PREDICTING THE HYPOXIC-VOLUME IN CHESAPEAKE BAY WITH...

PREDICTING THE HYPOXIC-VOLUME IN CHESAPEAKE BAYWITH THE STREETER–PHELPS MODEL: A BAYESIAN APPROACH1

Yong Liu, George B. Arhonditsis, Craig A. Stow, and Donald Scavia2

ABSTRACT: Hypoxia is a long-standing threat to the integrity of the Chesapeake Bay ecosystem. In this study,we introduce a Bayesian framework that aims to guide the parameter estimation of a Streeter–Phelps modelwhen only hypoxic volume data are available. We present a modeling exercise that addresses a hypothetical sce-nario under which the only data available are hypoxic volume estimates. To address the identification problemof the model, we formulated informative priors based on available literature information and previous knowl-edge from the system. Our analysis shows that the use of hypoxic volume data results in reasonable predictiveuncertainty, although the variances of the marginal posterior parameter distributions are usually greater thanthose obtained from fitting the model to dissolved oxygen (DO) profiles. Numerical experiments of joint parame-ter estimation were also used to facilitate the selection of more parsimonious models that effectively balancebetween complexity and performance. Parameters with relatively stable posterior means over time and narrowuncertainty bounds were considered as temporally constant, while those with time varying posterior patterns wereused to accommodate the interannual variability by assigning year-specific values. Finally, our study offers pre-scriptive guidelines on how this model can be used to address the hypoxia forecasting in the Chesapeake Bay area.

(KEY TERMS: hypoxia; Chesapeake Bay; Bayesian inference; Markov chain Monte Carlo; Streeter–Phelpsmodel; uncertainty analysis; eutrophication.)

Liu, Yong, George B. Arhonditsis, Craig A. Stow, and Donald Scavia, 2011. Predicting the Hypoxic-Volume inChesapeake Bay with the Streeter–Phelps Model: A Bayesian Approach. Journal of the American WaterResources Association (JAWRA) 1-15. DOI: 10.1111 ⁄ j.1752-1688.2011.00588.x

INTRODUCTION

Hypoxia (dissolved oxygen [DO] £2 mg ⁄ l) is a seri-ous problem for estuaries in the United States(Bricker et al., 2007) and has received considerablescientific and policy attention (e.g., NRC, 2000; Diazand Rosenberg, 2008). Low oxygen concentrations in

the Chesapeake Bay were first reported in the 1930s,with a significant increase in severity and spatialextent observed since the 1950s (Newcombe andHorne, 1938; Officer et al., 1984; Cooper and Brush,1991; Boesch et al., 2001). The causes and conse-quences of Chesapeake Bay hypoxia have beenthe focus of research and policy action, including aseries of intergovernmental agreements and nutrient

1Paper No. JAWRA-10-0127-P of the Journal of the American Water Resources Association (JAWRA). Received August 16, 2010; acceptedJune 7, 2011. ª 2011 American Water Resources Association. Discussions are open until six months from print publication.

2Respectively, Research Professor, College of Environmental Science and Engineering, The Key Laboratory of Water and SedimentSciences Ministry of Education, Peking University, Beijing 100871, China; Associate Professor, Department of Physical and EnvironmentalSciences, University of Toronto, Toronto, Canada M1C 1A4; Senior Research Scientist, NOAA Great Lakes Environmental ResearchLaboratory, Ann Arbor, Michigan 48105-2945; and Professor, School of Natural Resources & Environment, University of Michigan, AnnArbor, Michigan 48109 (E-Mail ⁄ Liu: [email protected]).

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JOURNAL OF THE AMERICAN WATER RESOURCES ASSOCIATION

AMERICAN WATER RESOURCES ASSOCIATION

reduction efforts (Boesch et al., 2007). An analysis ofthe relationship between hypoxia and nitrate loadingfor two periods, 1950-1979 and 1980-2001, demon-strated that the Bay’s ability to assimilate nitrogeninputs was reduced in the latter period (Boesch et al.,2001; Kemp et al., 2005). This loss of assimilativecapacity may indicate a state change, implying thatdisproportionately large nitrogen load reductions maybe required to bring hypoxia under control (Schefferand Carpenter, 2003).

For the Chesapeake Bay, and elsewhere, modelshave become important tools for guiding decisions onnutrient load reduction (NRC, 2000; Arhonditsis andBrett, 2004; Arhonditsis et al., 2006). In developingand assessing such models, there are three key issuesthat need to be addressed before the models can beused for decision making: (1) establishing a reliableand cost-effective model; (2) estimating model parame-ter values; and (3) assessing model uncertainties. Sev-eral types of models have been developed for theChesapeake Bay, ranging from complex, mechanisticmodels (e.g., Cerco and Cole, 1993; Cerco and Noel,2004) to simple statistical models (Hagy et al., 2004;Kemp et al., 2005); each with features that make themdifferentially amenable to effectively addressing thesethree issues. Complex models can be used to organizeexplicit, detailed assumptions about the relationshipsamong important ecosystem processes and to test thelogical consequences of those assumptions under dif-ferent circumstances. However, complex models areusually over-parameterized, requiring modelers toadopt a combination of literature values and expertjudgment to select key parameter values. Additionally,complex models usually carry a high computationaldemand, which inevitably makes it difficult toconduct thorough sensitivity or uncertainty analyses.Alternatively, statistical models are computationallymore efficient, but generally do not resolve physical,ecological, and biogeochemical processes. By acknowl-edging that models at both ends of the complexityspectrum have different strengths and weaknesses,simple biophysically based water quality models,tested against observations, were recommended bythe National Research Council (NRC, 2001) as a cost-effective way to inform decision making.

Consistent with that recommendation, Scavia et al.(2006) used a simple process-based approach, theStreeter–Phelps equation (Streeter and Phelps, 1925),to model Chesapeake Bay DO conditions. Scavia et al.(2006) used fixed a priori parameter values based ona combination of available information and expertjudgment, and the uncertainty associated with twoparameters (initial DO deficit and downstream advec-tion) was estimated using Monte Carlo simulations.Stow and Scavia (2009) introduced a Bayesian config-uration of the Chesapeake Bay model to estimate DO

profiles. Their work demonstrated that the Bayesianapproach is useful in a decision context because itcan combine both expert judgment and rigorousparameter estimation to yield model forecasts alongwith the underlying uncertainty.

While the Streeter–Phelps model has been shownuseful to predict longitudinal DO profiles in Chesa-peake Bay, goals for hypoxia management are definedin terms of reducing the volume of hypoxic water.Thus, it may not be necessary to predict the full DOprofile if the hypoxic volume can be accurately esti-mated. Additionally, in some systems such as thenorthern Gulf of Mexico, although hypoxic area is esti-mated annually, detailed DO profile measurementsare not generally available for calibrating the Streeter–Phelps model. Even without this information theStreeter–Phelps model has been successfully used topredict Gulf of Mexico hypoxia, but with calibrationdirectly to hypoxic area rather than to full DO pro-files (Scavia et al., 2003, 2004; Scavia and Donnelly,2007).

In this study, we introduce a Bayesian frameworkto guide the parameter estimation of a Streeter–Phelps model when only hypoxic volume data areavailable. In particular, we use the Chesapeake Bayas a case study to present a modeling exercise thataddresses a hypothetical scenario under which theonly information available is hypoxic volume esti-mates. Our analysis evaluates the credibility of theproposed statistical approach by undertaking a seriesof modeling experiments of individual and jointparameter estimation, whereby the model parameter-ization and the associated predictive uncertainty arecompared to those derived from conventionalStreeter–Phelps model fitted to DO profiles. Thepurpose of this study is to examine to what extentthe proposed Bayesian framework is suitable (1) toachieve realistic parameter values and acceptablepredictive capacity; (2) to gain insights into possibleregime shifts or other structural changes that mayhave occurred during the study period; and (3) to dic-tate areas where future model augmentations shouldfocus on. Finally, we offer prescriptive guidelines onhow this model can be used to address the hypoxiaforecasting in the Chesapeake Bay.

MATERIALS AND METHODS

Data Source

Data used in this study are the same as reportedin Scavia et al. (2006) for oxygen concentration pro-files and hypoxic volumes along the main stem of the

LIU, ARHONDITSIS, STOW, AND SCAVIA

JAWRA 2 JOURNAL OF THE AMERICAN WATER RESOURCES ASSOCIATION

Chesapeake Bay for 36 years from 1950 to 2003.There were 137 values reported for the DO profilesfor each of the 36 years, which were computed byinterpolating observations to populate a regular gridwith dimensions, first at 1-m resolution in the verti-cal and then at 1 km in the horizontal directionacross constant depths (Hagy et al., 2004). The Janu-ary-May average total nitrogen loads in 1945-1978from the Susquehanna River were computed usingempirical relationships from corresponding nitrateloads at Harrisburg, Pennsylvania (Hagy et al.,2004). The 1979-2004 loads were from USGS, baseddirectly on frequent measurements of SusquehannaRiver total nitrogen concentrations at Conowingo andwere derived using the seven-parameter log-linearregression model described by Cohn et al. (1989).

Model Equations

The model is an adaptation of the Streeter–Phelpsriver model which predicts downstream oxygen con-centrations from point sources of organic matter loads(Chapra, 1997). Chesapeake Bay is stratified verticallyin summer, with surface waters flowing seaward andbottom waters flowing landward. As such, a keyassumption for this application is that watershednutrient loads can be used to approximate a pointsource of organic matter deposited to sub-pycnoclinedepths at the southern end of the mid-Bay region(37�48¢N, 76�W; c. 220 km from the SusquehannaRiver mouth), and that the organic matter decays as itis transported up estuary below the pycnocline. Thenutrient nitrogen load was converted to organic carbonvia the Redfield C:N ratio (106:16 or 5.67 g C ⁄ g N),and then converted to a biologic oxygen demand viaO2:C (0.9, or 2.4 g O2 ⁄ g C). The validity of this assump-tion was first examined in Scavia et al. (2006) and wasfound to result in realistic representation of the ecolog-ical processes underlying the system dynamics. Thesteady-state solution used in Scavia et al. (2006) andStow and Scavia (2009) for Chesapeake Bay DOprofiles along downstream distance can be written as:

DO ¼ DOs �kdBODuðFÞ

kr � kde�kd

xv � e�kr

xv

� ��Die

�krxv þ e

ð1Þ

where DO is the measured DO concentration (mg ⁄ l),DOs is the saturation oxygen concentration (mg ⁄ l), kd

is the BOD decay coefficient (1 ⁄ day), kr is the reaer-ation coefficient (1 ⁄ day), BODu is the ultimate BOD(mg ⁄ l), F is the fraction of surface organic carbon pro-duction that settles below the pycnocline, x is the

downstream distance (km), Di is the initial DO deficitat x = 0 (mg ⁄ l), and e is model error. In the originalStreeter–Phelps formulation, v represents net down-stream advection. However, earlier predictions of theChesapeake Bay DO profiles were based on variant vacross years, while other parameters were retainedconstant. The selection of the parameter v, as opposedto the cross-pycnocline flux or the sub-pycnoclinedecay rate parameters, was justified by three basicreasons: (1) it was assumed that the other two parame-ters were much less likely to vary across time, (2) therewere good in situ measurements to compare thosemodeled flux rates with observations, and (3) the origi-nal physical meaning of v (i.e., downstream advection)in the steady-state application of this river model haslittle meaning (Scavia et al., 2006). In this regard, therole of v is conceptually similar to a time-variant struc-tural error, and therefore can potentially account forthe impact of un-modeled ecological processes and mayaccommodate structural shifts in the ecosystem func-tioning that have occurred during the study period.Aggregating those uncertainties into one coefficientallowed their characterization through Monte Carlosimulations in previous applications (Scavia et al.,2003; Scavia and Donnelly, 2007). In the present studythough, we will revisit the assumption of a temporallyvarying v. The impact of the alternative approach (i.e.,temporally constant v) on the model parameterizationwill be examined, along with the predictive uncer-tainty. It should also be noted that in this applicationkr reflects oxygen flux across the pycnocline, not withthe atmosphere. A multiplier (K) was added to equa-tion (1), replacing kr with kr*K, to scale the previouslycalculated cross-pycnocline flux coefficient kr whichvaries spatially along the system (Hagy et al., 2004),but not across years.

The observed annual hypoxic volumes V (km3) inthe system were linked with the predicted hypoxiclength Lm (km), using the regression equation origi-nally derived from existing measurements in thesystem (Scavia et al., 2006), i.e.,

If DOmh < 2 mg=L then wh ¼ 1; else wh ¼ 0;

Lm ¼X137

h¼1

lhwh

V ¼ 0:00391 L2m þ ev

ð2Þ

where lh is length of the segment h in which the pre-dicted oxygen concentration (DOmh) is below 2 mg ⁄ l;eV is the error term that reflects the discrepancybetween the ‘‘observed’’ hypoxic volume estimatesand those predicted from our model. In this study,we will examine the differences in the parameter

PREDICTING THE HYPOXIC-VOLUME IN CHESAPEAKE BAY WITH THE STREETER–PHELPS MODEL: A BAYESIAN APPROACH


estimation and the predictive uncertainty that mayarise when the calibration exercise is founded uponthe hypoxic volume data and the DO concentrationsare not part of the fitting process. The hypoxic vol-ume fitting is essentially an ad hoc feature to the DOprofile model, and thus the predictive uncertaintyassociated with the latter model is propagatedthrough the former one. Yet, our analysis addressesthe hypothetical scenario under which DO data donot exist and the corresponding error cannot be quan-tified, but rather is treated as an implicit componentof the fitting exercise.

Bayesian Framework for Hypoxia Modeling

Bayesian statistics provide rigorous methods foruncertainty analysis and key information for decisionmaking in the context of environmental management(Reckhow, 1994; Stow et al., 2006). All unknownparameters h are treated as random variables andtheir distributions are derived from past informationand current observations from the system (Borsuket al., 2001). Bayesian inference is based on Bayes’Theorem (Gill, 2002):

pðhjyÞ ¼ pðhÞpðyjhÞpðyÞ ¼ pðhÞpðyjhÞR

h pðhÞpðyjhÞdh/ pðhÞpðyjhÞ

ð3Þ

where p(h|y) is the posterior probability of h, whichis the conditional distribution of the parameters afterobservation of the data; h is the estimated parameter;p(h) is the prior probability of h; p (y|h) is the like-lihood function, which represents the probability forthe occurrence of the observations y given differentrealizations of the postulated mechanistic relation-ship between the response and predictor variables,i.e., Equation (1).

Model Fitting to DO Profiles. Transferringequation (1) into Bayes’ Theorem, the error term e wasassumed to be normally distributed with zero meanand variance r2, e � N(0, r2). The likelihood functionof equation (1) can be written as:

DOgh � Nðlgh; r2Þ

lgh ¼ DOS �kdBODuðFgÞ

kr � kde�kd

xgv � e�kr

xgv

� ��Die

�krxgv

ð4Þ

where h represents the 137 observed DO estimatesalong the transect; and g represents the 36 years

Model Fitting to Hypoxic Volume Data. Thelikelihood function for hypoxic volume, with the errorterm assumed to be normally distributed with zeromean and variance of r2

V is:

If lgh < 2 mg=L then wgh¼ 1; else wgh¼ 0

Lmg ¼X137

h¼1

lghwgh

Vg � Nð0:0039 L2mg; r

2VÞ

ð5Þ

In this analysis, model parameters were scruti-nized in two stages: (1) we started with values usedby Scavia et al. (2006) and systematically relaxed thefixed prior values for individual parameters; and then(2) we formed parameter subsets (refer to the Resultssection for details) to examine if the joint estimationof some parameters can further improve model per-formance while keeping the uncertainty of the under-lying model predictions at realistic levels. Selectedparameters were allowed to differ by year using aBayesian structure, founded upon the assumptionthat the year-specific estimates arise from a commonnormal distribution. For example, for a selectedparameter h, the year-specific values hi were allderived from a normal distribution of the form: hi �N(h.mean, h.sd), in which h.mean and h.sd representthe parameters of the common normal prior. Theh.mean values were set equal to the averages of theestimated values from Stow and Scavia (2009), whilethe ranges between the corresponding maximum andminimum values were assumed to contain 95% of theprior probability mass. Using this approach, weobtained the following informative normal priors:Di �N(0.30,1.20); F �N(0.75,0.13)I(0,1); K �N(0.6,0.20)I(0,1); kd �N(0.11,0.05)I(0,); and v �N(2.5, 0.77)I(0,), in which the numbers in the brackets representthe mean and the standard deviation of the corre-sponding normal distributions; and I(,) denotes thecensoring imposed to eliminate negative values (andvalues greater than one) during the Bayesian updat-ing of the model. With the parameters K and F, wealternatively used a number of beta distributions thatallocated a greater probability mass to parametervalues higher than 0.5, and the results remainedpractically unaltered. It should also be clarified thatthe fact that these informative priors are looselybased on the parameter estimates presented by Stowand Scavia (2009) may seem that we are ‘‘doubledipping’’ into the same dataset. Yet, the actual prac-tice followed merely aimed to avoid unnecessarydivergence of the Markov chain Monte Carlo (MCMC)sampling into ecologically unrealistic regions of theparameter space that were unlikely to hold true inthe real world. Namely, the informative priors for



each parameter were based on the corresponding val-ues across all the years of the study period, whichrepresented a quite broad range (see Figures 2-4 ofStow and Scavia, 2009). From our experience, rela-tively similar priors would have been derived, even ifwe had reviewed the pertinent literature and subse-quently used the framework presented by Arhonditsiset al. (2007). Thus, what may be criticized as ‘‘doublecounting’’ of the same piece of information is actuallya formal framework of using existing knowledge toachieve an ecologically meaningful foundation of ourmodel.

Markov chain Monte Carlo is a method to drawsamples from multidimensional distributions fornumerical integration (Qian et al., 2003). The ideaunderlying the MCMC implementation in Bayesianinference is to construct a Markov process whose sta-tionary distribution is the model posterior distribu-tion, and then run the process long enough toproduce an accurate approximation of this distribu-tion (Malve and Qian, 2006). There are many meth-ods (e.g., Gibbs sampler) for obtaining sequence ofrealizations from the posterior model distributions(Gelfand and Smith, 1990), but all of them are spe-cial cases of the general Metropolis-Hastings algo-rithm (Metropolis et al., 1953; Hastings, 1970). Weused WinBUGS (version 1.4.3; Lunn et al., 2000),called from R (version 2.6.0; R2WinBUGS version2.1-8; Gelman and Hill, 2007). The MCMC samplingwas carried out using four chains, each with 20,000iterations (first 10,000 discarded after model conver-gence); and samples for each unknown quantity weretaken from the next 10,000 iterations using a thin

equal to 40 to reduce serial correlation. Statisticalinference was based on the resulting 1,000 MCMCsamples. A potential scale reduction factor, Rhat,was produced in package R2WinBUGS to determinethe model convergence (at convergence, Rhat = 1.0).Rhat is approximately equal to the square root of thevariance if the mixture of all the chains divided bythe average within-chain variance; if it is greaterthan 1.0, the chains have not mixed well (Gelmanand Hill, 2007).

Three measures of fit were combined to test andcompare model results: (1) the deviance informationcriterion (DIC), a Bayesian measure of model fit thatpenalizes the complexity of the model structure, andtherefore a lower DIC value suggests a more favor-able model (Spiegelhalter et al., 2002); (2) modelresidual standard error; and (3) the coefficient ofdetermination, R2, between the original data andpredicted mean values. The former measure enablesthe selection of the most parsimonious models thateffectively balance between model-fit and complexity,while the R2 was mainly selected because of itscommon use in the modeling practice for assessinggoodness-of-fit.

For a Bayesian model with data y, unknownparameters h, and the likelihood function p(y|h), thedeviance is defined as (Spiegelhalter et al., 2002):

DðhÞ ¼ �2 log½pðy hj Þ� þ c ð6Þ

where c is a constant. The effective number of para-meters in the model is:

012345678

0 50 100 150 200

0123456789

0 50 100 150 200

012345678

0 50 100 150 2000123456789

0 50 100 150 200

1973

1997

1950

2003

Distance

ecnatsiDecnatsiD

Distance

FIGURE 1. Comparison Between Observed (red line) and Predicted Mean (purple line) Dissolved Oxygen (DO) Profiles in ChesapeakeBay Using Year-Specific K Estimates. The blue lines represent the 95% credible intervals. The green line corresponds to the DO

profile predicted from Scavia et al. (2006). Distance is kilometers downstream from the Susquehanna River.



1950 1960 1970 1980 1990 2000 2010Year

-2.0

-1.0

0.0

1.0

2.0

3.0

(a) (b)

(c) (d)

(e) (f)

(g) (h)

(i) (j)

Di v

alue

(mg/

L)

1950 1960 1970 1980 1990 2000 2010Year

-2.0

-1.0

0.0

1.0

2.0

3.0

4.0

5.0

Di v

alue

(mg/

L)

1950 1960 1970 1980 1990 2000 2010Year

0.20.30.40.50.60.70.80.91.0

K

1950 1960 1970 1980 1990 2000 2010Year

0.20.30.40.50.60.70.80.91.0

K

1950 1960 1970 1980 1990 2000 2010

Year

0.5

0.6

0.7

0.8

0.9

1.0

F

1950 1960 1970 1980 1990 2000 2010Year

0.5

0.6

0.7

0.8

0.9

1.0F

1950 1960 1970 1980 1990 2000 2010Year

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

k d (1

/day

)

1950 1960 1970 1980 1990 2000 2010

Year

0.00

0.05

0.10

0.15

0.20

0.25

k d (1

/day

)

1950 1960 1970 1980 1990 2000 2010Year

0.00

1.00

2.00

3.00

4.00

5.00

v (km

/day

)

1950 1960 1970 1980 1990 2000 2010

Year

0.00

1.00

2.00

3.00

4.00

5.00

6.00

v (km

/day

)

FIGURE 2. Comparison of the Year-Specific Estimates for the Parameters Di, K, F, kd, and v Fitted to the Hypoxic Volume(left panels) and the Dissolved Oxygen (DO) Profile Data (right panels). The red circles correspond to the parameter values

reported in Scavia et al. (2006). The blue diamonds and dashed lines correspond to the posterior mean values along withthe 95% credible intervals. The two dashed horizontal lines represent the 95% credible intervals of the prior distributions.



pD ¼ DðhÞ �DðhÞ ð7Þ

where h is the expectation of h. DðhÞ is the expecta-tion of D(h):

DðhÞ ¼ Eh½DðhÞ� ð8Þ

Moreover, DIC is calculated as:

DIC ¼ pDþDðhÞ ð9Þ

Thus, with this Bayesian model comparison, wefirst assessed model fit or model ‘‘adequacy’’ (sensuSpiegelhalter et al., 2002), DðhÞ, and then we penal-ized complexity, pD.

R2 is defined as (Gelman and Hill, 2007):

R2 ¼ 1� SSE

SST¼ 1�

Pni¼1

ðyi � y0iÞ2

Pni¼1

ðyi � yiÞ2ð10Þ

1950 1960 1970 1980 1990 2000 2010Year

1.02.03.04.05.06.07.08.09.0

10.011.012.0

Hyp

oxia

vol

ume(

km3 )

1950 1960 1970 1980 1990 2000 2010Year

1.02.03.04.05.06.07.08.09.0

10.011.012.0

Hyp

oxia

vol

ume(

km3 )

(a) Original prior (b) Double precisions

1950 1960 1970 1980 1990 2000 2010Year

1.02.03.04.05.06.07.08.09.0

10.011.012.0

Hyp

oxia

vol

ume(

km3 )

(c) Half precisions

FIGURE 4. Sensitivity of the Hypoxic Volume (km3) Predictions to the Prior Distributions Assigned to the Parameter Di. the OriginalPrior Based on the Informative Normal Distribution of Di �N(0.30,0.70) Was Examined With Double (1.40) and Half Precision (0.35).

1950 1960 1970 1980 1990 2000 2010Year

1.02.03.04.05.06.07.08.09.0

10.011.012.0

Hyp

oxia

vol

ume(

km3 )

1950 1960 1970 1980 1990 2000 2010Year

0.01.02.03.04.05.06.07.08.09.0

10.011.012.013.0

Hyp

oxia

vol

ume(

km3 )

(a) (b)

FIGURE 3. Comparison Between Observed (red circles) and Predicted Mean Hypoxic Volume (black triangles) Values inChesapeake Bay Using Year-Specific Estimates for the Parameters (A) Di, F and (B) Di, kd, and F. The lower and upper barsrepresent the 2.5 and 97.5% credible bounds. The blue diamonds correspond to the predicted values in Scavia et al. (2006).



where, SSE and SST are the sum of squared errorsand total sum of squares, respectively; yi and y0

iare

the original data and predicted mean values, respec-tively; yi is the mean of the observations yi; and n isthe number of observations. It should be notedthough that the Bayesian approach generates a pre-dictive distribution and not a single value for eachvariable (y0

i), and thus the use of R2 is essentially a

non-Bayesian (‘‘point’’) assessment of the model per-formance. For the same reason, the posterior meanvalues of the model error terms r and rV should notbe confused with the point estimate of the errorderived when using the mean model outputs (e.g., anR2 = 1 does not imply that the mean value of r or rV

is zero).

RESULTS

Individual Parameter Estimation

In the first phase of our analysis, we selected fiveparameters to estimate them individually with theDO profile and the hypoxic volume datasets. In theformer case, we fitted the model to the DO observa-tions along the system, while the latter inverse solu-tion exercise was only based on the observed annualhypoxic volume estimates. For these modeling exper-iments, one parameter was estimated and the otherfour were assigned the values of an earlier calibra-tion of the model presented by Scavia et al. (2006).We then evaluated the relative influence of eachparameter on model performance to determine theoptimal parameter combinations that will be exam-ined in the second phase of our analysis. Di, v, K, kd,and F are the model parameters being estimated,while DOs and BODu represent error-free boundaryconditions and inputs of the model, respectively.

For the hypoxic volume data, the use of year-specific Di, kd, and K estimates provided the lowestmodel residual standard error and highest R2 values(Figure 1 and Table 1). Further, the derivation ofyear-specific posteriors for v and F was also associ-ated with broader prediction uncertainty bands of thehypoxic volume levels, while kd resulted in the nar-rowest (Supporting Information). A different way ofviewing this result is that there is a wider range ofpredicted outputs that have been assigned ‘‘some’’likelihood during the updating of the v and F distri-butions. The fact that the MCMC search resulted ina ‘‘flatter’’ predictive posterior can conceivably beexplained by – at least – two factors: (1) an erroneousprior specification of the relative plausibility of thedifferent parameter values that may lead to an

exploration of suboptimal regions of the model poster-ior; and ⁄ or (2) the relationship between the specificparameters and the corresponding model output issteep and approximately linear (or even higher order)within the parameter range examined and as suchevery incremental change is accompanied by a signifi-cant change of the hypoxic volume predictions duringthe MCMC sampling. The latter finding can be inter-preted as evidence that the model is more sensitive tothe prior specifications of the parameters v and F,and that the predictive uncertainty can be signifi-cantly controlled by formulating more articulatepriors for these parameters based on site-specificinformation.

When the model was fitted to the DO profiles, theyear-specific K estimates resulted in the lowest modelresidual standard error, and highest R2 values, whilethe assumption of a year-specific Di gave the worst fit(Table 1). In all the five cases, most DO observationsare included within the 95% credible intervals (e.g.,Figure 1 for the case of year-specific K estimation).Posterior predictions and the deterministic solutionspresented by Scavia et al. (2006) were comparable forsome years (e.g., 1950, 1987, 1999, 2002), but thedeterministic approach resulted in notably lower DOprofiles for others (e.g., 1973, 1997, 2003). By con-trast, the Bayesian 2.5 and 97.5% uncertainty boundsinclude most of the observations as well as Scaviaet al.’s (2006) predictions.

To gain insight into how the inference changesdepending on the data used to calibrate the model,we compared the parameter estimates between themodel fit against the DO profile and the hypoxic vol-ume data (Figure 2). Among the five parameters, theyear-specific K estimates have the narrowest uncer-tainty bands when using the hypoxic volume data to

TABLE 1. Goodness-of-Fit of the Hypoxic Volume and DissolvedOxygen (DO) Profile Models Using Year-Specific Estimates

for the Parameters Di, v, K, kd, and F.

EstimatedParameter R2

Model ResidualStandard Error

Mean 2.5% 97.5%

Hypoxic Volume ModelDi 1.00 0.20 0.06 0.57v 0.97 1.27 0.79 1.94K 1.00 0.58 0.11 0.97kd 0.99 0.13 0.07 0.22F 0.98 1.71 1.22 2.36

DO Profile ModelDi 0.56 1.50 1.47 1.53v 0.76 1.20 1.17 1.22K 0.78 1.03 1.01 1.05kd 0.64 1.69 1.65 1.72F 0.75 1.21 1.19 1.24



parameterize the model, and the same was true forkd with both the hypoxic volume and DO profile data-sets. Parameter estimates for the hypoxic volumedata were comparable to those used in Scavia et al.(2006); especially those derived for Di (Figure 2a), K(Figure 2c), and v (Figure 2i). By contrast, the F andkd posterior estimates were correspondingly higherand lower compared to the values reported by Scaviaet al. (2006). We also note the relatively constant F,kd, and K values over time when fitting the model tothe hypoxic volume data, which renders support tothe assumption of keeping them constant across the36 years of the study period.

Figure 2 also illustrates more dramatic differencesbetween the posterior estimates from the DO profiledata and those derived from the original work (Scaviaet al., 2006). The posterior year-specific Di estimateshave temporal patterns similar to those reported inScavia et al. (2006) but with generally higher values(Figure 2b). Most F posteriors were lying within the0.90-1.00 range, with main exceptions being the 1965and 2001 estimates (Figure 2f). The posterior esti-mates for K vary between 0.30 and 0.70; roughlyaround the constant 0.60 reported in Scavia et al.(2006) (Figure 2d). Similarly, the original kd value of0.09 falls within the 0.04–0.13 range derived fromthe Bayesian updating (Figure 2h). Bayesian-estimated year-specific v values are higher than the1.0–4.0 values reported in Scavia et al. (2006), with anotably consistent temporal pattern that rangesbetween 2.4 and 4.9, that is, a systematic decrease inthe early period was followed by a consistent increasein the most recent years (see also following discus-sion) (Figure 2j).

The discrepancy between the prior and posteriordistributions offers insights into the effects of theBayesian updating on the parameter estimation pro-cess (Arhonditsis et al., 2007). That is, how much canwe learn for the different parameters when havingthe hypoxic volume dataset at hand? We found thatthe coefficients of the variation (or ‘‘relative standarddeviation’’, CV = standard deviation ⁄ mean value) ofthe posterior distributions were significantly reducedfor all the parameters considered, but there was sub-stantial difference among the parameters. Notably,the Di and K posterior CV values were reduced from3.98 to 0.396 and from 0.33 to 0.042, respectively.The latter result reflects the significant reduction ofthe Di (74%) and K (89%) standard deviations. For F,kd, and v, we found a relatively smaller reduction ofthe parametric uncertainty after the model updatingprocess. Namely, the F, kd, and v CV values werereduced from 0.170, 0.455, and 0.307 to 0.118, 0.265,and 0.167, respectively. While the differences inthe uncertainty reduction among the different param-eters partly reflect the soundness of their prior

characterization, these results can also be used to dic-tate which parameters of our mathematical modelcan be reasonably delineated through an inverse solu-tion exercise and which ones warrant direct estima-tion in the field (assuming that they represent realecological process and they are not ‘‘effective’’ param-eters used for modeling purposes).

Joint Parameter Estimation With the Hypoxic VolumeDataset

The lessons learned from the individual parameterestimation were used to optimize the size of the cali-bration vector examined, thereby avoiding an over-inflation of the predictive uncertainty during the jointparameter estimation exercise. The main criterionused to decide which parameters should be kept con-stant in the subsequent numerical experiments ofjoint parameter estimation was primarily the stabil-ity of the central tendency of the posterior parameterdistributions and secondarily the ‘‘narrowness’’ of thecorresponding credible (uncertainty) intervals. Thatis, we selected parameters with relatively similarmost likely values over time and high confidence (orlow uncertainty) about these values. Based on theprevious results, the parameter K appears to fullysatisfy this criterion when using the hypoxic volumedata. Relatively stable posterior means were alsofound for the parameter F, but the correspondingcredible intervals were quite wide. The parameter kd

was another candidate for being treated as constantthat has the additional advantage of maintaining thetrend of temporal stability with both the hypoxic vol-ume and the DO data. Yet, the high model error (andrelatively low R2) resulting from the latter case (seeTable 1) suggests that the values assigned to the restof the parameters restricted this univariate search ina suboptimal region of the model posterior and thusour exercise may not reveal the full range of dynam-ics associated with the parameter kd. Because we alsoassumed that the scaling factor K likely demonstratesless temporal variability, we selected this parameterover the sub-pycnocline decay rate kd to remain con-stant for the subsequent numerical experiments. Inparticular, we assigned the original value derived byScavia et al. (2006) when estimating different combi-nations of the remaining parameters (Di, v, kd, F).

Subsequently, we conducted nine numerical experi-ments: (1) joint estimation of year-specific Di and Fvalues; (2) joint estimation of year-specific Di, kd, andF values; (3) joint estimation of year-specific Di, v,and F values; (4) joint estimation of year-specific vand temporally constant Di, F values; (5) joint esti-mation of year-specific F and temporally constant Di,v values; (6) joint estimation of year-specific Di and



temporally constant F, v values; (7) joint estimationof year-specific kd and temporally constant F, Di val-ues; (8) joint estimation of year-specific F and tempo-rally constant kd, Di values; and (9) joint estimationof year-specific Di and temporally constant kd, F val-ues. In all cases, the parameter not being part of thecalibration exercise was set equal to the value usedin Scavia et al. (2006).

Among the nine experiments, the R2, DIC, andmodel residual standard error values indicate thatexperiments (1), (3), and (9) were associated with thebest model performance (Table 2, Figure 3, and Sup-porting Information). Notably, when using the DICas the primary criterion, the ninth experimentappears to overwhelmingly outperform the rest of theparameter combinations examined, while the fifthone resulted in the worst fit to the observed data.The latter finding challenges the practice of treatingthe parameters v and Di as temporally constant, sug-gesting that it is certainly more appropriate to baseour hypoxic volume predictions on year-specificestimates for these particular parameters. We alsocompared the posterior estimates from the best per-forming parameter combinations with those reportedby Scavia et al. (2006). The purpose was to explorethe variability of the parameter estimates and todetermine which ones should be retained constant infuture modeling exercises. For example, the joint esti-mation of year-specific Di and F (first experiment)provided Di values higher than in Scavia et al.(2006), while the F estimates were generally aroundthe 0.85 value reported in Scavia et al. (2006). Whenestimating year-specific Di, v, and F simultaneously(third experiment), the predicted annual hypoxic vol-umes are quite close to the observed ones based againon F values that range around the 0.85 level,although the Di and v values differ significantly fromthose reported in Scavia et al. (2006). Joint estima-tion of year-specific Di and temporally constant kd

and F values (ninth experiment) provides kd and Fposterior estimates that are slightly lower and

year-specific Di values that are higher compared toScavia et al. (2006). Other interesting comparisonsare provided in the Supporting Information section.

Robustness Analysis to the Specificationof the Parameter Priors

We examined the robustness (i.e., sensitivity) ofthe hypoxic volume predictions to the specificationof the parameter priors by doubling and cutting inhalf the precision of the original priors. For illustra-tion purposes, two cases are presented herein: esti-mation of Di alone (Figure 4) and joint estimation ofyear-specific v with temporally constant Di and Fvalues (Figure 5). When doubling the precision ofthe parameter priors (i.e., cutting the variance inhalf), the parameter space becomes narrower andthus the MCMC sampling focuses on suboptimalregions. Consequently, the overall model perfor-mance became relatively worse, as indicated by theR2 values and the wider credible intervals of thepredicted hypoxic volumes. When the precisionswere reduced (i.e., increasing the variance), theprior parameter space was expanded, which in prin-ciple increases the odds of locating the global optimaof the model. The fact that there was no significantimprovement of model performance increases ourconfidence that the original specification of the infor-mative priors did permit sufficient coverage of themodel likelihood and that the posterior inferencesdrawn herein were not biased from the selection ofthe parameter priors.

DISCUSSION

Given the growing evidence that hypoxia is per-haps one of the greatest threats to the integrity of

TABLE 2. Goodness-of-Fit of the Hypoxic Volume Model Based on Different Parameter Combinations.

Estimated Parameters R2 DIC

Model Residual Standard Error

Mean 2.5% 97.5%

1. Year-specific Di and F 1.00 388 0.13 0.04 0.442. Year-specific Di, kd, and F 0.99 585 0.45 0.24 0.913. Year-specific Di, v, and F 1.00 564 0.44 0.04 1.934. Year-specific v, temporally constant Di and F 0.98 374 1.10 0.30 2.155. Year-specific F, temporally constant Di and v 0.25 601 2.31 1.84 2.946. Year-specific Di, temporally constant F and v 0.29 182 2.28 1.76 2.967. Year-specific kd, temporally constant F and Di 0.95 514 0.74 0.50 1.068. Year-specific F, temporally constant kd and Di 0.86 557 1.09 0.80 1.489. Year-specific Di, temporally constant kd and F 1.00 44 0.25 0.05 0.69

Note: DIC, deviance information criterion.



aquatic ecosystems, the question of model credibilityand the development of rigorous methods for assess-ing the uncertainties associated with parametervalues and predictions have recently received consid-erable attention in hypoxia modeling. Scavia et al.(2006) used a deterministic Streeter–Phelps model topredict Chesapeake Bay DO concentrations andhypoxia volume conditions, and assessed the uncer-tainty associated with the initial DO deficit (Di)and the calibration term (v) with Monte Carlo simu-lations. More recently, Stow and Scavia (2009) intro-duced a Bayesian configuration of the same model todemonstrate a systematic multiple parameter estima-tion and to assess the uncertainty underlying the DOprofile model predictions. The present analysisextended the use of this Bayesian framework to fitthe model against hypoxic volume data and examinedhow parameter estimation and predictive uncertaintycan vary depending on the type of data used tocalibrate the model.

Examination of the hypoxic volume case stemsfrom the pragmatic need to overcome the lack ofDO data (e.g., when only hypoxic volume or area esti-mates are available), and to improve the predictivepower of hypoxia modeling given that the decision-making process is often guided from the hypoxiavolume projections rather than the forecasted DOconcentrations. The switch from DO concentrations to

hypoxic volume data though raises a basic problem inthat model fitting is now based on one datum peryear, that is, annual hypoxic volume, and thereforethe empirical estimation of year-specific parametersbecomes problematic. In this regard, the dilemmaarising is that the time-varying parameters are morerealistic and can offer insights with respect to regimeshifts or temporal changes of the ecosystem function-ing but, from a statistical inference standpoint, thefitting process lacks replication. To address this prob-lem, we opted for informative priors founded upon lit-erature information and previous knowledge from thesystem (Arhonditsis et al., 2007). The incorporation ofadditional information along with the dataset duringthe model fitting process aimed to reduce the dispar-ity between what we wanted to learn from the modeland what was actually observed is the primary rea-son for the poor model identifiability (Reichert andOmlin, 1997). While the adoption of informativepriors is arguably a valuable way to alleviate theproblem of model overidentification, the impact of thesomewhat controversial inclusion of prior informationon the robustness of the model results was furtherassessed by relaxing the confidence placed on thesepriors.

Our first set of numerical experiments (i.e., indi-vidual parameter estimation) indicated that the cali-bration of the Streeter–Phelps model against the

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FIGURE 5: Sensitivity of the Hypoxic Volume Predictions to the Prior Distributions Assigned to the Parametersv, Di, F. (A) The original priors are based on the following normal distributions: Di � N(0.30,0.70), F � N(0.75,60)I(0,1),

and v � N(2.5, 1.7)I(0,); the original precisions are replaced by double (B), and half precisions (C).



hypoxic volume data provides a good fit along withreasonable predictive uncertainty, although the vari-ances of the marginal posterior parameter distribu-tions were greater than those obtained from fittingthe model to the observed DO profiles. While the lat-ter result may cast doubt on the ability of thisapproach to offer insights into the ecosystem func-tioning, the substantial degree of updating of some ofthe parameter posteriors (i.e., the initial DO deficitand the scaling factor K) suggests that our knowledgedid improve relative to what we knew prior to thecalibration. Furthermore, the limited uncertaintyreduction of kd suggests that the relatively narrowprior characterization was sound; and thus is ade-quate for achieving optimal model fit. Our analysisalso indicates that the hypoxic volume predictions aremore sensitive to the prior specifications of someparameters (F, v), and therefore the predictive uncer-tainty can be significantly controlled by formulating,whenever possible, prior distributions with centraltendency and dispersion values that more realisti-cally represent the dynamics of the system. Forexample, it seems feasible to delineate an articulateprior for the fraction of surface organic carbon pro-duction that settles below the pycnocline (F) in theChesapeake Bay, but it may be more difficult withthe net downstream advection v. The ambiguity asso-ciated with the interpretation of the parameter v canconceivably be addressed by a spatially explicit ana-lytical solution of the original differential equation ofthe Streeter–Phelps model, which will relax theassumption of constant flow velocity along the sys-tem. The existing hydrodynamic work in the area caneffectively meet the data requirements of the refor-mulated model (e.g., flow profiles), whereas an error-in-variables (or measurement error) model may thenbe used to accommodate the significantly reduced(relative to the present approach) temporal variabilityof the parameter v.

The results derived from the joint estimation ofdifferent parameter combinations can be used todetermine which parameters should have year-specific or temporally constant character, and thusto develop more parsimonious models that effec-tively balance between performance and complexity.Parameters with relatively stable posterior meansover time and, if possible, narrow credible intervalscan be considered as temporally constant, whilethose with time-varying posterior patterns can beused to accommodate the interannual variability byassigning year-specific values. For example, similarto the Scavia et al. (2006) study, we found that theposteriors of the fraction of the surface organiccarbon production that settles below the pycnocline(F), the scaling factor (K) and the BOD decay coeffi-cient (kd) displayed little variation over the 36

years when fitted against the hypoxia volume data(Figure 2); so we may be able to hold them con-stant for hypoxia forecasts without missing much ofthe interannual system variability. By contrast, theposteriors of the initial DO deficit (Di) and the cali-bration term (v) varied significantly over time, pro-viding evidence that the assignment of temporallyconstant values is probably inappropriate. Moreimportantly, the year-to-year variations of their val-ues likely reflect a gradual change in the prevailingconditions in the system and ⁄ or a possible regimeshift.

In this regard, our approach can conceivably assistwith future hypoxia forecasting, as the time-varyingparameter estimates can be linked to potentiallyimportant ecological ⁄ physical process changes.Namely, our model parameterization revealed anincreasing trend for the upstream DO deficit,whereas the v values were characterized by aU-shape pattern with a global minimum in the early1980s. A possible explanation for why v decreased isthat during the earlier period, increased surface pro-duction was actually moving down estuary past theboundary of the Streeter–Phelps model and outside ofthe mesohaline region that influences hypoxia (Liuand Scavia, 2010). This is also consistent with Har-ding and Perry’s (1997) assertions that showed Chl aconcentrations in the mesohaline portion of the Bay(inside the boundary of our model) remained rela-tively constant after the 1950s while concentrationsin the polyhaline region (outside the boundary of themodel) increased; presumably in response to increas-ing loads (see Figures 1 and 2 of Harding and Perry1997). In contrast to the earlier period, the v increaseafter the 1980s is consistent with empirical evidencethat, for a given freshwater discharge rate, the recentobserved hypoxic volume in Chesapeake Bay havebeen two or more times greater than that of 1949 tothe 1980s. The distinctly higher Di values during thesame period are also on par with the latter assertion,suggesting that a threshold of nutrient loading aswell as in the ecosystem buffering capacity has beenreached. Apparently, the Bay has become more sus-ceptible to the development of hypoxic conditionsafter the 1980s, and since then the volume of hypoxicwaters has remained consistently high regardless ofthe hydraulic loading (Boesch et al., 2001).

This kind of structural changes and ⁄ or regime shiftwill definitely affect the forecasting statements sup-ported by the model. For example, we can expecthigher v and Di values for forecasting purposes inrecent years than in the 1950s-1970s, that is, for agiven nutrient load, higher v and Di values will leadto higher hypoxic volume projections, which impliesthat a higher loading reduction will be necessary forachieving the targeted hypoxic volume goal (Liu and



Scavia, 2010). The present modeling exercise has,thus, demonstrated the challenges and risks ofhypoxia control under the likelihood of ecosystemregime shifts. We also note that our illustration pos-tulates conditional independence among the year-spe-cific values of the different parameters, which maynot be the best strategy for forecasting purposes. Analternative approach would have been the sequentialupdating of the model in time (i.e., the posteriors ofone year provide the priors for the next year),whereby the updated information about the parame-ters will be ‘‘discounted’’ by adding a stochastic dis-turbance term to represent the aging of informationwith the passage of time. This type of dynamic model-ing framework may support probabilistic risk assess-ment that more effectively integrates past experienceand present information with the future response ofthe system (Dorazio and Johnson, 2003; Qian andReckhow, 2007).

Another critical test of the credibility of thehypoxia volume model is its ability to accuratelyreproduce DO concentrations in the estuary, giventhat the DO data are not part of the fitting process.To assess the extent to which this premise holds true,we illustrate the comparison between the predictedDO profiles from the two approaches when estimatingyear-specific v and temporally constant Di and F(Figure 6). Despite the accurate hypoxia volume

predictions presented herein, we noted that there areclear differences between the two predicted DOprofiles. Namely, the calibration of the model againstthe hypoxia volume data tends to result in higherdiscrepancies from the observed DO levels along theestuary and relatively similar conclusions weredrawn from the rest of the numerical experiments (notpresented here). That is, for some years, the hypoxiavolume model accurately reproduces the length of theestuary below 2 mg ⁄ l but displaces the profile up ordown estuary. The question this raises is ‘‘Does thisdisplacement matter if the prediction goal is forhypoxia volume?’’ We suggest that, in most cases, itdoes not. If, however, the spatial distribution of lowoxygen is of primary interest, other models with morerealistic physics will likely be required.

The Bayesian inference techniques can help withthe quantification of the predictive uncertainty whichis important information for model-based decisionmaking. Bayes’ Theorem provides a rigorous frame-work to estimate predictive uncertainty based on agiven model structure and data. For the purpose ofprediction, the Bayesian approach generates a poster-ior predictive distribution that represents the currentestimate of the value of the response variable, takinginto account both the uncertainty about the para-meters and the uncertainty that remains when theparameters are known (Arhonditsis et al., 2007,

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FIGURE 6. Comparison Between Observed (red line) and Predicted Mean (purple and black lines) Dissolved Oxygen (DO) Profiles in Chesa-peake Bay Using Year-Specific v Estimates, While Keeping Di and F Temporally Constant. Purple and black lines correspond to the predictedDO data from DO profile model and hypoxic volume model, respectively. The green line corresponds to the DO profile predicted from Scaviaet al. (2006), while the horizontal dashed line represents the hypoxia threshold, 2 mg ⁄ l. Distance (X-axis) is kilometers downstream from theSusquehanna River; Y-axis is DO (mg ⁄ l).



2008a,b). Therefore, estimates of the uncertainty ofBayesian model predictions are more realistic thanthose based on the classical procedures. Predictionsare expressed as probability distributions, therebyconveying significantly more information than pointestimates in regards to uncertainty. The presentationof the model outputs as probabilistic assessment ofwater quality conditions will make the model resultsmore credible and appealing to decision makers andstakeholders. Thus, the often deceptive deterministicstatements are avoided and the water quality goalsare set by explicitly acknowledging an inevitable riskof non-attainment, the level of which is subject todecisions that reflect different socioeconomic valuesand environmental concerns (Arhonditsis et al.,2008a,b). Furthermore, the Bayesian (iterative) nat-ure of the proposed approach is conceptually similarto the policy practice of adaptive management, i.e.,an iterative implementation strategy that is recom-mended to address the often-substantial uncertaintyassociated with water quality model forecasts, avoidthe implementation of inefficient and flawed manage-ment plans. Adaptive implementation or ‘‘learningwhile doing’’ augments initial model forecasts of man-agement schemes with post-implementation monitor-ing, i.e., the initial model forecast serves as theBayesian prior, the post-implementation monitoringdata serve as the sample information (the likelihood),and the resulting posterior probability (the integra-tion of monitoring and modeling) provides the basisfor revised (and improved) management actions(Zhang and Arhonditsis, 2008).

In conclusion, we used a Bayesian configurationof the modified Streeter–Phelps equation to predictthe DO profiles and the hypoxic volume in the Ches-apeake Bay for 36 years between 1950 and 2003.Our analysis suggests that fitting the model againstboth DO and hypoxic volume data provides realisticpredictions along with uncertainty bounds ofthe spatiotemporal hypoxia patterns, although theinference regarding the parameter posteriors cansignificantly vary depending on the type of data andcalibration parameters used. Although not illus-trated here, the present modeling exercise can alsoprovide the foundation of posterior simulations fordetermining the optimal nutrient loading reductions(with a realistic margin of safety) that can ulti-mately allow the system to comply with the targetedhypoxia volume goal. We also advocate use of infor-mative priors, whenever available knowledge per-mits, to alleviate the identification problems thatarise when switching from DO profile to hypoxia vol-ume data. Finally, the numerical experiments ofjoint parameter estimation presented herein canbe particularly useful for optimizing the modelcomplexity.

SUPPORTING INFORMATION

Additional results of our modeling analysis can befound in the online version of this article.

Figure S1. Comparison between observed andpredicted hypoxic volume (km3) values using year-specific parameter estimates. The lower and upperbars represent the 2.5 and 97.5% credible bounds.The red circles correspond to the observed values.

Figure S2. Hypoxic volume predictions based onjoint estimation of Di, v, and F.

Figure S3. Hypoxic volume predictions using vari-ous combinations of Di, v, kd, and F.

Figure S4. Parameter estimation with theStreeter–Phelps model fitted to the hypoxic volumedata: (a) joint estimation of year-specific Di and Fvalues; (b) joint estimation of year-specific Di andtemporally constant kd and F values. The red dotscorrespond to the Bayesian estimated values and thelines correspond to the values presented by Scaviaet al. (2006).

Figure S5. Joint estimation of the parameters Di,v, and F with the Streeter–Phelps model fitted to thehypoxic volume data.

Figure S6. Hypoxic volume predictions basedon various combinations of Di, kd, and F. The red cir-cle corresponds to the observed data. The diamondcorresponds to the values presented by Scavia et al.(2006). The triangle corresponds to the predictionswhen estimating year-specific kd values and keepingconstant F and Di. The cross corresponds to the pre-dictions when estimating year-specific F values andkeeping constant kd and Di. The square correspondsto the predictions when estimating year-specific Di

values and keeping constant kd and F.Table S1. Goodness-of-fit to the observed DO

profiles when the hypoxic volume dataset is used tocalibrate the Streeter–Phelps model.

Please note: Neither AWRA nor Wiley-Blackwell isresponsible for the content or functionality of anysupporting materials supplied by the authors. Anyqueries (other than missing material) should bedirected to the corresponding author for the article.

ACKNOWLEDGMENTS

The authors gratefully acknowledge the comments from twoanonymous reviewers and Dr. James D. Hagy III for use of thedata amassed in his dissertation. This work is contribution number116 of the Coastal Hypoxia Research Program and was supportedin part by grant NA05NOS4781204 from NOAA’s Center for Spon-sored Coastal Ocean Research, and in part by Scientific ResearchFoundation for the Returned Overseas Chinese Scholars, SEM andChina National Water Pollution Control Program (2008ZX07102-001). This is GLERL contribution number 1591.



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