Developing Total Maximum Daily Loads Under Uncertainty: Decision Analysis and the Margin of Safety

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UCOWRJournal of Contemporary Water researCh & eduCation

Developing Total Maximum Daily Loads underUncertainty: Decision Analysis and the Margin of Safety

Daniel P. Ames1 and Upmanu Lall2

1Idaho State University, Pocatello, ID; 2Columbia University, New York, NY

Universities CoUnCil on Water resoUrCes JoUrnal of Contemporary Water researCh & edUCation

issUe 140, pages 37-52, september 2008

Section 303(d) of the U.S. Clean Water Act (CWA) (U.S. Code 1972) details requirements for individual states and the

U.S. Environmental Protection Agency (EPA) to quantify existing contaminant levels and to take measures to improve water quality in impaired and threatened water bodies. These requirements include listing all impaired water bodies and conducting Total Maximum Daily Load (TMDL) analyses for all listed water bodies (33 U.S. Code 1313).

TMDLs must be developed with stakeholder participation and consensus that often highlights conflicts between environmental and economic objectives (Chen et al. 2004). Local and regional socio-economic impacts, limited scientific ability to evaluate and predict future water quality, hydroclimatic and ecological variation, and rapid changes in regional demographics and land use are difficult to assess. Indeed, implementing regulations needs to be a dynamic process (Maguire 2003); TMDL decisions are made under significant uncertainty with various associated risks that must be evaluated over time.

A plan must be developed to reduce pollutant input to the stream to a level below the TMDL with some margin of safety, by allocating the assimilative capacity of the stream for a particular pollutant among all sources. A report released by the National Research Council examining the scientific basis of the TMDL program, specifically suggests use of a, “Bayesian framework to determine preliminary probability distributions of impairment that can help direct monitoring efforts and reduce the quantity of monitoring data needed for making listing decisions at a given level of reliability”

(National Research Council 2000). The report also specifically calls for a reconsideration of the use of margin of safety in the TMDL program, such that is becomes based on uncertainty analysis, rather than arbitrary assignment. These recommendations are addressed in this paper within a Bayesian Decision Network framework.

TMDL Margin of Safety RequirementsGuidance documents provided by EPA (EPA

1991, 1997) represent the TMDL allocation problem as shown in Equation 1.

TMDL = ΣLA+ΣWLA+ MOS (1)

where: TMDL = Allowable total maximum daily load = Assimilative capacity for a particular waterbody and contaminant LA = Pollutant load allocation for nonpoint sources WLA = Pollutant load allocation for point source dischargers MOS = Margin of safety

The Clean Water Act requirement to take into account, “any lack of knowledge concerning the relationship between effluent limitations and water quality,” (section 303(d)(1)(c)) presents an interesting opportunity to explore decision analysis in a risk management context. Of particular interest is the “margin of safety” shown in Equation 1. The margin of safety is presented in the Clean Water Act as follows:

Such load shall be established at a level

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necessary to implement the applicable water quality standards with seasonal variations and a margin of safety which takes into account any lack of knowledge concerning the relationship between effluent limitations and water quality (CWA 303(d)(1)(c), emphasis added).

Guidance from EPA regarding the margin of safety is generally sparse. One EPA TMDL guidance document states that the margin of safety “is normally incorporated into the conservative assumptions used to develop TMDLs…” (EPA 1991). It is not clear from this guidance what makes an assumption conservative enough to satisfy margin of safety requirements. Presumably this decision is left to an EPA regulator tasked with approving the TMDL. Other EPA guidance suggests that, “If the margin of safety needs to be larger than that which is allowed through the conservative assumptions, additional margin of safety can be added as a separate component of the TMDL.” Here again is missing any guidance as to when conservative assumptions are not enough, or how much margin of safety needs to be added.

In practice there is no consistent meaningful application of a margin of safety. Of 13 approved TMDLs listed on an EPA web site (http://www.epa.gov/OWOW/tmdl/case.html ), six give no mention of a margin of safety, one arbitrarily set the margin of safety at 50 percent of the maximum load, and the remaining six simply include the following statement:

The margin of safety . . . was incorporated through the conservative assumptions used during TMDL development. If these conservative assumptions had been deemed insufficient, an additional margin of safety would have been added. (Colorado Department of Environmental Quality 1997).

These examples of guidance and sample TMDLs from EPA do not explicitly address the Clean Water Act requirement that the margin of safety should explicitly account for uncertainty. Conservative assumptions are only useful in a decision-making context when one can quantify how conservative the assumptions are. Additionally, the arbitrary addition of a number or a percentage to the anticipated load does not explicitly account for

uncertainty in the load estimate itself.The Clean Water Act implies that lack of

knowledge concerning the relationships in a watershed may result in an unsatisfactory estimate of contaminant load. This uncertain estimate carries with it the possibility or risk of violation of the water quality standard. The margin of safety is intended to reduce this risk of violation to some acceptable level. If one can quantify the risk of violating a stream standard given different management options, then risk serves as a means to evaluate alternatives.

Each state and the EPA will potentially invest hundreds of millions of dollars in the TMDL program over the next several years (EPA 1996). Given the current five-year renewal cycle for TMDLs, this expenditure could continue indefinitely. A risk-based approach to the margin of safety has the potential to improve TMDL decision-making and ultimately reduce costs. This approach will be demonstrated through a synthetic TMDL case study in a following section.

TMDL Challenges and NeedsThe TMDL program faces several challenges.

One of the most significant of these is associated with socioeconomic analysis. The local and regional socioeconomic impacts of regulating land use and other activities as part of a TMDL are as yet unclear. This is problematic because the Clean Water Act specifically requires that an assessment be made of the “economic and social costs of meeting Clean Water Act objectives in each state” as well as the “economic and social benefits of such achievement.”

Another significant TMDL challenge is associated with uncertainty in data and models. In many cases, there is rather limited data on water quality and quantity and associated causative factors. The scientific ability to assess current and predict future water quality is limited given this paucity of data as well as the wide range of hydroclimatic and ecological variation and rapid changes in regional demographics and land use. Because of this, TMDL decisions are typically made under considerable uncertainty.

In light of these requirements and challenges, there is a need for consistent and broadly applicable TMDL guidance. EPA has developed several

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Developing Total Maximum Daily Loads Under Uncertainty

Journal of Contemporary Water researCh & eduCation

guidance documents, such as Guidance for Water Quality based Decisions: The TMDL Process (EPA 1998a). Additionally, several states have also developed guidelines (e.g., Idaho Department of Environmental Quality 1999). These documents provide general principles for monitoring, priority ranking, and targeting of water bodies and TMDL development. EPA has also produced and released a GIS-based computer program, Better Assessment Science Integrating point and Nonpoint Sources (BASINS) (EPA 2001), which can be used to model contaminant input to streams from point and nonpoint sources. These and other EPA modeling tools (such as those listed in EPA 1998b) are provided to assess current and predict future in-stream water quality.

Although EPA documents and models provide general guidance, they do not explicitly address the social and economic impact of TMDL allocation decisions or risk and uncertainty. A decision analytic framework for TMDLs that addresses these issues using probability analysis and Bayesian networks is explored in the remainder of this paper.

A Bayesian Network Framework for TMDL Risk Assessment and Decision Analysis

Bayesian Networks for TMDL AnalysisBayesian networks (Pearl 1988) are graphical,

probabilistic models that explicitly account for risk and uncertainty in complex systems where causes and effects can be identified. A directed acyclic graph is used to represent the cause and effect relationships between variables in the system. Relationships between variables are defined through conditional probability distributions. This type of probabilistic framework lends itself to modeling complex systems of interrelated variables where relationships between variables are defined with some uncertainty. Additionally, Bayesian networks can be used to link decision variables to multiple separate endpoints. In this way, the probable effect of decisions can be propagated throughout the network and quantified at every variable. Equally useful is the ability of a Bayesian network to propagate the probable effect of an observation of any variable in the system to

all other variables. A more complete introduction to Bayesian networks and their use is given in Pearl (1988) and Jensen (1996).

There is some precedent for using Bayesian networks in watershed management, though not specifically for TMDLs. Haas and Cleaves (1997) applied a Bayesian network-based water eutrophication model on the Mokelumme River watershed in northern California. Additionally, work by Reckhow (1994, 1999) and Borsuk et.al (2001, 2004) demonstrates the application of Bayesian network uncertainty analysis on the Neuse River watershed.

One of the motivations for using a Bayesian network in TMDL analysis is to produce results that can be interpreted in the context of risk and uncertainty. The TMDL equation (1) is usually interpreted in a very deterministic manner, suggesting that the total allocated pollutant load can never exceed the water quality standard. However, most state water quality regulations do allow for some exceedance of water quality standards (e.g. “only one out of the most recent ten observations shall exceed the water quality standard.”) This inconsistency between TMDL guidance and water quality standards needs to be rectified. One approach is to recast Equation 1 in terms of risk of violating the standard:

P[(ΣLA + ΣWLA) > TMDL] < LAR (2)

Here, the probability, P, of the total load from point (WLA) and nonpoint (LA) sources exceeding the TMDL, or water quality standard, must be less than some level of acceptable risk (LAR). This type of risk analysis is explored further in the context of a synthetic case study in a following section.

Another motivation for applying Bayesian networks to TMDLs is because they provide a clear representation of the interconnections between management options, intermediate variables, water quality measures, and economic and ecological outcomes of interest.

Deriving the Structure of a Bayesian Network for TMDL Analysis

The first step to developing a Bayesian network model for TMDL decision analysis is to identify

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the outcomes or endpoints of interest. These may include economic costs and benefits to specific stakeholder groups as well as significant physical, chemical, or biological indicator variables. Next, management alternatives that may have an effect on these endpoints must be identified. Finally, a conceptual, causal network model of the system is developed that clearly identifies the physical and mechanistic connections between management alternatives, pollutant sources, natural and anthropogenic influences, intermediate variables, and important endpoints and outcomes of interest. This initial structure must be then be tested for validity and modified as needed so that all conditionally independent variables are represented as such.

The drainage network of a watershed forms a natural structure around which to build a Bayesian network. Control points at stream confluences can be useful intermediate nodes in the network when they represent the state of the contaminant at that point. Figure 1 shows a network of control points extracted from a portion of the hydrologic network of the Teton River in eastern Idaho. Such a diagram defines the flow of information through the physical network to control points of interest for decision-making.

With the addition of management options, socioeconomic, ecological, and other endpoints to the network diagram, the Bayesian network becomes a complete graphical representation of the cause and effect relationships associated with a particular set of management decisions. Figure 2 shows how the Teton River Bayesian network might look for management of sediment loads. In this network diagram, rectangular nodes represent three management options. These include “BMP Implementation,” “Reduce Road Density,” and “Reduce Road Usage.” The first management option, “BMP Implementation” is an agricultural “best management practice” in which farmers along Darby Creek control access of cattle to the stream. The two remaining management options relate to Forest Service management of unpaved roads used by recreators in the upper part of the Teton River. In this example, road density and road usage directly impact bank stability, which, in turn, impacts sediment loads in the upper Teton River.

Diamond-shaped nodes represent endpoint

outcomes of decisions. Agricultural costs and benefits result from BMP management and recreational costs and benefits result from management of Forest Service roads. Finally, attainment of the Salmonid spawning beneficial use of the Teton River at control points 3 and 4 is also represented in the network.

The size of a Bayesian network, and hence the number of variables that need to be characterized to populate it, grows rapidly as one moves downstream or considers more processes. However, recognizing that the state of the process at a particular node is known conditional on the state of the upstream nodes, it is possible to “cut” the intermediate nodes out of the system. This allows one to focus in on the direct problems of consequence. For example, if all that is needed is an estimate of the management options impact on Salmonid spawning, then the Bayesian network in Figure 2 could collapse down into four nodes: the three management options and the beneficial use attainment node. However, it is likely that an analyst will also be interested in the risk of violation of the sediment water quality standard at the control points. In this case the full network may be needed.

Note that even when a large network representation is produced, each node only depends on a few others. In this way, the Bayesian network is used to decompose a large and complex problem into a series of smaller problems that can be solved sequentially, and potentially more easily.

Populating a Bayesian Network for TMDL Analysis

Once a Bayesian network structure is defined, several sources of information can be used to populate the conditional probability distributions that connect network nodes. Potential sources of information include: at-site data or data from similar water bodies, results of mechanistic model simulations, expert opinion, and stakeholder surveys. Each of these sources of information will have varying degrees of reliability that may need to be explicitly accounted for in the conditional probability distributions.

At-Site DataA primary goal of TMDL development is

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to accurately characterize the relationships between the physical, biological, chemical, and socioeconomic aspects of the system. Such a characterization should provide decision-makers with an assessment of the probability of beneficial use attainment and stakeholder costs and benefits under current conditions as well as under different management actions, climate conditions and changes in demographics or land use. In an ideal case, characterization of the system through conditional probability distributions would be completed though the use of historical at-site data. For example, measurements of the target contaminant at the control point under a variety of historical conditions could be used to estimate the conditional probability distribution relating those conditions to contaminant levels.

Mechanistic ModelsIn most cases, adequate at-site data for such a

characterization are not available. Often this leads to the use of mechanistic watershed and receiving waters loading models to estimate contributions from different sources at specified control points under existing or projected conditions. Such models are intended to quantitatively describe

different components of the hydrologic balance in the watershed including overland flow, infiltration, subsurface flow and channel flow. Physical, chemical and biological processes attendant to the fate and transport of contaminants through the hydrologic system can also be modeled. A computer model such as QUAL2E can be used for in-stream routing of contaminants (Brown 1987). Ames et al. (2005) use such an approach to derive conditional probability distributions for Phosphorous in East Canyon Creek, Utah using Monte Carlo simulations.

Unfortunately, scarcity of data often makes the calibration and validation of such models difficult. Natural variability of meteorological parameters used to drive such models compounds the uncertainty associated with their use. Consequently such models are best used in a diagnostic rather than predictive context. Specifying scenarios for natural variability (e.g., climate) and for management options (e.g., land use) and their respective likelihood can provide a diagnostic context. Factors that complicate the use of deterministic mathematical models include, but are not limited to, natural and human induced variability in loadings, seasonal variation in

Figure 1. Extraction of a network of control points from a hydrologic network. A portion of the hydrologic network of the Teton River watershed in eastern Idaho is shown in (A). A network of control points for probability modeling of a particular contaminant (e.g. sediment) extracted from the hydrologic network is shown in (B). This is extended to a Bayesian network in Figure 2.

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anticipated biological response, and the possibility of catastrophes or rare occurrences not accounted for in the model.

Regional DataIn cases where at-site data are limited and

mechanistic models are not suitable, information from similar water bodies can be used as part of a regional analysis to estimate the probable at-site conditions. This approach involves the use of a database of regional historical data and a set of nonparametric algorithms and classification methods to identify causal relationships in similar watersheds.

The following steps could be followed to do this kind of analysis: (1) assemble a database on past loads and river flows over the river network; (2) identify key physical attributes of the watershed or reach in terms of drainage area, soil type, vegetative cover, land use, mean discharge, point loads, etc.; (3) identify similar reaches based on these attributes at other locations in the regional database; (4) develop probabilistic relationships

for potential loading by sources given the selected attributes using similar reaches. An analysis such as this was used to estimate the concentration of phosphorous in tributaries to the Teton River based on physical characteristics of the tributary sub-watersheds.

Point Source LoadsCharacterization of point source loads can

be done through the use of National Pollution Discharge Elimination System permit monitoring records at point source discharge facilities. These records can be used to estimate the average load of a contaminant to the receiving water as well as seasonal, diurnal, and annual variations. A probability distribution derived from such data may need to be augmented with data from similar point sources. For example, it may be that an estimate of the load from a point source under a new management alternative is needed. In this case, one could use data from a similar point source where the management option has been implemented (e.g., a similar wastewater treatment

Figure 2. A possible Bayesian network for sediment management in the Teton River watershed. Management options (rectangular), endpoints (diamond-shaped), and intermediate nodes (rounded rectangles) complete the network.

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plant that is already using a proposed treatment method.) Another challenge for point source load estimation is accounting for potential upsets or equipment failures. These low probability events have the potential for being the most damaging to a natural system.

Socioeconomic ImpactsIn many cases, standard measures of economic

welfare can be used as a metric to assess stakeholder cost or benefit levels. For example, profit levels for a factory owner or rancher can be used to quantify the costs or benefits resulting from specific management actions. Additionally, measures of consumer welfare can be used to quantify the benefits or costs incurred by recreational users of a water body as a result of different water quality levels. For example, the number of fish in the stream could be an indicator that contributes to recreational user welfare. Socioeconomic analysis can also be used to develop models of human behavior in response to uncertain variables. In this way, the operation of a farm or factory in response to climate or prices, for example, could be modeled in the Bayesian network as a predictor of pollutant loading from those sources.

Use of a Bayesian Network for TMDL Analysis

A Bayesian network framework for TMDL assessment provides a structured way to deal with issues of uncertainty and equitability as part of a collaborative management process. A key precept of this approach is that the focus is on the main interrelationships between variables, recognizing that available information is and will be incomplete. Thus the sensitivity of outcomes to decisions can be analyzed. For instance, a decision may be to collect additional data on nonpoint source contaminant loads. The question is whether it is useful to invest in this data collection effort. Given an initial assessment of the nonpoint source load and a proposed monitoring plan, the Bayesian network may be used to estimate the likelihood that that the collection of additional data will significantly change the nonpoint source load estimate. This is a type of sensitivity analysis where the effects of changes in one part of the

system (e.g., fertilizer application on a farm) are propagated probabilistically throughout the system (e.g., to an estimate of stream loading). Such an analysis can be particularly useful when data collection funds are limited and must be directed where the resulting information is most likely to be important.

Take as another example the decision to establish a riparian buffer of a certain width along a stream segment to reduce the amount of nonpoint source pollutants reaching the stream. For such a decision, an analyst may choose complex modeling of surface-subsurface flow interactions, leaching processes, overland flows and pollutant transport for a variety of climatic scenarios. In a probabilistic modeling framework, the analyst could also use the results from prior applications of such models or historical at-site data or data from other stream reaches where a similar situation exists. Whatever the source of information chosen, the focus is organizing and analyzing system interconnections in the context of a decision process rather than on detailed modeling of the ambient processes themselves.

A Bayesian network approach focuses on risks associated with watershed management activities. Such risk can be defined in terms of cost (what is the probability that the management plan will cost a certain amount?), standard violation (what is the probability that temperature will exceed some value for a particular period of time?), or ultimate stream health (what is the probability that observed numbers of macro-invertebrates will be less than expected?) In each case, the goal of watershed management is to reduce risk to a level that is acceptable to the affected parties — local stakeholders and regulators alike.

A complete Bayesian network can be used to evaluate the cost, benefit, and risk associated with management options. Costs and benefits derived by or allocated to each stakeholder for each management option can be identified. Ultimately, decision-makers are presented with the probability of success (or risk of failure) of each management option and can presumably make decisions that have a low risk of failure and a high probability of benefiting stakeholders in an equitable manner.

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A Synthetic TMDL Care StudyConsider a watershed with a single stream

as illustrated in Figure 3. Assume that the only designated beneficial use of the stream is “coldwater biota” and that it has historically supported a blue ribbon trout fishery. Additionally, assume that the stream is 303(d) listed for only one pollutant, biochemical oxygen-demanding organic matter (BOD). BOD loadings to the stream tend to reduce the amount of dissolved oxygen available to trout, thereby impacting the trout fishery. For water quality accounting purposes, the stream has been divided into three reaches corresponding to three divisions in the watershed (Figure 3). A TMDL would potentially have to be written for each of these 3 reaches. For the purposes of this case study, only conditions at the outlet of the watershed will be considered.

Stakeholders and Competing InterestsThree sources of BOD loading to the stream

are considered. These include natural unregulated sources, primarily in the Forest Service lands; a diffuse, nonpoint source along the length of Reach 2 associated with cattle at the ranch; and one point source, the sugar beet factory. It is assumed that there is no interest in trying to control the natural sources of BOD to the stream, but that some regulation of loads due to the ranch and the sugar beet factory may be needed to maintain dissolved oxygen water quality standards at the watershed outlet. The possibility of such regulation on the rancher and factory owner gives them a strong interest to participate in the TMDL development process for this stream.

Another significant stakeholder group in this scenario is composed of recreational anglers who use the blue ribbon trout fishery in Reach 3. This stakeholder group does not significantly impact water quality, but benefits from water quality improvements and is highly supportive of actions that would reduce loadings of BOD to the stream. In addition to the factory owner, the rancher, and the trout anglers, federal and state agencies such as the EPA and state Department of Environmental Quality are also stakeholders in this process. Typically the primary interest of the agency stakeholders is successful completion

of the TMDL process, rather than some particular interest associated with sugar beets, trout, cattle, or BOD.

Management AlternativesThe set of load allocation management

alternatives should include all options and combinations of options available to the decision-makers in the system. This set of all management options is likely to be large and of high dimension. Some options may not result in BOD allocations that satisfy the TMDL requirement, or do so only at prohibitive cost to the stakeholder community. This cannot be fully determined until the impact of each option is assessed through modeling of the physical and socioeconomic system. The challenge for the EPA, Department of Environmental Quality and stakeholders is to identify management options that ex ante appear to provide a high likelihood of meeting the management objectives.

Other considerations for reducing the set of options to be considered may be political. For instance, best management practices (BMPs) can be implemented through incentive-based EPA and National Resource Conservation Service programs. If BMPs on the cattle ranch appear to have a high probability of reducing BOD and satisfying the TMDL, they are a logical choice to be considered in the set of management options that are examined in detail.

It may be the case that few legally enforceable load allocation options are available. In this event, other considerations can be used to focus on the management options that are most likely to provide a solution to the TMDL management problem. For example, informal estimates of point and nonpoint BOD loading or informed guesses as to the costs of reducing BOD loading can reduce the set of management options to be considered.

For this case study, only management alternatives at the ranch and at the sugar beet factory are considered. In particular, the rancher has two alternatives, either do nothing (status quo) or implement a BMP. The proposed BMP, as shown in Figure 3, involves building fences to create a riparian buffer zone of a specified width along the stream to protect it from direct inputs of cattle manure. Management alternatives available to the factory owner include operating

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under existing NPDES permits (status quo) or building a wastewater treatment facility. The proposed wastewater treatment facility requires a significant expenditure by the factory owner to treat effluent waters, reducing BOD loadings. Potential management scenarios for the TMDL include all four combinations of the available management alternatives at the ranch and factory. Each of these scenarios translates into a different risk for violation of the dissolved oxygen standard in the stream and also to markedly different costs and benefits to the different stakeholders.

Case Study Bayesian NetworkA simple Bayesian network representation

of this example case study is shown in Figure 4. Here the management alternatives described previously are shown as rectangular nodes in the diagram labeled Ranch BMP and Factory Mgmt. Ranch BMP has two potential states, Status Quo

and Impose Riparian Buffer. If the option Impose Riparian Buffer is selected, the probable BOD loading from the ranch decreases. This presumably results in the probability of higher dissolved oxygen in the stream, an improved trout fishery and an increased likelihood of attaining the coldwater biota beneficial use. Dissolved oxygen in the stream is also affected by BOD loading from the factory. As described previously, the management alternative at the factory is installation of a wastewater treatment system to reduce effluent BOD. Costs and benefits realized by the factory owner, the rancher, and the angling community are also represented as nodes in the network.

Risk AnalysisThe TMDL equation (1) is a simple summation of

waste load allocations between point and nonpoint sources of contaminants. A margin of safety is built in to account for unforeseen variability and changes in the system. This margin of safety is typically a set value that is meant to assure that the total load allocation to point and nonpoint sources never exceeds the TMDL. As discussed previously, this is often a relatively arbitrary value with little meaningful interpretation. A potentially more useful approach is to consider a risk of violation metric instead of the margin of safety. Risk of violation is a measure of the probability of violating the water quality standard at a particular control point given different management scenarios (as in Equation 2). The Bayesian network shown in Figure 4 can be used for such an analysis by propagating uncertainty from decisions, through key variables, and to critical endpoints.

Assume that the Bayesian network for this case study has been populated with continuous conditional probability distributions at every node, derived from either statistical analyses of historical water quality and source data, or by using a physical-chemical transport model together with stochastic climate and economic scenarios. Using the Bayesian network, the resultant probability density function (PDF) of dissolved oxygen at the watershed outlet can be computed for different management alternatives. From this PDF, one can estimate the risk of violating the in-stream water quality standard for dissolved oxygen as illustrated below.

Figure 3. Diagram of a synthetic watershed for TMDL analysis. In this watershed, a single stream passes through U. S. Forest Service lands, cattle grazing lands and urban lands. A nonpoint source (cattle ranch) and a point source (sugar beet factory) are identified. Ad-ditionally, two management alternatives are shown. These include forming a riparian buffer in the grazing lands to separate cattle from the stream, and building a wastewater treatment plant at the sugar beet factory.

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Consider dissolved oxygen concentrations under the hypothetical situation that there is no ranch and only the factory point source is of interest. The estimated PDF of dissolved oxygen at the watershed outlet under this scenario is shown in Figure 5. Note that in this and all subsequent figures, probability distributions were simulated as mixtures of Beta distributions for illustration purposes. The average dissolved oxygen in this case is 6.6 mg/L. Assuming that the water quality standard for dissolved oxygen is 5 mg/L, then the standard is violated about 3.5 percent of the time. This may be due to occasional exceedance of the waste storage capacity at the factory, or due to malfunctions or other unanticipated problems.

Given this scenario, and a typical TMDL interpretation, one might compute the margin of safety as 1.6 mg/L (the average dissolved oxygen concentration minus the dissolved oxygen standard). The question then becomes, is this sufficient margin of safety? As discussed previously, there is no clear guidance in the TMDL program to answer this question. One could argue that this is a 32 percent margin of safety (1.6/5) and is amply sufficient. On the other hand, as shown in Figure 5, this only represents the average dissolved oxygen as compared to the standard. In fact, even with the 32 percent margin of safety, there is still a 3.5 percent risk of violation.

Now consider the hypothetical situation where there is no factory and one need only consider nonpoint contributions from the ranch. The estimated dissolved oxygen PDF for this case is shown in Figure 6. Here the average dissolved oxygen concentration at the watershed outlet is 7 mg/L, slightly better than the previous case. For a ranch such as this, large BOD loads are likely associated with large runoff events that may, in turn, be driven by short intense storms (e.g., 100-year return period, 1 hour duration) or by continuous rainfall over a longer duration (e.g., a 10-year, 24-hour rainstorm). The PDF of dissolved oxygen in this case exhibits considerably more variation. The risk of violating the dissolved oxygen standard is about 11 percent, although the margin of safety is 2 mg/L or 40 percent. This suggests that an improved margin of safety does not always correspond with a reduction of risk.

The combined effect of the two sources (under status quo conditions) is shown in Figure 7. Here the average dissolved oxygen at the watershed outlet is 6.2 mg/L (24 percent margin of safety) yet the risk of violating the standard is nearly 23 percent. While BOD loads from the point and nonpoint source are largely independent in this example, the increased frequency of violation in this case reflects the additive effects of the two loadings and nonlinearity in transport and reaeration processes

Figure 4. Bayesian network representation of case study TMDL. In this Bayesian network, decision nodes for ranch and factory management result in BOD loadings to the stream and, ultimately dissolved oxygen and trout fishery conditions. Attainment of the beneficial use of the stream, as well as costs and benefits to stakeholders are shown as outcomes in the diagram.

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in the stream.Note that although in each of these scenarios

average dissolved oxygen satisfies the water quality standard, it is the low frequency, low dissolved oxygen events that constrain fish production in the stream and ultimately reduce recreational benefit for stakeholders. On the basis of the high risk of violating the dissolved oxygen standard, controls on BOD at both the ranch and factory should be considered. Referring to the Bayesian network for this case study, four management scenarios are available. These include (1) factory status quo and ranch status quo (Figure 7); (2) factory wastewater treatment with ranch status quo; (3) factory status quo with ranch BMP; and (4) both factory wastewater treatment and ranch BMP.

The estimated PDF of dissolved oxygen under scenario (2) is shown in Figure 8. This management scenario results in a PDF similar to the ranch-only case, with an average dissolved oxygen of 7 mg/L and a frequency of violation of 12 percent (margin of safety is 40 percent). The reduced likelihood of low frequency factory “upsets” is reflected in this estimate. The estimated PDF of dissolved oxygen under scenario (3), is shown in Figure 9. Here, a riparian buffer restricts animals to a specified distance from the stream. BOD loading into the stream from overland or subsurface flows diminishes rapidly as the land

sources are moved away from the stream. The result is a marked reduction in the risk of violation from 23 to 5 percent. It is important to note that the average dissolved oxygen is still 7 mg/L (margin of safety of 40 percent) but the risk of violating the standard has been significantly reduced. Under management scenario (4), the risk of violating the dissolved oxygen standard is reduced to 1 percent with an associated average dissolved oxygen of 7.2 mg/L (margin of safety of 44 percent). The PDF for this case is shown in Figure 10. It can be assumed that most of the residual risk is related to factory upsets.

This case study illustrates that TMDL management, load allocation, and margin of safety issues are better understood in the context of risk reduction rather than through estimates of average conditions relative to some standard. Appropriate understanding and management of the variability of the process is likely the key to effective policy implementation. If the management goal were simply to reduce risk to one percent or less, then management scenario (4) would be selected. A more reasonable management goal would consider the costs and benefits of risk reduction on the stakeholder community. Such an analysis may take advantage of opportunities for watershed trading, where stakeholders benefiting from management alternatives (e.g., anglers) bear a portion of the cost associated with implementing them. Also, it

Figure 5. Estimated PDF of dissolved oxygen (DO) at the watershed outlet due to releases from the sugar beet factory only. Factory releases generally do not violate the DO standard of 5 mg/L. However, approximately 3.5 percent of the time, unexpected loadings or reductions in treatment plant capacity can lead to serious violations.

Prob

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may be more cost-effective for the factory owner to supplement the cost of BMPs at the ranch, rather than to bear the cost of building a wastewater treatment facility.

Evaluation of Management OptionsThe process of evaluating and implementing

a TMDL management strategy is complex and can be divisive to the stakeholder community. Particular management options are likely to favor one segment of the stakeholder population while penalizing others. Additionally, individual stakeholders have different preferences for water resource management and may have varying political influence over the final management decision.

To simplify the process of evaluating management options, suppose that each option can be usefully ranked using two key characteristics. The first characteristic is the risk of water quality standard violation under each management option. The second characteristic is the impending welfare changes associated with each management option. Also, assume that the objective of the stakeholder community is to determine which management option satisfies the dissolved oxygen TMDL requirement with reasonable probability and at an acceptable cost to the stakeholder community.

Management options that are incapable of reducing the risk of violating the dissolved oxygen standard to an acceptable level, or that impose an unacceptable cost on the stakeholder community may be eliminated from further consideration.

As mentioned previously, pollutant-trading plans can potentially be implemented to achieve economic equity and efficiency. For instance, anglers may be able to compensate the rancher to add a riparian zone. Angling benefits accrue in proportion to BOD reduction. Compensation offered would be contingent on the perceived benefits to the fishery and the expected cost of establishing the riparian buffer. The sugar beet factory owner may also be motivated to compensate the rancher for providing a riparian buffer zone. The corresponding reduction in the upstream BOD load may provide the factory owner with additional assimilative capacity for BOD at his discharge point.

In summary, the evaluation of management options involves the following major steps: (1) identification of structural and operational costs, (2) estimation of the efficacy of the management option in reducing the frequency and severity of adverse impacts, (3) estimation of the benefits (including non-monetary) to different stakeholders, (4) identification of the willingness of stakeholders to “trade” and reach a consensus viewpoint, and (5)

Figure 6. Estimated PDF of critical dissolved oxygen (DO) at the watershed outlet due to ranch operations only. BOD loading from the ranch is largely driven by precipitation. The resulting DO variability consequently has a positively skewed PDF, and the DO standard of 5 mg/L is violated about 11 percent of the time.

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recommendation of a load allocation, monitoring and trading plan.

Summary and ConclusionsBOD loading, dissolved oxygen concentrations,

flows and other important variables associated with water quality management undergo considerable variation in both time and space in response to

climate variations, changes in seasonal output from point and nonpoint sources and changes in land use. Only some of this variation can be regulated — much of it will always be present. Because of this, a probabilistic TMDL management framework based on the reduction of risk is called for. Bayesian networks have been presented here as a suitable framework to support such an

Figure 7. Estimated PDF of dissolved oxygen (DO) at the outlet due to factory and ranch operations. Note that the combined effect of the two sources leads to a violation of the DO standard of 5 mg/L about 23 percent of the time.

Figure 8. Estimated PDF of dissolved oxygen (DO) at the watershed outlet under scenario (2), a treatment plant at the factory. The DO standard of 5 mg/L is still violated about 12 percent of the time.

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Ames and Lall50


analysis. An approach for risk-based analysis using Bayesian networks has been presented. In some cases, economics and other cost and benefit measures such as those shown in Figure 4 can assist with determining the level of acceptable risk. In other cases the level of acceptable risk is or will be legislated.

Many important TMDL issues remain. How does one select critical monitoring locations and time sampling frequencies? How does one establish a level of acceptable risk? What will be the impact of climate change or major unforeseen changes in watershed land use? Should management policies explicitly consider rare natural occurrences, or

Figure 9. Estimated PDF of dissolved oxygen (DO) at the watershed outlet under scenario (3), a riparian buffer at the ranch. The DO standard of 5 mg/L is still violated about 5 percent of the time.

Figure 10. Estimated PDF of dissolved oxygen (DO) at the watershed outlet under scenario (4), a riparian buffer at the ranch and a treatment facility at the factory. The DO standard of 5 mg/l is violated about 1 percent of the time.

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should they be tied to routine conditions? The case study presented in this paper is a stylized example that overlooks these and many other complexities of the TMDL process. Nonetheless, it shows the potential utility for a new TMDL decision-making and watershed management framework based on uncertainty and risk analysis.

Author Bios and Contact InformationDaniel P. aMes is an Assistant Professor in the Department of Geosciences at Idaho State University. His research interests include GIS-based watershed modeling, decision support systems, Bayesian decision networks, geospatial hydrology, and development of open source geographic information systems. He can be contacted at [email protected].

uPManu lall is the Alan and Carol Silberstein Professor of Earth and Environmental Engineering and of Civil Engineering and Engineering Mechanics in the Department of Earth and Environmental Engineering at Columbia University. His research interests include hydroclimatology, nonlinear dynamics, applied statistics, natural hazards, water systems, risk management, and water technologies for developing countries. He can be contacted at [email protected].

References

Ames, D. P., B. T. Neilson, D. K. Stevens, and U. Lall. 2005. Using Bayesian networks to model watershed management decisions: an East Canyon Creek case study. Journal of Hydroinformatics 7(4): 267-282.

Borsuk, M. E., C. A. Stow, and K. H. Reckhow. 2004. A Bayesian network of eutrophication models for synthesis, prediction, and uncertainty analysis. Ecological Modelling 173: 219-239.

Borsuk, M., R. Clemen, L. Maguire, and K. Reckhow. 2001. A Multiple-Criteria Bayes Net Model of the Neuse River Estuary. Group Decision and Negotiation 10:355-373.

Brown, L. C. 1987. Uncertainty analysis in water quality modeling using QUAL2E: Case Studies. Available at: <http://www.epa.gov/OWOW/tmdl/case.html> (Dec. 29, 2001).

Chen C.W., J. Herr, and L. Weintraub. 2004. DecisionDecision support system for stakeholder involvement. Journal of Environmental Engineering-ASCE

130(6): 714-721. doi: 10.1061/(ASCE)0733-9372(2004)130:6(714).

Colorado Department of Environment Quality. 1997. Denver Metro, The South Platte River Segment TMDL.

Federal Register: August 23, 1999. “Proposed Revisions to the Water Quality Planning and Management Regulation: Proposed Rule.” 40 CFR 130. 64 (162): Part II.

Federal Register: August 23, 1999. “Revisions to the National Pollutant Discharge Elimination System Program and Federal Antidegradation Policy in Support of Revisions to the Water Quality Planning and Management Regulation, Federal Register: August 23, 1999 (Volume 64, Number 162) Page 46057-46089, Part III, Environmental Protection Agency 40 CFR Part 122 et al.

Haas, T. C., and D. Cleaves. 1997. Modeling Waterbody Eutrophication with a Bayesian Belief Network . Available at: <http://www.uwm.edu/People/haas/eutro.ps> (July 14, 2008).

Idaho Department of Environmental Quality. 1999. State of Idaho Guidance for Development of Total Maximum Daily Loads.

Jensen, F .V. 1996. Introduction to Bayesian Networks. Springer-Verlag New York, Inc., Secaucus, NJ.

Maguire, L. A. 2003. Interplay of science and stakeholder values in Neuse River total maximum daily load process. Journal of Water Resources Planning and Management ASCE. 129(4): 261-270.

National Research Council. 2000. Assessing The TMDL Approach To Water Quality Management. Committee to Assess the Scientific Basis of the Total Maximum Daily Load Approach to Water Pollution Reduction, Water Science and Technology Board, Division on Earth and Life Studies, National Research Council. National Academy Press. Washington, D.C.

Pearl, J. 1988. Probabilistic reasoning in intelligent systems: Networks of plausible inference. San Mateo, CA: Morgan Kaufmann.

Reckhow, K. H. 1994. Water quality simulation modeling and uncertainty analysis for risk assessment and decision making, Ecological Modeling 72: 1-20.

Reckhow, K. H. 1999. Water Quality Prediction, Mechanism, and Probability Network Models. Canadian Journal of Fisheries and Aquatic Sciences 56:1150-1158.

Revisions to the Water Quality Planning and Management Regulation; Proposed Rule. 40 CFR 122. 64 (162): Part III.

United States Code. Clean Water Act Section 303(d) [33

Ames and Lall52


USC 1313] Available at: <http://www4.law.cornell.edu/uscode/33/ch26.html> (Dec. 29, 2001).

U.S. Environmental Protection Agency. 1991. Guidance for Water Quality-Based Decisions: The TMDL Process. Assessment and Watershed Protection Division. EPA 440/4-91-001 April, 1991. Available at: <http://www.epa.gov/OWOW/tmdl/decisions/> (Dec. 29, 2001).

U.S. Environmental Protection Agency. 1996. TMDL Development Cost Estimates: Case Studies of 14 TMDLs. Office of Water. EPA-R-96-001.169.

U.S. Environmental Protection Agency. 1997. Technical Guidance Manual for Developing Total Maximum Daily Loads. Office of Water. EPA823-B-97-002.

U.S. Environmental Protection Agency. 1998a. Decisions in the TMDL Program. Available at: <http://www.epa.gov/OWOW/tmdl/decisions/dec1c.html> (Dec. 29, 2001).

U.S. Environmental Protection Agency. 1998b. Compendium of Tools for Watershed Assessment and TMDL Development. Available at: <http://www.epa.gov/OWOW/tmdl/comptool.html> (Dec. 29, 2001).

U.S. Environmental Protection Agency. 2001. BASINS. Available at: <http://www.epa.gov/OST/BASINS> (Dec. 29, 2001).

Developing Total Maximum Daily Loads Under Uncertainty: Decision Analysis and the Margin of Safety

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