1. Introduction

Mercury is recognized internationally as an important pollutant, since mercury and its compounds are persistent, bioaccumulative and toxic, and they pose human and ecosystem risks. Mercury can cycle in the environment in all media as part of both natural and anthropogenic activities. Due to such complicated cycling, successful control of mercury pollution in water depends on actions at various points in the mercury cycle. An important aspect of this cycling is the bioaccumulation of mercury along the aquatic food chain. Mercury, in the form of methylmercury, bioaccumulates up the aquatic food chains so that organisms in higher trophic levels have higher mercury concentrations [1]. As a result, contaminated fish consumption is the most predominant path of human exposure to mercury. The primary targets for toxicity of mercury and mercury compounds are the nervous system, kidney, and developing fetus [2].

Since methyl mercury is the primary bioaccumulative form of mercury, controlling the conversion of mercury in water bodies to methyl mercury is a possible option to control bioaccumulation. This work looks into the option of using time dependent liming strategy of lakes and rivers to control water pH. To make the analysis more robust, it proposes to derive liming strategies in the presence of time dependent uncertainty. Optimal control theory, in combination with the real options theory to effectively deal with various sources of uncertainty, is to be used for the derivation of the liming strategies. Such an analysis is expected to make the liming operation more reliable, thereby presenting one more tool to manage the harmful effects of mercury pollution.      

2. Theory and Approach The presence of mercury in water is harmful mainly because of its bioaccumulative nature. Among the various forms of mercury in water, methyl mercury (MeHg) is the most important one since it has the greatest bioaccumulative potential. As a result, even though the concentration of MeHg in freshwater ecosystems is typically less than 10%, almost all the mercury found in top level species is MeHg [2]. Thus, methylation of mercury to MeHg is a key step in the bioaccumulation of mercury in the aquatic food chain [3]. The concentration of MeHg in water depends on the equilibrium between the methylation and demethylation reactions, which occur in the water column as well as the sediments. The exact mechanism of the methylation reaction is not clear. The proposed mechanisms include abiotic [4] and biotic [5] (in the presence of enzyme producing bacteria) which is more prominent. However, studies have shown that low water pH (acidic lake) aids the methylation reaction [5, 6]. Therefore, control of lake pH is an effective method to control the bioaccumulation of MeHg. Lake liming is one of the options to control the pH of a lake, and it is practiced in the European countries [7]. The process of liming is the addition of limestone (calcite), primarily calcium carbonate, to neutralize the acidic water. However, the task of lake liming is complicated due to the presence of various kinds of uncertainties, such as lack of information on the exact pH of the lake, seasonal variations in lake pH, and topological effects of liming. As a result, the application of liming in North American lakes has been restricted. This calls for the incorporation of effective uncertainty modeling techniques.  

3. Model

Ottosson and Hakanson [8] present a simplified model to simulate lake pH in the presence of natural inputs and liming actions. It is a mixed model including both statistical regressions, based on catchment and lake morphometric data, and dynamic, mass-balance interactions. It is driven by data on the amount of lime and the month of liming. The predictive power of the model has been tested using independent data from various lakes. The model is represented by a set of algebraic and ordinary differential equations. The details are given in [8].

There are various possible sources of uncertainties in this setup, both, time dependent as well as time independent. This includes: spatial and temporal uncertainties about the lake pH values; uncertainties about the effect of lime addition and uncertainty about the effect of liming on lake biota. The first step towards effectively dealing with uncertainties is the reliable representation of these uncertainties.

            Real options theory presents different ways to represent and forecast uncertainty using stochastic processes. Wiener process, also known as Brownian motion, is an example of a simple continuous time and continuous state stochastic process stochastic process which has been used to model a variety of continuous stochastic processes [9, 10].

Owing to their success, this work will use such stochastic equations to represent various uncertainties that might be encountered in the lake liming operation.

            To ascertain that liming actions are not adversely affecting the lake biota, the work proposes to use Fisher information (FI) based measure to quantify the biodiversity and hence the sustainability of the lake biota. The application of FI based measures for sustainability quantification has been illustrated in [11, 12].  

4. Methodology

To account for the complex nature of liming operation, and to take care of the inherent time dependent variability, this work proposes to use optimal control theory to derive time dependent liming strategy. Optimal control theory is designed to optimize a time dependent performance objective of the system. It has been extensively used in control applications for natural systems, primarily because of its generality and its ability to handle any type of system (including nonlinear). For deterministic model, Pontryagin's maximum principle will be used to formulate and solve the control problem.

The presence of uncertainty in the model complicates the control problem solution. We plan to combine optimal control theory and real options theory to improve time dependent decision making under uncertainty. In real options theory, the idea is to optimize not only the decisions but also the timing of the decisions to optimize the given objective. Efficient techniques from this theory, when combined with the established methods in optimal control theory, will lead to reliable tools in dealing with time dependent uncertainties.

5. Summary

The work proposes to control the bioaccumulation of mercury along the aquatic food chain in lakes/rivers by controlling the water pH. This is to be achieved through time dependent liming actions, derived using optimal control theory. The analysis is extended to incorporate uncertainties, which includes effective modeling using stochastic processes and efficient solution techniques using real options theory. The analysis is expected to make the liming operation more reliable, thereby presenting one more tool to manage the harmful effects of mercury pollution.

References:

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[12] Shastri, Y. & Diwekar, U. (2006b). Sustainable ecosystem management using optimal control theory: Part 2 (Stochastic systems). Journal of Theoretical Biology (Accepted for publication).