Water Resour Manage (2012) 26:1513–1535 DOI 10.1007/s11269-011-9970-3
Value of Information as a Context-Specific Measure of Uncertainty in Groundwater Remediation Xiaoyi Liu · Jonghyun Lee · Peter K. Kitanidis · Jack Parker · Ungtae Kim
Received: 12 May 2011 / Accepted: 22 December 2011 / Published online: 14 January 2012 © Springer Science+Business Media B.V. 2012
Abstract The remediation of groundwater sites has been recognized as a difficult and expensive task for years. One of the challenges is that the success of remediation is usually contingent upon an appropriate level of characterization of the physical, chemical, and biological site properties. For example, thermal treatment cannot be economically applied if the location of a non-aqueous phase liquid (NAPL) source is unknown. Both characterization and remediation are expensive. Thus, efforts need to be prioritized and optimized taking effects of uncertainty into consideration. Traditional measures of uncertainty, such as variance and correlation coefficients, do not fully depict the significance of uncertainty. For example, a small error in a parameter to which performance is sensitive may affect the prospect for remediation success much more than a large error in a parameter that has minor influence. In this paper, we quantify uncertainty as the expected increase in the cost of achieving clean-up objectives that is associated with uncertainty in performance prediction models, i.e., the minimum expected cost attainable with the present state of uncertainty minus the expected cost achievable if uncertainty were fully or partially removed. This measure, a.k.a., the value of information (VOI), is context-specific, i.e., it is dependent on site conditions and remediation strategies as well as specific remediation objectives and unit costs. We consider clean-up objectives, cost formulations, and sensitivity of costs to uncertainty in parameters, measurements, and the model itself and seek to minimize expected cost under conditions of incomplete information. We present
X. Liu (B) · J. Lee · P. K. Kitanidis Department of Civil and Environmental Engineering, Stanford University, Stanford, CA, USA e-mail:
[email protected] Present Address: X. Liu Lawrence Berkeley National Laboratory, Earth Science Division, Berkeley, CA 94720, USA J. Parker · U. Kim Department of Civil and Environmental Engineering, University of Tennessee, Knoxville, TN, USA
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results from a synthetic case study of dense non-aqueous phase liquid (DNAPL) plume treatment. The results quantify the cost attributable to uncertainty, thus setting an upper limit on how much one should pay for characterization, and helping decision makers to decide whether the data should be collected or not. Keywords Groundwater remediation · Optimization · Value of information · Calibration · Uncertainty quantification 1 Introduction Groundwater remediation, especially remediation of DNAPL contaminated sites, is usually difficult and expensive, and a successful, cost-effective remediation relies on accurate site characterization. There are a number of ways to characterize a DNAPL site. For example, EPA (2004) summarized 45 current geophysical and nongeophysical DNAPL site characterization technologies; Powell and Silfer (2005) reviewed two in-situ NAPL characterization techniques, partition interwell tracer testing (PITT) and use of a rapid optical screening tool (ROST) with case studies; Rao et al. (2000) assessed several tracer techniques for NAPL source zone characterization with an emphasis on the reliability of those methods, and Griffin and Watson (2002) compared several field techniques for DNAPL site characterization and proposed a procedure for DNAPL identification and delineation. Among the available methods, some directly measure DNAPL saturation in the aquifer; some obtain indirect measurements that can be converted to DNAPL saturation; and the others do not generate DNAPL saturation measurements but quantities such as DNAPL flux out of the source zone. Other model-based methods use calibration (Basu et al. 2008; Christ et al. 2006; Newman et al. 2006; Parker et al. 2008) or inverse modeling (Dridi et al. 2009; Illman et al. 2010; Yeh and Zhu 2007; Zelt et al. 2006; Zhu et al. 2009) to tune the model and at the same time, to characterize the site. Indeed, the choice of characterization methods should depend on the goals of the study. For instance, if the purpose is to use an upscaled model (Christ et al. 2006; Falta et al. 2005a, b; Park and Parker 2005; Parker et al. 2008) to predict the DNAPL remaining in the source zone and dissolved contaminant concentrations in the aquifer, deriving a high-resolution 3D distribution of DNAPL in the source zone is not necessary. In general, accurate characterization of DNAPL site is a difficult task for scientists and engineers. For example, Hayden et al. (2006) showed that even low grade heterogeneity such as simple aquifer layering could significantly complicate source zone characterization. Thus, no matter what kind of characterization technology is used, uncertainty is always an important factor that needs to be carefully considered (Abriola 2005). Consequently, a number of methods have also been developed for remediation optimization under uncertainty (Gorelick 1990; Mantoglou and Kourakos 2007; Mylopoulos et al. 1999). As an unavoidable component in the analysis of environmental problems and decision making of environmental restoration activities, uncertainty is usually quantified with variance or higher moments of unknown parameters. However, effects of parameter variance or other moments is context non-specific, meaning that it is not affected or related to the specific use of the parameter and it does not depict
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the severity of the prediction uncertainty for a specific problem. In fact, parameter uncertainty has no value by itself and “no theory that involves just the probabilities of outcomes without considering their consequences could possibly be adequate in describing the importance of uncertainty to a decision maker” (Howard 1966). For example, a small error in a parameter (hydraulic conductivity) to which performance (contaminant removed) is sensitive may affect the prospect for remediation success (pump-and-treat) much more than a large error in an insensitive parameter (porosity). Further, in environmental decision processes, it is difficult to judge how much better one strategy is than another based only on statistical moments. This issue is of particular interest in data collection for environmental problems, especially in the subsurface where measurements are difficult and expensive to acquire. Before one starts collecting measurements, a cost-benefit analysis is required to justify the cost of collection. New measurements reduce the variance of parameters by conditioning; however, quantification of reduction in variance is not enough to answer the question whether measurements should be collected or not. To answer this question, we need to estimate the worth of the information content in data, i.e., the value of information (VOI). In general terms, VOI is the difference between the cost when specified information is not used and that when the information is used. Theories on VOI have been discussed in the literature (Gould 1974; Howard 1966; McCarthy 1956) and expanded to many fields such as medical sciences (Ades et al. 2004; Felli and Hazen 1998), economics (Hanemann 1989), and operations research (Cachon and Fisher 2000; Gavirneni et al. 1999). Howard (1966) shows that a numeric value can be assigned to the reduction of uncertainty and establishes a theory on the value of information. However, as Gould (1974) points out, the value of information does not necessarily increase as the risk of uncertainty increases. In fact, VOI is application dependent (Stephens 1989) and is highly contingent upon the form of the specific cost/gain function. In other words, it is use-specific, or context-specific. In rare cases, information is perfect, i.e., all the uncertainty has been removed and the outcomes are predictable. In most situations, however, information is imperfect. For example, the weather forecasts do not give exact predictions about the weather tomorrow, but there is a certain probability that the forecast is wrong. In environmental and earth science problems where measurements usually contain a significant amount of noise, most of the information we have is imperfect. Yokota and Thompson (2004) and Bratvold et al. (2009) provide detailed reviews on application of VOI in environmental health risk management and the oil industry, respectively, and Dawdy (1979) presents an early review on the worth of hydrologic data for network design. Indeed, there are also a number of attempts in assessing the value of information in hydroscience. For example, Slack et al. (1975) and Klemeš (1977) use VOI in flood analysis and reservoir optimization. Ben-Zvi et al. (1988), Borisova et al. (2005), Feyen and Gorelick (2005), James and Gorelick (1994), Reichard and Evans (1989), and Wagner et al. (1992) applied VOI in contaminant remediation and groundwater management. However, in general, these applications are incomplete in at least one of the following aspects. 1) simplified cost functions are assumed and none or limited optimization and decisions are included; 2) the definition of uncertainty is simplified, meaning no characterization is included and known distributions are used for unknown parameters, or no correlation between multiple parameters is considered; 3) the information evaluated is either practically
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unavailable or only the value of perfect information is assessed; 4) measures such as statistical moments not pertinent to a specific cost function are used. Due to these limits, it is difficult to expand those practices to more general applications. In addition, there is a lack of comprehensive discussion on the application of VOI in practical environment and earth science problems, where perfect information is generally unavailable; the system state is usually difficult to accurately characterize; and nonlinearity prevails in forward simulation, cost function, and regulatory constraints. All these add difficulties to the problem. In this paper, we first review the definition of VOI and then analyze its properties with one-dimensional and multi-dimensional quadratic cost functions. Then we propose a general and flexible Monte Carlo framework for evaluation of a contextspecific measure of uncertainty, VOI, and we apply it in the context of groundwater remediation. This framework does not introduce additional assumptions on the forward model, characterization scheme, or cost function, and can be easily integrated into high-performance computing (HPC) environments with so-called “embarrassingly parallel” schemes. We analyze the difficulties in implementing the framework, such as computational intensity, that arise from repetitive execution of calibration, integration, and optimization, and we tackle these problems by using an importance re-sampling technique (bootstrap filter) and Monte Carlo simulation. Finally, with a DNAPL remediation model, we illustrate the use of this framework to assess the value of perfect information for unknown parameters, and imperfect information from measurements of partitioning tracer tests for a synthetic DNAPL remediation problem.
2 Value of Information The value of information (VOI) is the minimum expected cost when specified information is not used minus that when the information is used, and reflects the maximum price that one should pay for the information. Since in most situations, the information can have many outcomes, VOI is expressed as a statistical expectation, or the expected value of information. For simplicity, the term VOI will be used in the rest of this paper to describe the expected value of information. The way to assess VOI can be formulated as: VOI = min {Es [c(s, x)]} − Ey min Es|y [c(s, x)] , (1) x
x
where x represents the decision variables (controls); s is a random vector of the unknown parameters, hence representing the current state of characterization; c(s, x) is the cost function; y represents the information or the data that we are evaluating; s | y is a conditional random vector representing the state of characterization after using the information; and Es [c(s, x)] returns the expected value of the function in the brackets with respect to the subscribed random variable (vector) immediately succeeding to E. To illustrate, for example, s can be parameters related to hydrogeology and DNAPL source distribution, x can be decisions related to certain remediation actions, and y can be additional measurements obtained from actual or proposed monitoring wells.
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To illustrate the use of this formula, let us now take a 1-D example problem. Suppose our task is to remove DNAPL in a source zone where the DNAPL amount is s, and the cost function for removing x DNAPL is: c(s, x) = 2x + (s − x)2 I(x
