Matthew J. Tonkin

A hybrid regularized inversion methodology for highly parameterized environmental models

[1] A hybrid approach to the regularized inversion of highly parameterized environmental models is described. The method is based on constructing a highly parameterized base model, calculating base parameter sensitivities, and decomposing the base parameter normal matrix into eigenvectors representing principal orthogonal directions in parameter space. The decomposition is used to construct super parameters. Super parameters are factors by which principal eigenvectors of the base parameter normal matrix are multiplied in order to minimize a composite least squares objective function. These eigenvectors define orthogonal axes of a parameter subspace for which information is available from the calibration data. The coordinates of the solution are sought within this subspace. Super parameters are estimated using a regularized nonlinear Gauss-Marquardt-Levenberg scheme. Though super parameters are estimated, Tikhonov regularization constraints are imposed on base parameters. Tikhonov regularization mitigates over fitting and promotes the estimation of reasonable base parameters. Use of a large number of base parameters enables the inversion process to be receptive to the information content of the calibration data, including aspects pertaining to small-scale parameter variations. Because the number of super parameters sustainable by the calibration data may be far less than the number of base parameters used to define the original problem, the computational burden for solution of the inverse problem is reduced. The hybrid methodology is described and applied to a simple synthetic groundwater flow model. It is then applied to a real-world groundwater flow and contaminant transport model. The approach and programs described are applicable to a range of modeling disciplines. Copyright 2005 by the American Geophysical Union.

Calibration‐constrained Monte Carlo analysis of highly parameterized models using subspace techniques

We describe a subspace Monte Carlo (SSMC) technique that reduces the burden of calibration-constrained Monte Carlo when undertaken with highly parameterized models. When Monte Carlo methods are used to evaluate the uncertainty in model outputs, ensuring that parameter realizations reproduce the calibration data requires many model runs to condition each realization. In the new SSMC approach, the model is first calibrated using a subspace regularization method, ideally the hybrid Tikhonov-TSVD superparameter"" approach described by Tonkin and Doherty (2005). Sensitivities calculated with the calibrated model are used to define the calibration null-space, which is spanned by parameter combinations that have no effect on simulated equivalents to available observations. Next, a stochastic parameter generator is used to produce parameter realizations, and for each a difference is formed between the stochastic parameters and the calibrated parameters. This difference is projected onto the calibration null-space and added to the calibrated parameters. If the model is no longer calibrated, parameter combinations that span the calibration solution space are reestimated while retaining the null-space projected parameter differences as additive values. The recalibration can often be undertaken using existing sensitivities, so that conditioning requires only a small number of model runs. Using synthetic and real-world model applications we demonstrate that the SSMC approach is general (it is not limited to any particular model or any particular parameterization scheme) and that it can rapidly produce a large number of conditioned parameter sets.

Efficient nonlinear predictive error variance for highly parameterized models

Predictive error variance analysis attempts to determine how wrong predictions made by a calibrated model may be. Predictive error variance analysis is usually undertaken following calibration using a small number of parameters defined through a priori parsimony. In contrast, we introduce a method for investigating the potential error in predictions made by highly parameterized models calibrated using regularized inversion. Vecchia and Cooley (1987) describe a method of predictive error variance analysis that is constrained by calibration data. We extend this approach to include constraints on parameters that lie within the calibration null space. These constraints are determined by dividing parameter space into combinations of parameters for which estimates can be obtained and those for which they cannot. This enables the contribution to predictive error variance from parameterization simplifications required to solve the inverse problem to be quantified, in addition to the contribution from measurement noise. We also describe a novel technique that restricts the analysis to a strategically defined predictive solution subspace, enabling an approximate predictive error variance analysis to be completed efficiently. The method is illustrated using a synthetic and a real-world groundwater flow and transport model.

Approaches to highly parameterized inversion: A guide to using PEST for model-parameter and predictive-uncertainty analysis

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Importance of considering intraborehole flow in solute transport modeling under highly dynamic flow conditions

Correct interpretation of tracer test data is critical for understanding transport processes in the subsurface. This task can be greatly complicated by the presence of intraborehole flows in a highly dynamic flow environment. At a new tracer test site (Hanford IFRC) a dynamic flow field created by changes in the stage of the adjacent Columbia River, coupled with a heterogeneous hydraulic conductivity distribution, leads to considerable variations in vertical hydraulic gradients. These variations, in turn, create intraborehole flows in fully-screened (6.5m) observation wells with frequently alternating upward and downward movement. This phenomenon, in conjunction with a highly permeable aquifer formation and small horizontal hydraulic gradients, makes modeling analysis and model calibration a formidable challenge. Groundwater head data alone were insufficient to define the flow model boundary conditions, and the movement of the tracer was highly sensitive to the dynamics of the flow field. This study shows that model calibration can be significantly improved by explicitly considering (a) dynamic flow model boundary conditions and (b) intraborehole flow. The findings from this study underscore the difficulties in interpreting tracer tests and understanding solute transport under highly dynamic flow conditions.

Approaches to the simulation of unconfined flow and perched groundwater flow in MODFLOW.

Various approaches have been proposed to manage the nonlinearities associated with the unconfined flow equation and to simulate perched groundwater conditions using the MODFLOW family of codes. The approaches comprise a variety of numerical techniques to prevent dry cells from becoming inactive and to achieve a stable solution focused on formulations of the unconfined, partially-saturated, groundwater flow equation. Keeping dry cells active avoids a discontinuous head solution which in turn improves the effectiveness of parameter estimation software that relies on continuous derivatives. Most approaches implement an upstream weighting of intercell conductance and Newton-Raphson linearization to obtain robust convergence. In this study, several published approaches were implemented in a stepwise manner into MODFLOW for comparative analysis. First, a comparative analysis of the methods is presented using synthetic examples that create convergence issues or difficulty in handling perched conditions with the more common dry-cell simulation capabilities of MODFLOW. Next, a field-scale three-dimensional simulation is presented to examine the stability and performance of the discussed approaches in larger, practical, simulation settings.

KT3D_H2O: a program for kriging water level data using hydrologic drift terms.

It is often necessary to estimate the zone of contribution to, or the capture zone developed by, pumped wells: for example, when evaluating pump-and-treat remedies and when developing wellhead protection areas for supply wells. Tonkin and Larson (2002) and Brochu and Marcotte (2003) describe a mapping-based method for estimating the capture zone of pumped wells, developed by combining universal kriging (kriging with a trend) with analytical expressions that describe the response of the potentiometric surface to certain applied stresses. This Methods Note describes (a) expansions to the technique described by Tonkin and Larson (2002); (b) the concept of the capture frequency map (CFM), a technique that combines information from multiple capture zone maps into a single depiction of capture; (c) the development of a graphical user interface to facilitate the use of the methods described; and (d) the integration of these programs within the MapWindow geographic information system environment. An example application is presented that illustrates ground water level contours, capture zones, and a CFM prepared using the methods and software described.

Approaches in highly parameterized inversion - GENIE, a general model-independent TCP/IP run manager

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Source Screening Module for Contaminant Transport Analysis Through Vadose and Saturated Zones

At complex sites there may be many potential sources of contaminants within the vadose zone. Screening-level analyses are useful to identify which potential source areas should be the focus of detailed investigation and analysis. A source screening module (SSM) has been developed to support preliminary evaluation of the threat posed by vadose zone waste sites on groundwater quality. This tool implements analytical solutions to simulate contaminant transport through the unsaturated and saturated zones to predict time-varying concentrations at potential groundwater receptors. The SSM integrates several transport processes in a single simulation that is implemented within a user-friendly, Microsoft Excel™ - based interface.