(423b) Optimal Control of Lake pH for Mercury Bioaccumulation Control

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:

[1] DeSimone, R., Penley, M., Charbonneau, L., Smith, G.,
Wood, J., Hill, H., Pratt, J., Ridsdale, S., & Williams, R. (1973). The
kinetics and mechanism of cobalamine-dependent mthyl and ethyl transfer to
mercuric ion. Biochimica et Biophysica Acta, 304, 851?863.

[2] USEPA (1997). Mercury study report to congress.
Report to congress: EPA-452/R-97-003, United States Environmental Protection
Agency.

[3] Sorensen, J., Glass, G., Schmidt, K., Huber, J., &
Rapp, G. (1990). Airborne mercury deposition and watershed characteristics in
relation to mercury concentrations in water, sediments, plankton and fish of
eighty northern Minnesota lakes. Environmental Science and Technology,
24, 1716?1727.

[4] Nagase, H., Ose, Y., Sato, T., & Ishikawa, T.
(1982). Methylation of mercury by humic substances in an aquatic environment. Science
of Total Environment, 32, 147?156.

[5] Winfrey, M. R. & Rudd, J. W. M. (1990).
Environmental factors affecting the formation of methylmercury in low pH lakes.
Environmental Toxicology and Chemistry, 9, 853?869.

[6] Xun, L., Campbell, N., & Rudd, J. (1987).
Measurements of specific rates of net methyl mercury production in the water
column and surface sediments of acidified and circumneutral lakes. Canadian Journal
of Fisheries and Aquatic Sciences, 44, 750?757.

[7] Hakanson, L. (2003). Consequences and correctives
related to lake acidification, liming and mercury in fish A case-study for lake
Huljesjn, Sweden, using the LakeWeb-model. Environmental Modeling and
Assessment, 8, 275?283.

[8] Ottosson, F. & Hakanson, L. (1997). Presentation
and analysis of a model simulating the pH response of lake liming. Ecological
Modelling, 105, 89?111.

[9] Dixit, A. & Pindyck, R. (1994). Investment
under uncertainty. Princeton University Press, Princeton, New Jersey.

[10] Diwekar, U. (2003). Introduction to Applied
Optimization. Kluwer Academic Publishers, Dordrecht.

[11] Shastri, Y. & Diwekar, U. (2006a). Sustainable
ecosystem management using optimal control theory: Part 1 (Deterministic
systems). Journal of Theoretical Biology (Accepted for publication).

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