification in this implementation. TROUT LAKE ISTHMUS PROJECT ILLUSTRATION The method described in this work was evaluated on field data collected by the USGS Wisconsin Water Science Center and the UW-Madison Center for Limnology in the Trout Lake basin of northern Wisconsin (Figure 1). A groundwater flow and transport modeling project is focused on the isthmus between Big Muskellunge and Crystal Lakes. These two lakes are hydraulically connected and their stages tend to vary in concert so the flow regime through the groundwater of the isthmus is modeled as steady-state. Data Types and Availability Three sets of data are used for this project; 18 head, δ O concentrations and tritium 18 concentrations. The δ O concentrations, like the head data, are modeled as steady-state whereas tritium is modeled as transient. The head data were collected in a single sampling event in May 1999 and are representative of conditions over the time period from 1950-2008. The head data display a dramatic drop (about one meter) Figure 1: Location of Trout Lake Basin. over a spatial distance of one or two meters, indicating a relatively narrow band of very low hydraulic conductivity. This is consistent with the glacial depositional history and this information is incorporated into the inversion both by supplementing raw head observations with head-difference observations, and by assigning zones to segregate the extremely low hydraulic conductivity band from the surrounding sediments. Parameters are still free to vary within their zones, subject to the smoothness imparted by the prior information term, but correlation is censored across zonal boundaries. 18
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The stable δ O data indicate provenance with less negative values indicating depletion of O relative to 18 18 O. Depletion occurs during evaporation to the atmosphere; thus, higher δ O compositions indicate
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MODFLOW and More 2008: Ground Water and Public Policy - Conference Proceedings, Poeter, Hill, & Zheng - www.mines.edu/igwmc/
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water that enters the subsurface through a surface water body. More negative δ O values indicate water recharged terrestrially with little exposure to the atmosphere after falling as precipitation. The recurring seasonal nature of isotopic composition in precipitation and the observation of sharp contacts between 18 lake-derived versus terrestrially recharged water allows transport of δ O to be modeled in steady-state. MT3D does not provide a steady-state option, so the model is run to a large time until concentrations are stable when concentrations are reported. Once water loses direct contact with the atmosphere in the 16 18 groundwater, depletion of O ceases and δ O can be modeled as a conservative solute. Tritium data record the transient passage of the worldwide atmospheric pulses of tritium derived from aboveground nuclear testing that peaked in 1963. Three sampling events were performed at the isthmus in 1992, 1998, and 2006. The discrete history of the tritium source necessitates transient modeling and tritium is radiogenic so its reaction rate for decomposition must also be considered. Inverse Model Setup Inverse modeling to estimate hydraulic conductivity parameters at the isthmus was performed with a prototype of the code being implemented in PEST. A two-dimensional cross section model was created with the lakes represented by constant head boundaries. A buffer of homogeneous layers extended to a distant groundwater divide upstream from Crystal Lake and beyond Big Muskellunge Lake downstream. The boundaries and the homogeneous layers are not depicted in Figures 2 and 3. The distant boundaries are modeled as no-flow at the upgradient divide and as a downgradient general head boundary to simulate regional flow conditions. The bottom of the model is a no-flow boundary consistent with low hydraulic conductivity bedrock that underlies glacial sediments and the top of the model is unconfined. Vertical and horizontal hydraulic conductivity were estimated independently resulting in a total of 2,344 parameters to estimate with porosity and dispersivity held constant. The number of observations is 145 resulting in a highly underdetermined problem. The prior covariance structure selected was a linear variogram which should provide the smoothest solution consistent with the data. Zonal boundaries were enforced based on inspection of the head data that indicated the presence of a silt layer. Outside the isthmus and away from the observations, parameters for ten homogenous zones were also estimated. These zones were most relevant for transport as they control the rate of entry of solutes into the aquifer system and were estimated as discrete values. Across all zonal boundaries, zero correlation was assumed leaving them independent from one another. For the three distributed zones within the isthmus, distinct variogram parameters were estimated allowing the level of smoothness to vary among the zones.
Figure 2: Best estimate of vertical and horizontal hydraulic conductivity based on 18 head, δ O and tritium data.
Figure 3: Rejected intermediate estimate of vertical and 18 horizontal hydraulic conductivity based on head, δ O and tritium data. Note different scale than Figure 2.
Estimation Results 18
Parameter estimation was performed first only using head data, then adding δ O data, and finally incorporating tritium. We discuss the results incorporating all available data. Figure 2 shows the estimated hydraulic conductivity parameters. Remarkably, a homogeneous field is estimated despite the freedom for a potentially complex and variable answer, consistent with the maximum entropy property of the method.
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MODFLOW and More 2008: Ground Water and Public Policy - Conference Proceedings, Poeter, Hill, & Zheng - www.mines.edu/igwmc/
Horizontal hydraulic conductivity exceeds vertical in each zone as expected in a glacial outwash environment, and the anisotropy ratio in the shallowest zone is consistent with independent tracer test calculation (Kenoyer, 1988). Figure 3 shows an intermediate rejected solution with unrealistically high parameter values and more heterogeneity than the best estimate. The fit is nearly identical in these two cases, but the solution chosen by the algorithm is the simplest solution consistent with the data. CONCLUSIONS The Bayesian geostatistical inverse method is a valuable approach to obtain both a most probable and best fit parameter estimate for an inverse problem. The illustration on the Trout Lake isthmus model shows that, even when given the flexibility to provide a complex solution, the simplest solution consistent with the data is favored. This flexibility allows for a robust search of parameter space without overfitting. The major roadblocks to more widespread adoption of Bayesian geostatistical inversion were addressed. Computational expense due to sensitivities is mitigated through implementation of adjoint-state versions of MODFLOW-2005 and MT3D. Other computational impediments and the lack of a practical tool are addressed through the implementation into the popular parameter estimation software package PEST. REFERENCES Banta, E. R., Poeter, E. P., Doherty, J. E., and Hill, M. C. 2006. JUPITER: Joint Universal Parameter IdenTification and Evaluation of Reliability-- An application programming interface (API) for model analysis, Reston, VA, U.S. Geological Survey Techniques and Methods Book 6, Section E, Chapter 1: 268. Clemo, T., 2007. MODFLOW-2005 Groundwater Model - User Guide to the Adjoint State Based Sensitivity Process (ADJ), Technical Report, BSU CGISS 07-01, Boise State University. Clemo,T. (in preparation) Joint adjoint state sensitivity calculations for MT3DMS and MODFLOW-2005, Technical Report, Boise State University. Doherty, J., 2007. PEST, Model Independent Parameter Estimation, Updated User Manual: 5th Edition. Brisbane, Queensland Australia, Watermark Numerical Computing. Harbaugh, A. W., Banta, E.W., Hill, M.C., and McDonald, M. G., 2000. MODFLOW-2000. the U.S. Geological Survey Modular Ground-Water Model--User Guide to Modularization Concepts and the Ground-Water Flow Process, United States Geological Survey: Open File Report 00-92: 121. Harbaugh, A. W., 2005. MODFLOW-2005, the U.S. Geological Survey modular ground-water model -- the Ground-Water Flow Process, Reston, Virginia, U.S. Geological Survey Techniques and Methods: 6-A16: 256. Kenoyer, G. J., 1988. Tracer Test Analysis of Anisotropy in Hydraulic Conductivity of Granular Aquifers, Groundwater Monitoring and Remediation 8(3): 67-70. Kitanidis, P. K., 1995. Quasi-linear geostatistical theory for inversing, Water Resources Research 31(10): 2411-2419. Kitanidis, P. K. and Vomvoris, E.G., 1983. A Geostatistical Approach to the Inverse Problem in Groundwater Modeling (Steady State) and One-Dimensional Simulations, Water Resources Research 19(3): 677-690. Li, W. and Cirpka, O.A., 2006. Efficient geostatistical inverse methods for structured and unstructured grids, Water Resources Research 42(6): W06402. Nowak, W., Tenkleve, S., and Cirpka, O.A., 2003. Efficient computation of linearized cross-covariance and auto-covariance matrices of interdependent quantities, Mathematical Geology 35(1): 53-66. Nowak, W. and Cirpka, O.A., 2004. A modified Levenberg-Marquardt algorithm for quasi-linear geostatistical inversing, Advances in Water Resources 27(7): 737-750. Poeter, E. P., Hill, M.C., Banta, E.R., Mehl, S., and Christensen, S., 2005. UCODE_2005 and Six Other Computer Codes for Universal Sensitivity Analysis, Calibration, and Uncertainty Evaluation, U. S. Geological Survey Techniques and Methods 6-A11: 283. Zheng, C. and Wang, P.P., 1999. MT3DMS: A modular three-dimensional multispecies transport model for simulation of advection, dispersion, and chemical reactions of contaminants in groundwater systems; Documentation and user's guide. Vicksburg, MS, U.S. Army Engineer Research and Development Center No. SERDP-99-1.
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