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Bayesian Implementation of a Chesapeake Bay Hypoxia Model
Primary Investigator:
Craig Stow - NOAA/GLERL
Co-Investigators:
Don Scavia* - University of Michigan, SNRE*
NOAA Research Area:
Advancing understanding of ecosystems to improve resource management
Performance Objective:
Increase number of regional coastal and marine ecosystems delineated with approved indicators of ecological health and socioeconomic benefits that are monitored and understood.
Research Milestones:
Meet annual targets for the number of coastal, marine, and Great Lakes ecological characterizations that meet management needs.
Executive Summary of Rationale
Models to predict hypoxia that include quantified estimates of predictive uncertainty are important tools to assist decision-making. The Bayesian approach facilitates incorporation of data from many years and allows parameter estimates (predictions) to differ with time. Predictive uncertainty is described probabilistically allowing utility to be evaluated over the range of likely outcomes. Examination of parameter evolution with time provides a basis for inference regarding whether fundamental changes in the underlying processes affecting hypoxia are occurring..
Proposed Work
Current/Ongoing
This project was begun last year and has produced one accepted manuscript: Stow, C. A. and D. Scavia. Modeling Hypoxia in the Chesapeake Bay: Ensemble estimation using a Bayesian hierarchical model. Journal of Marine Systems. In press. and two talks at the November 2007 Estuarine Research Federation Conference in Providence, RI. This year to model will be updated into a multilevel/hierarchical framework where the new structure will represent time periods corresponding to apparent alternate stable states. Forecasts can then be made over a range of BOD loads using a weighted average of parameter distributions reflecting a degree-of-belief that the system will be in one of these states.
Scientific Rationale
Chesapeake Bay has experienced summertime bottom water hypoxia since at least the 1950s. Several models have been used to assist management decision-making to reduce hypoxia; the complexity of these models has ranged from a complex three-dimensional dynamic model to simpler statistical relationships. Each of these approaches has advantages and disadvantages and there is an ongoing debate among modelers regarding the relative utility of complex vs. simple models. More recently Scavia et al. (2006) used a compromise approach, applying a simple process-based model to describe Chesapeake Bay dissolved oxygen patterns. Scavia et al. (2006) applied the Streeter-Phelps dissolved oxygen model:
where DO = the dissolved oxygen concentration (mg/L), DOs = the saturation oxygen concentration, k1 = the BOD decay coefficient (1/day), k2 = the reaeration coefficient (1/day), BODu = the ultimate BOD (mg/L), x = the downstream distance (km), v = stream velocity (km/day), Di = the initial DO deficit (mg/L), and e is the model error term.
Figure 1.
This model describes the DO depletion that occurs when BOD inputs initially remove oxygen from a stream and subsequent recovery as reaeration occurs (Figure 1). In their application Scavia et al. (2006) used fixed a priori values for the model inputs. We propose to extend this work by developing a Bayesian version of the model, using the available data to predict the model inputs and their uncertainty. It is of particular interest to evaluate how some of these inputs have changed with time, as suggested by the D.O. data. D.O. measurements along a fixed 137 station transect (~220 km) are available for 36 years from 1950 – 2003.
With the assumption that e ~ N(0, o2) we get a likelihood function for each year:
that can be incorporated into Bayes Theorem:
where π(θ| y) is the posterior probability of θ the probability of the parameter vector, θ, after observing the data, y), π(θ) is the prior probability of θ , (the probability of θ before observing y), and f (y | θ) is the likelihood function. To estimate values for all 36 years of datawe will develop a hierarchical model where the model parameters for each year are assumed to arise from a common prior distribution. This prior distribution also provides a basis for future prediction if year-specific parameter distributions are not considered appropriate. Adaptation of this model into a Bayesian framework will provide a basis to evaluate future BOD loading scenarios and their uncertainty.
Governmental/Societal Relevance
Hypoxia is increasingly recognized as a worldwide problem. Models that can predict hypoxia and associated uncertainties are essential tools to assist in management decision-making.
Relevance to Ecosystem Forecasting
Water quality models provide an essential framework for scientific assessment in support of water quality management and decisions such as Total Maximum Daily Load (TMDL) determinations. Models allow decision-makers to evaluate the logical outcomes of alternative management actions based on informed speculation about system behavior captured in a set of equations. Given a choice of models, a decision-maker is likely to choose the model that predicts most accurately. If a model were available that was 100% accurate, this model would be a clear choice over one that was, say, 50% accurate. With 100% accuracy, management actions could be chosen based only on the societal value of the consequences of those actions. A model with only 50% accuracy is still useful but applying such a model requires hedging decisions by the relative probabilities of a range of possible outcomes and the societal value of those outcomes. Even models of relatively low predictive accuracy can be useful if the accuracy is appropriately quantified. Thus, model uncertainty quantification provides information useful in both model selection and application. However, decision-makers are often confronted with models and given no information regarding forecast uncertainty. How then, can these models be appropriately used for decision-making? Incorporation of the Chesapeake Bay model into a probabilistic framework provides a basis to quantify the predictive uncertainty so that managers can use these uncertainties for more informed decision-making.
Cited References
Scavia, D.,E.L.A. Kelly, and J.D. Hagy. 2006.A simple model for forecasting the effects of nitrogen loads on Chesapeake Bay hypoxia. Estuaries and Coasts, 29: 674-684.
*Link leads off GLERL's website
