Abstract Alluvial systems are dynamic, structurally complex environments. Groundwater that is extracted from these systems is widely used for domestic consumption. Numerical models play an important role in the management of this extraction. To provide support for decision‐making, models must assimilate data that inform the spatial and temporal dynamics of the interactions between alluvial surface water (SW) and groundwater (GW). In theory, assimilation of noble gas tracer data into numerical models that solve the advection‐dispersion equation as it pertains to gas tracer concentrations can reduce the uncertainties of management‐salient predictions made by these models. To date, however, this has not been undertaken; nor has its decision‐support efficacy been tested. A modeling framework is introduced whereby data space inversion, a highly efficient method for evaluation of post‐data‐assimilation uncertainties of complex simulators, is used to evaluate benefits accrued through assimilation of noble gas tracer concentrations (i.e., 222Rn; 37Ar; 4He) along with more classical observations. The ability of noble gas concentrations to reduce the uncertainties of predictions made using an integrated surface‐subsurface hydrological model is rigorously evaluated in simulation experiments that span a range of plausible hydraulic and hydrological conditions. Outcomes of these experiments demonstrate that measurements of 222Rn and 4He concentrations are able to constrain temporally‐variable predictions, such as the magnitudes of SW‐GW exchange fluxes, as well as solute breakthrough curves that reflect the residence times of potential pollutants following an aquifer contamination event. 37Ar concentrations can also provide important information on SW‐GW interactions, especially the residence times of bank filtrate.

Integrity of simulator-based Bayesian analysis requires adequate representation of prior parameter probabilities, and quantification and reduction of posterior predictive uncertainties through history-matching. In many groundwater management contexts, hydrogeological complexity and long numerical model run times can render both of these tasks difficult. We present three new technologies that can make simulator-based Bayesian analysis that is undertaken in complex hydrogeological environments more effective and more tractable. These are demonstrated using a case study where groundwater head, streamflow and groundwater age data are assimilated in order to assess groundwater vulnerability to anthropomorphic deterioration of its quality. Bayesian analysis begins by generating ensembles of realizations of hydraulic property and other parameters used by a multi-layer groundwater model. The first technology supports this first step, by ensuring that respect for complex hydrogeology is embodied in nonstationary representations of hydraulic properties, as well as in stochasticity of so-called “hyperparameters” which govern their spatially variable geostatistics. The second and third technologies support data assimilation in two different ways, both of which are numerically cheap. One of these options, Ensemble Space Inversion (ENSI) requires adjustment of parameter fields in order for model outputs to match field measurements. The other option, Data Space Inversion (DSI) avoids parameter field adjustment through construction of direct statistical linkages between model-generated counterparts to field measurements and groundwater predictions of management interest. This statistical model is then history-matched in lieu of the numerical model. Deployment of both of these strategies at our case study site yields similar results. They reveal the likely existence of young water at depth over large parts of a regional aquifer system. This has repercussions for the quality of extracted water, and for land management in recharge areas.

Practical application of some new technologies, and the light they shed on model design and model problem decompositionNew robust efficient modelling technologies are available to quantify the uncertainties associated with model predictions. We adopt a decision support modelling framework which uses a combination of two of these new technologies, Data Space Inversion and Ensemble Space Inversion. Using this framework helps answer model design and deployment questions that are critical for a specified decision. These questions include:What contributes to the uncertainty of what could go wrong with this decision? Where is the information that may reduce this uncertainty? How can this information be best harvested – what model structure, parameterisation, observation weighting strategy, and technologies are most appropriate? How are the consequences of information insufficiency best expressed? We demonstrate how this modelling framework reveals the predictive accuracy costs of over-fitting to some types of data. We also identify for a specific prediction which alternative model structures and inversion methods are more appropriate given alternative data sets.

Groundwater modelers frequently grapple with the challenge of integrating aquifer test interpretations into parameters used by regional models. This task is complicated by issues of upscaling, data assimilation, and the need to assign prior probability distributions to numerical model parameters in order to support model predictive uncertainty analysis. To address this, we introduce a new framework that bridges the significant scale differences between aquifer tests and regional models. This framework also accounts for loss of original datasets and the heterogeneous nature of geological media in which aquifer testing often takes place. Using a fine numerical grid, the aquifer test is reproduced in a way that allows stochastic representation of site hydraulic properties at an arbitrary level of complexity. Data space inversion is then used to endow regional model cells with upscaled, aquifer-test-constrained realizations of numerical model properties. An example application demonstrates that assimilation of historical pumping test interpretations in this manner can be done relatively quickly. Furthermore, the assimilation process has the potential to significantly influence the posterior means of decision-pertinent model predictions. However, for the examples that we discuss, posterior predictive uncertainties do not undergo significant reduction. These results highlight the need for further research.

Categorical parameter distributions consisting of geologic facies with distinct properties, for example, high-permeability channels embedded in a low-permeability matrix, are common at contaminated sites. At these sites, low-permeability facies store solute mass, acting as secondary sources to higher-permeability facies, sustaining concentrations for decades while increasing risk and cleanup costs. Parameter estimation is difficult in such systems because the discontinuities in the parameter space hinder the inverse problem. This paper presents a novel approach based on Traveling Pilot Points (TRIPS) and an iterative ensemble smoother (IES) to solve the categorical inverse problem. Groundwater flow and solute transport in a hypothetical aquifer with a categorical parameter distribution are simulated using MODFLOW 6. Heads and concentrations are recorded at multiple monitoring locations. IES is used to generate posterior ensembles assuming a TRIPS prior and an approximate multi-Gaussian prior. The ensembles are used to predict solute concentrations and mass into the future. The evaluation also includes an assessment of how the number of measurements and the choice of the geological prior determine the characteristics of the posterior ensemble and the resulting predictions. The results indicate that IES was able to efficiently sample the posterior distribution and showed that even with an approximate geological prior, a high degree of parameterization and history matching could lead to parameter ensembles that can be useful for making certain types of predictions (heads, concentrations). However, the approximate geological prior was insufficient for predicting mass. The analysis demonstrates how decision-makers can quantify uncertainty and make informed decisions with an ensemble-based approach.

Hydrological numerical modelling is generally designed to provide predictions of uncertain quantities in a decision-support context. In the implementation of decision-support modelling, data assimilation and uncertainty quantification are often the most difficult and time-consuming tasks. This is because the imposition of history-matching constraints on model parameters usually requires a large number of model runs. Data Space Inversion (DSI) provides an alternative (and highly model-run-efficient) method for predictive uncertainty quantification that avoids the need for parameter estimation. It does this by evaluating covariances between model outputs used for history matching (e.g. hydraulic heads) and model predictions based on model runs that sample the prior parameter probability distribution. By focusing on the direct relationship between model outputs under historical conditions and predictions of system behaviour under future conditions, DSI avoids the need to estimate or adjust model parameters. This is advantageous when using such as Integrated Surface and Subsurface Hydrologic Models (ISSHMs). These models are characterised by long run times, a penchant for numerical instability and/or complex parameterisation schemes that are designed to maintain geological realism. This paper demonstrates that DSI provides a robust and efficient means of quantifying the uncertainties of complex model predictions, at the same time as it provides a basis for complementary linear analyses that can explore issues such as data worth. DSI is applied in conjunction with an ISSHM representing a synthetic but realistic stream-aquifer system. Predictions of interest are fast travel times and surface water infiltration. Linear and nonlinear estimates of prediction uncertainty based on DSI are validated against a more traditional approach to prediction uncertainty quantification which requires adjustment of a large number of parameters. A DSI-generated surrogate model is then used to investigate the effectiveness and efficiency of existing and possible future monitoring networks. This demonstrates the benefits of using DSI in conjunction with a complex numerical model to quantify prediction uncertainty and support data worth analysis in complex hydrogeological environments.

Processing of aquifer test drawdowns to obtain estimates of transmissivity, and sometimes storativity, is an integral part of hydrogeological site investigations. Analysis of these data often relies on an assumption of hydraulic property uniformity. Aquifer properties are often estimated by fitting a Theis curve to measured drawdowns. Where an aquifer exhibits heterogeneity, quantities that are forthcoming from such analyses are assumed to represent spatially-averaged properties. However the nature of the averaging process, and the area over which averaging takes place are unknown. In this study we derive spatial averaging functions that link inferred hydraulic properties to real-world hydraulic properties. These functions employ Fréchet integrals derived by previous investigators that link observation well drawdowns to aquifer properties under an assumption of mild aquifer heterogeneity. It is shown that these hydraulic property spatial averaging functions are complex, especially at times that immediately follow the commencement of pumping. Furthermore, they cross hydraulic property boundaries, so that estimates of storativity can be contaminated by heterogeneities in real-world transmissivity, and vice versa . Because of its greater averaging area at later times, estimates of transmissivity are generally more immune to the effects of local hydraulic property heterogeneity than are those of storativity. They are therefore more reflective of broadscale real-world hydraulic properties, particularly those that prevail in areas that are removed from the immediate vicinity of the pumping and observation wells.

This paper describes a methodology for undertaking probabilistic investigations into the locations at which contaminants have leaked into a groundwater system. The methodology is built with highly parameterized, stochastic history‐matching in mind. It is able to reduce uncertainties associated with estimates of subsurface hydraulic properties at the same time as it reduces uncertainties associated with inferences of contaminant sources. Particles are used to simulate contaminant movement. This reduces simulator execution time while increasing simulator stability. Borehole measurements of groundwater chemistry are endowed with a binary classification that indicates the presence, or otherwise, of a contaminant plume. This classification is transferred to passing particles as a detect or nondetect status awarded to their trajectories. Because this status is continuous with respect to model parameters, the latter can be adjusted in order to ensure that the same trajectory cannot possess both a detect status and a nondetect status. Particle trajectory statuses can be assigned to model cells from which they are released. By calculating cell statistics using a large number of history‐match‐constrained, stochastic parameter fields, probability maps can be drawn. We illustrate two of these. The first maps the probability that a contaminant sourced at a particular location will go undetected by the current observation network. The second maps the probability that a contaminant source cannot exist at a particular location. The method is extended to examine the worth of supplementing an existing observation network with new wells.

Abstract. It is incumbent on decision-support hydrological modelling to make predictions of uncertain quantities in a decision-support context. In implementing decision-support modelling, data assimilation and uncertainty quantification are often the most difficult and time-consuming tasks. This is because the imposition of history-matching constraints on model parameters usually requires a large number of model runs. Data space inversion (DSI) provides a highly model-run-efficient method for predictive uncertainty quantification. It does this by evaluating covariances between model outputs used for history matching (e.g. hydraulic heads) and model predictions based on model runs that sample the prior parameter probability distribution. By directly focusing on the relationship between model outputs under historical conditions and predictions of system behaviour under future conditions, DSI avoids the need to estimate or adjust model parameters. This is advantageous when using integrated surface and sub-surface hydrologic models (ISSHMs) because these models are associated with long run times, numerical instability and ideally complex parameterization schemes that are designed to respect geological realism. This paper demonstrates that DSI provides a robust and efficient means of quantifying the uncertainties of complex model predictions. At the same time, DSI provides a basis for complementary linear analysis that allows the worth of available observations to be explored, as well as of observations which are yet to be acquired. This allows for the design of highly efficient, future data acquisition campaigns. DSI is applied in conjunction with an ISSHM representing a synthetic but realistic river–aquifer system. Predictions of interest are fast travel times and surface water infiltration. Linear and non-linear estimates of predictive uncertainty based on DSI are validated against a more traditional uncertainty quantification which requires the adjustment of a large number of parameters. A DSI-generated surrogate model is then used to investigate the effectiveness and efficiency of existing and possible future monitoring networks. The example demonstrates the benefits of using DSI in conjunction with a complex numerical model to quantify predictive uncertainty and support data worth analysis in complex hydrogeological environments.

We present an efficient method for the detection of buried paleo-channels in alluvial sand and gravel (ASG) aquifers. Buried paleo-channels in ASG aquifers are typically highly conductive for groundwater flow and responsible for preferential flow paths capable of transporting contaminants faster than the surrounding sediments. However, it is notoriously difficult to detect and reproduce buried paleo-channel like connected structures of increased hydraulic conductivity with commonly used techniques. Consequently, these anisotropic structures are rarely considered in the delineation of wellhead protection zones and the models used for groundwater management. To alleviate this problem, we propose a new framework based on a combination of hydraulic and tracer-based measurements and subsequent calibration of a fully coupled surface water-groundwater model against these observations. Tracer-based observations consist of radioactive tracers ( 222 Rn, 37 Ar, 3 H/ 3 He), which allow characterization of groundwater residence times, and of atmospheric noble gases, which allow tracking recently infiltrated river water through an ASG aquifer. Calibration is based on an innovative pilot point inversion that considers the spatially-varying directionality of the alluvial sediments and facilitates the identification of buried paleo-channel like connected structures. The proposed approach is more efficient compared to other existing methods for paleo-channel detection, as complex and often poorly constrained geostatistical simulations are not required. The applicability framework is demonstrated on a real-world wellfield in an ASG aquifer in Switzerland.

The primary tasks of decision-support modelling are to quantify and reduce the uncertainties of decision-critical model predictions. Reduction of predictive uncertainty requires assimilation of information. Generally, this information resides in two places: 1) expert knowledge emerging from site characterization and 2) field measurements of present and historical system behavior. The former is uncertain and should therefore be expressed stochastically in a model. The range of parameter and predictive possibilities can then be constrained through history-matching. Implementation of these Bayesian principles places conflicting demands on the level of model structural complexity. A high level of structural complexity can facilitate expression of expert knowledge by establishing model details that are recognizable by site experts, and through supporting model parameters that bear a close relationship to real-world hydraulic properties. However, such models often run slowly and are numerically delicate; history-matching therefore becomes difficult or impossible. In contrast, if endowed with enough parameters, structurally simple models facilitate the achievement of a good fit between model outputs and field measurements. However, the values with which parameters are endowed may bear a looser relationship with real-world properties and are therefore less receptive to information born of expert knowledge. The model design process is therefore one of compromise. In this paper we describe a methodology that reduces the cost of compromise by allowing expert knowledge of system properties to inform the parameters of a structurally simple model. The methodology requires the use of a complementary model of strategic, but not excessive, structural complexity that is stochastic, fast-running and requires no history-matching. We demonstrate the approach using a real-world case in which modelling is used to support management of a stressed coastal aquifer. We empirically validate the approach using a synthetic model.

Abstract Alluvial sand and gravel (ASG) aquifers are highly heterogeneous and exhibit strong, spatially variable anisotropy, often interspersed by buried paleo‐channels of increased hydraulic conductivity. Groundwater flow and solute transport is often characterized by preferential flow caused by anisotropic properties in ASG aquifers. Connected ASG subsurface structures such as buried paleo‐channels, however, are difficult to reproduce with commonly used techniques, and anisotropy is rarely considered in applied groundwater models. To ease the notoriously difficult problem of how to consider anisotropy, we propose a novel modeling framework based on calibration of an integrated surface‐subsurface hydrological model via spatially varying, preferred anisotropy pilot point inversion. The inversion leverages hydraulic and tracer‐based observations representing multiple spatial and temporal scales. We demonstrate the applicability of the framework on a real‐world ASG site used for drinking water production, and we quantify the information content of observations to identify connected paleo‐channels and provide guidance for optimal field‐data acquisition.

We present case studies on two methods for quantifying uncertainty in predictions for highly parameterised geothermal models. One method is fully linear, while the other is nonlinear as it requires sampling model parameters and additional simulations but can handle non-differentiable quantities of interest. We test both methods on a Leyte (Philippines) model and find similar results, though the linear method is computationally cheaper. We then consider models for BacMan (Philippines) and Ohaaki (New Zealand). Here the quantity of interest does not satisfy smoothness requirements, and we apply only the simulation-based method. This method achieves reasonable results but at the cost of many rejected samples.

Management of groundwater resources is increasingly reliant on numerical simulation. Unfortunately, decision-support modelling is often conducted under the premise that predictive reliability increases with modelling complexity. In truth, while modelling complexity can support quantification of predictive uncertainty, the latter is a function of data availability. Excessive complexity can often erode, rather than enhance, a model’s ability to quantify and reduce the uncertainties of decision-critical predictions by reducing its capacity to assimilate prediction-salient information. We submit that a groundwater model is more productively viewed as a data assimilator for decision-pertinent information than as simulating subsurface processes, even though the latter role (though imperfect) underpins the former. Assimilated data may, or may not, allow rejection of the hypothesis that a certain course of management action will have adverse consequences. Either way, the decision-making process requires that this hypothesis be tested. In the following document, we outline how decision-support environmental modelling can be implemented with the scientific method, and discuss how uncertainties of decision-salient predictions can be addressed with appropriate model complexity so that stakeholder expectations are better aligned with what models can and cannot deliver to the decision-making process.

This paper explores the adequacy of steady-state-only calibration as a precursor to use of a groundwater model for decision-support. First, it reviews metrics by which a decision-support model should be judged. On the basis of these metrics, it establishes the shortcomings that a decision-support model may incur through foregoing transient calibration. These are 1) failure to reduce the uncertainties of management-salient model predictions to the extent that available data allows, and 2) creation of unquantifiable bias in management-salient predictions. Two methodologies for quantification of these deficiencies are proposed. The first of these addresses uncertainty reduction. This is relatively easy to implement, as it requires only that sensitivities of pertinent model outputs to a model's parameters be calculated. The second methodology addresses predictive bias. Implementation of this second methodology is more expensive as it requires repeated calibration of a steady state model against stochastic realizations of a transient model.These methods are demonstrated using a synthetic case which explores the viability of steady-state-only calibration of models deployed to examine the impacts of pumping on stream flows and groundwater levels. It is demonstrated that, for some predictions of management interest, steady-state-only calibration is more than sufficient for this kind of decision-support modelling.

© Flinders University 2021. Copyright: This work is copyright. Apart from any use permitted under the Copyright Act 1968, no part may be reproduced by any process, nor may any other exclusive rights be exercised, without the permission of Flinders University, GPO Box 2100, Adelaide 5001, South Australia.

Realistic environmental models used for decision making typically require a highly parameterized approach. Calibration of such models is computationally intensive because widely used parameter estimation approaches require individual forward runs for each parameter adjusted. These runs construct a parameter-to-observation sensitivity, or Jacobian, matrix used to develop candidate parameter upgrades. Parameter estimation algorithms are also commonly adversely affected by numerical noise in the calculated sensitivities within the Jacobian matrix, which can result in unnecessary parameter estimation iterations and less model-to-measurement fit. Ideally, approaches to reduce the computational burden of parameter estimation will also increase the signal-to-noise ratio related to observations influential to the parameter estimation even as the number of forward runs decrease. In this work a simultaneous increments, an iterative ensemble smoother (IES), and a randomized Jacobian approach were compared to a traditional approach that uses a full Jacobian matrix. All approaches were applied to the same model developed for decision making in the Mississippi Alluvial Plain, USA. Both the IES and randomized Jacobian approach achieved a desirable fit and similar parameter fields in many fewer forward runs than the traditional approach; in both cases the fit was obtained in fewer runs than the number of adjustable parameters. The simultaneous increments approach did not perform as well as the other methods due to inability to overcome suboptimal dropping of parameter sensitivities. This work indicates that use of highly efficient algorithms can greatly speed parameter estimation, which in turn increases calibration vetting and utility of realistic models used for decision making.

First posted December 22, 2020 For additional information, contact: Director, Upper Midwest Water Science CenterU.S. Geological Survey8505 Research WayMiddleton, WI 53562 PEST++ Version 5 extends and enhances the functionality of the PEST++ Version 3 software suite, providing environmental modeling practitioners access to updated Version 3 tools as well as new tools to support decision making with environmental models. Version 5 of PEST++ includes tools for global sensitivity analysis (PESTPP-SEN); least-squares parameter estimation with integrated first-order, second-moment parameter and forecast uncertainty estimation (PESTPP-GLM); an iterative, localized ensemble smoother (PESTPP-IES); and a tool for management optimization under uncertainty (PESTPP-OPT). Additionally, all PEST++ Version 5 tools have a built-in fault-tolerant, multithreaded parallel run manager and are model independent, using the same protocol as the widely used PEST software suite.PEST++ Version 5 is consistent with PEST++ Version 3 conventions and design philosophy. The software’s emphasis continues to target efficient and optimized algorithms that have proven beneficial in decision-support settings and can accommodate large, highly parameterized problems. Expanded and new capabilities are now available to express uncertainty using Monte Carlo and analytical uncertainty approaches and allow evaluation of thousands to millions of parameters. New management optimization capabilities in Version 5 also allow environmental models to be used to answer management questions using multiple societal constraints in a risk-based framework.The PEST++ Version 5 software suite can be compiled for Microsoft Windows® and Unix-based operating systems such as Apple and Linux®; the source code is available with a Microsoft Visual Studio® 2019 solution; and CMake support for all three operating system is also provided. PEST++ Version 5 continues to build a foundation for an open-source framework capable of producing model-independent, robust, and efficient decision-support tools for large environmental models. The functionality of each of the PEST++ tools are demonstrated on a simple example problem. Implications of decisions used when using the PEST++ suite tools are also discussed.

Categorical parameter distributions are common-place in hydrogeological systems consisting of geologic facies/categories with distinct properties, e.g., high-permeability channels embedded in a low-permeability matrix. Parameter estimation is difficult in such systems because the discontinuities in the parameter space hinder the inverse problem. Previous research in this area has been focused on the use of stochastic methods. In this paper, we present a novel approach based on Traveling Pilot points (TRIPS) combined with subspace parameter estimation methods to generate realistic categorical parameter distributions that honor calibration constraints (e.g., - measured water levels). In traditional implementations, aquifer properties (e.g., hydraulic conductivity) are estimated at fixed pilot point locations. In the TRIPS implementation, both the properties associated with the pilot points and their locations are estimated. Tikhonov regularization constraints are incorporated in the parameter estimation process to produce realistic parameter depictions. For a synthetic aquifer system, we solved the categorical inverse problem by combining the TRIPS methodology with two subspace methods: Null Space Monte Carlo (NSMC) and Posterior Covariance (PC). A posterior ensemble developed with the rejection sampling (RS) method is compared against the TRIPS ensembles. The comparisons indicated similarities between the various ensembles and to the reference parameter distribution. Between the two subspace methods, the NSMC method produced an ensemble with more variability than the PC method. These preliminary results suggest that the TRIPS methodology has promise and could be tested on more complicated problems.

We present a framework for design and deployment of decision support modeling based on metrics which have their roots in the scientific method. Application of these metrics to decision support modeling requires recognition of the importance of data assimilation and predictive uncertainty quantification in this type of modeling. The difficulties of implementing these procedures depend on the relationship between data that is available for assimilation and the nature of the prediction(s) that a decision support model is required to make. Three different data/prediction contexts are identified. Unfortunately, groundwater modeling is generally aligned with the most difficult of these. It is suggested that these difficulties can generally be ameliorated through appropriate model design. This design requires strategic abstraction of parameters and processes in a way that is optimal for the making of one particular prediction but is not necessarily optimal for the making of another. It is further suggested that the focus of decision support modeling should be on the ability of a model to provide receptacles for decision-pertinent information rather than on its purported ability to simulate environmental processes. While models are compromised in both of these roles, this view makes it clear that simulation should serve data assimilation and not the other way around. Data assimilation enables the uncertainties of decision-critical model predictions to be quantified and maybe reduced. Decision support modeling requires this.