How “Mechanistic” are Mechanistic Water Quality Models?

Mechanistic surface water quality models have been developed by scientists and engineers as mathematical descriptions of hydrologic and ecologic processes. Mechanistic modelers have tended to concentrate on the mathematical expression of theory, probably as a consequence of: (1) scientific interest and challenge, (2) a belief that the theory was reasonably well-understood and that this understanding could be expressed mathematically, (3) limited available data to fit and evaluate models, and (4) limited resources to collect additional data. For these reasons, model coefficients and reaction rates in mechanistic models generally are intended to characterize actual processes and are not (prior to model “tuning”) intended to be empirically-fitted constants (which might be considered an “effective” value for a model parameter).

Since the parameters of mechanistic models are intended to describe real processes, it may be assumed that an experimental study of a particular process can yield a parameter estimate that can be directly inserted into the model. In some cases, it is acknowledged that a reaction rate or coefficient in a model is affected by certain conditions in a waterbody (e.g., turbulence), and thus adjustments must be made to the experimentally-based value. However, if the model truly is a complete mechanistic description of the system of interest, then adjustment should be unnecessary; this is the underlying belief of modelers who advocate development of “physically-based” models.

However, given the relative simplicity of all simulation models in comparison to the complexity of nature, it seems reasonable to question the legitimacy of any "mechanistic" mathematical description of surface water quality. Further, given data limitations and scientific knowledge limitations, it seems reasonable to question even the goal to strive for a model that need not be calibrated. The correctness of model structure, the knowledge of the model user, and the availability of experimental and observational evidence all influence parameter choice for mechanistic models. Unfortunately, too often knowledge and data are extremely limited, making choice of parameters and choice of important model processes guesswork to a distressingly large degree. The example presented below is not re-assuring with respect to these two issues: (1) scientific support for the selection of model parameters, and (2) scientific support for the specification of appropriate model functional relationships.

One of the basic functions in an aquatic ecosystem model is phytoplankton settling. An early example of its use is in the model proposed by Chen and Orlob (1972):

   

  where:

         V = segment volume (m3)

         C1 = phytoplankton concentration (g/m3)

         Q = flow volume (m3/t)

         E = diffusion coefficient (m2/t)

         A = segment surface/bottom area (m2)

         µ1 = phytoplankton growth rate (t-1)

        R1 = phytoplankton respiration rate (t-1)

         s1 = phytoplankton settling rate (t-1)

         M1 = phytoplankton mortality rate (t-1)

         µ2 = zooplankton growth rate (t-1)

         C2 = zooplankton concentration (g/m3)

         F2,1 = fractional feeding preference

Other examples are quite similar; a common alternative approach is that phytoplankton settling is sometimes treated as a velocity term with an areal loss:

     phytoplankton settling (mass/time) = v1AC1                                                       

To understand some of the problems with the current approach for parameter determination in mechanistic surface water quality models, it is useful to examine this process further. For that purpose, "phytoplankton settling velocity" provides a good example. Phytoplankton, or algae, are important in aquatic ecosystems, and thus one or more phytoplankton compartments are found in most mechanistic surface water quality models concerned with nutrient enrichment. Phytoplankton settling is one of the key mechanisms for removal of phytoplankton from the water column.

Stoke's law provides the starting point for the mathematical characterization of phytoplankton settling. Few models, however, employ Stoke's law; instead a simple constant settling velocity (in units of length/time) expression is commonly used. To apply a model with this settling velocity term, a modeler must either measure phytoplankton settling directly, or select a representative value from another study. Since field measurement of phytoplankton settling is a difficult task, use of literature-tabulated values is standard practice.

Probably the most thorough listing of suggested values for phytoplankton settling velocity continues to be Bowie et al. (1985), which presents a thorough table of reported values, by algal type (see the table below). Bowie et al. note that under quiescent conditions in the laboratory, phytoplankton settling is a function of algal cell radius, shape, density, and special cell features such as gas vacuoles and gelatinous sheaths. For natural water bodies, water turbulence can be quite important. In two- or three-dimensional models with hydrodynamic simulation, turbulence is accounted for in the model equations; in zero- or one-dimensional models, the effect of turbulence on phytoplankton settling must usually be incorporated into the choice of settling velocity.

That information is typically the extent of technical guidance considered by modelers when selecting this parameter using a reference like the table from Bowie et al. The range of options in the table is substantial, even within a single category (e.g., diatoms) for algal type. The algal cell size, shape, and other features mentioned in the previous paragraph can vary from species to species within a single type category, so this may be responsible for some of the variability in the table. However, even if the modeler who must choose a point estimate has data that identify dominant species in a water body at a particular time and location, dominance is apt to change with time and location. Further, models contain at most only a few distinct phytoplankton compartments, so a choice must still be made concerning species to be modeled and their characteristics.

Examination of the original references from which the table was created does little to enlighten the parameter selection process. Most of the references summarized in the table do not present observational studies of phytoplankton; rather, they are simulation model studies, and the value for phytoplankton settling velocity listed in the table is the value chosen for the model. In some of the references checked, little or no basis was provided for the choice. When a rationale for choice was given, it was usually to adopt or adjust the few values presented in the literature from experimental studies, or to adopt a value from another modeling study. In one way or another, it appears that virtually all of the values presented in the table have some dependency on the early experimental work of Smayda and Boleyn (1965) and other work by Smayda.

Unfortunately, evaluation studies of simulation models have provided little insight on good point estimates for this parameter. Observational data on surface water quality are almost always inadequate for testing functional relationships and assessing parameter choices. Typical observational data sets are noisy, with few measurements of each of only a few variables. In the case of phytoplankton settling velocity, observational data are apt to consist of phytoplankton cell densities at various dates, times, and areal locations, but probably not depths. Since phytoplankton are also removed from the water column through consumption by higher food chain organisms, the observational data do not permit separate identification of the removal mechanisms.

Given this situation, modelers have relied almost exclusively on the few experimental studies in the laboratory and their judgment concerning adjustments to these values. For one-dimensional models without explicit modeling of hydrodynamics, the chosen value may be as much as an order of magnitude higher than the laboratory values. Two- or three-dimensional models with hydrodynamics may incorporate the unadjusted laboratory value. After early modeling studies presented chosen values, these values were sometimes adopted in subsequent studies without comment (in effect, "default" values were identified). Thus, there is probably much less information in the columns of the table than implied by the number of values reported.

In summary, the choices for phytoplankton settling velocity appear to be based on ad hoc adjustments to a few values measured under controlled conditions. There is virtually no field confirmation of choices made for parameters individually (as opposed to collectively). This situation is fairly typical of the state-of-the-art in mechanistic surface water quality simulation modeling.

Bowie, G.L., Mills, W.B., Porcella, D.B., Campbell, C.L., Pagenkopf, J.R., Rupp, G.L., Johnson, K.M., Chan, P.W.H., Gherini, S.A., and Chamberlin, C.E., 1985. Rates, Constants, and Kinetics Formulations in Surface Water Quality Modeling. U.S. Environmental Protection Agency, EPA/600/3-85/040.

Chen, C.W and Orlob, G.T., 1972. Ecologic Simulation for Aquatic Environments. Office of Water Resources Research, US Dept. of Interior. Washington, DC.

Smayda, T.I. and Boleyn, B.J., 1965. Experimental observations on the floatation of marine diatoms. Part I: Thalassiosira naria, T. rotula and Nitzschia seriata. Limnol. and Oceanogr., 10:499-510.

Dealing Effectively with Uncertainty

Are we better off knowing about the uncertainty in outcomes from proposed actions? That is, will our decisions generally be better if we have some idea of the range of possible outcomes that might result? I have always thought so, and yet current practice in water quality modeling and assessment suggests that others feel differently or perhaps believe that uncertainty is small enough so that it can be safely ignored.

Consider my experience from many years ago. While in graduate school, I became involved in a proposed consulting venture in New Hampshire. As a young scientist, I was eager to “shake up the world” with my new scientific knowledge, so I suggested to my consulting colleagues that we add uncertainty analysis to our proposed 208 (remember the Section 208 program?) study. Everyone agreed; thus we proposed that uncertainty analysis be a key component of the water quality modeling task for the 208 planning process. Well, after we made our presentation to the client, the client’s first question was essentially, “The previous consultants didn’t acknowledge any uncertainty in their proposed modeling study, what’s wrong with your model?” This experience made me realize that I had much to learn about the role of science in decision making and about effective presentations!

While this story may give the impression that I’m being critical of the client for not recognizing the ubiquitous uncertainty in environmental forecasts, in fact I believe that the fault primarily lies with the scientists and engineers who fail to fully inform clients of the uncertainty in their assessments. Partially in their defense, water quality modelers may fail to see why decision makers are better off knowing the forecast uncertainty, and perhaps modelers may not want to be forced to answer the embarrassing question like that posed to me years ago in New Hampshire.

For this situation to change, that is, for decision makers to demand estimates of forecast error, decision makers first need: (1) motivation - that is, they must become aware of the substantial magnitude of forecast error in many water quality assessments, and (2) guidance – they must have simple heuristics that will allow them to use this knowledge of forecast error to improve decision making in the long run. Once this happens, and decision makers demand that water quality forecasts be accompanied with error estimates, water quality modelers can support this need through distinct short-term and long-term strategies.

Short-term approaches are needed due to the fact that most existing water quality models are incompatible with complete error analysis as a result of overparameterization; thus short-term strategies should be proposed for: (1) conducting an informative, but incomplete error analysis, and (2) using that incomplete error analysis to improve decision making. In the long-term, recommendations can be made to: (1) restructure the models so that a relatively complete error analysis is feasible, and/or (2) employ Bayesian approaches that are compatible with adaptive management techniques that provide the best approach for improving forecasts over time.

In the short-term, if knowledge, data, and/or model structure prevents uncertainty analysis from being complete, is there any value in conducting an incomplete uncertainty analysis? Stated another way, is it reasonable that decision making will be improved with even partial information on uncertainties, in comparison to current practice with no reporting of prediction uncertainties? Often, but not always, the answer is “yes,” although the usefulness of incomplete uncertainty characterization, like the analysis itself, is limited.

Using decision analysis as a prescriptive model, we know that uncertainty analysis can improve decision making when prediction uncertainty is integrated with the utility (or loss, damage, net benefits) function to allow decision makers to maximize expected utility (or maximize net benefits). When uncertainty analysis is incomplete (and perhaps more likely, when the utility function is poorly characterized) the concepts of decision analysis may still provide a useful guide.

For example, triangular distributions could be assessed for uncertain model terms, and assuming that parameter covariance is negligible (which unfortunately may not be the case), then limited systematic sampling (e.g., Latin hypercube) could be used to simulate the prediction error. The result of this computation could be either over/under estimation of error, but it does provide some indication of error magnitude. However, this information alone, while perhaps helpful for research and monitoring needs, is not sufficient for informed decision making. The approximate estimates of prediction uncertainty need to be considered in conjunction with decision maker attitudes toward risk for key decision variables.

Implicit in this attitude toward risk is an expression of preferences concerning tradeoffs. For example, are decision makers (or stakeholders, or other affected individuals/groups) risk averse with respect to ecological damage, such that they are willing to increase project costs in order to avoid species loss? If a reasonable quantification of prediction uncertainty were available for the decision attribute - loss of an endangered species, then the prediction might be expressed as “there’s a 40% chance of loss of this species with plan A, but only a 5% chance of loss with plan B.” When costs of the plans are also considered, the tradeoff between species loss and cost is augmented by awareness of risk that comes from the prediction uncertainty characterization. Risk is not evident from deterministic (point) predictions of the decision attributes, so the decision is likely to be better informed with the risk assessment that is made possible with prediction uncertainty.

In the long run, a better strategy is to restructure the models, emphasizing the development of models that are compatible with the need for error propagation and adaptive assessment/management. Bayesian (probability) networks are particularly suitable for this task (see http://kreckhow.blogspot.com/2013/07/bayesian-probability-network-models.html), as are simulation techniques that address the problem of equifinality resulting from overparameterized models (see: http://kreckhow.blogspot.com/2013/06/an-assessment-of-techniques-for-error.html).

No one can claim that scientific uncertainty is desirable; yet, no one should claim that scientific uncertainty is best hidden or ignored. Estimates of uncertainty in predictions are not unlike the point estimates of predicted response. Like the point predictions, the uncertainty estimates contain information that can improve risk assessment and decision making. The approaches proposed above will not eliminate this uncertainty nor will it change the fact that, due to uncertainty, some decisions will yield consequences other than those anticipated. They will, however, allow risk assessors and decision makers to use the uncertainty to structure the analysis and present the scientific inferences in an appropriate way. In the long run, that should improve environmental management and decision making.

Are Periodic Beach and Shellfish Bed Closures a Cost of Development?

In the past ten years, I have assessed beach and shellfish bed closures due to exceedances of indicator organism criteria in both fresh and salt water. In each case, the problem appears to be caused by factors/sources that are unlikely to be effectively controlled to eliminate the exceedances. In some cases, the proliferation of birds, such as Canada Geese and gulls, which have expanded their numbers due to their successful adaptation to our urban/suburban environment, is a key factor. In other cases, subsurface drainage to lower water tables in estuarine environments to allow new residential development leads to a first flush following rainfall that causes a short-term exceedance of bacterial indicator organisms. In still other situations, many older developed areas around fresh and salt waterbodies are served by storm drains that result in bacterial exceedances following storms. The nature of each of these sources, and perhaps the lack of public will to incur the substantial cost to reduce these exceedances, may mean that these problems are part of modern life.

A pragmatic strategy has emerged to allow us to “live in harmony” with these realities. State agencies are using local rain gauges to estimate when individual rainstorms are of sufficient magnitude to result in exceedances of indicator organism concentrations in surface water bodies. Once “exceedance rainfall” is observed, temporary closures are posted. After a few days, the state agency will sample the affected surface waters, and if indicator organism concentrations are below the water quality criterion (standard), then the area is re-opened for the designated uses.

Unfortunately, permanent solutions to this problem are likely to have highly uncertain effectiveness and may be quite costly. So, while we may not like the current approach, as it may suddenly disrupt plans for recreation, we may be unwilling to assume the cost of change. Ultimately, this may be another example where the goals of the Clean Water Act (to eliminate the discharge of pollutants into the nation’s waters, and to achieve water quality levels that are fishable and swimmable) are unlikely to be attained.

The Role of Scientists in Decision Making

I have been working recently on a project to assess the “adequacy” of data to inform water quality management and decision making. Is this an appropriate task for a scientist? That is, should a scientist assess the adequacy of data/information to make water quality management decisions? Should a scientist recommend water quality standards? My answer to both questions is no, these are not appropriate tasks for scientists. What does my answer imply about the role of scientists in water quality management and decision making?

Scientists have an important responsibility in the interpretation of science to inform environmental management and policy development. For example, as a scientist, I know the strengths and weaknesses in scientific knowledge in my area of study, and thus I can readily identify gaps in that knowledge. Therefore I and my scientific colleagues should be consulted in an evaluation of general scientific research needs in our areas of expertise. That point seems obvious.

But, I am also a private citizen with personal beliefs, values, and preferences. How do I express those? Should those personal preferences affect my scientific input in support of management and policy? While it has been my experience that people often expect scientists to provide policy recommendations, I think that, upon reflection, most people would prefer a scientific assessment untainted by personal values. In effect, people would prefer a scientific expert to assess the impact of various courses of action, but not to recommend a course of action. In principle, assessment involves scientific expertise but not personal values or preferences. For example, assessments may be summarized by statements such as "if you do A, then my scientific analysis indicates that X is expected to happen" or "if you do B, then my scientific analysis indicates that Y is expected to happen," and so on.

If, however, I were to say "I recommend that you do C," then behind that recommendation, besides science, is my preference for the tradeoffs implicit in recommended action C. What are these tradeoffs? They involve all things affected by the decision, such as: Who pays? How much? What water quality conditions are achieved? As a scientist, I do not have the right to make decisions on those tradeoffs; that right is granted to elected or appointed public officials in most cases. I may have an opinion on the tradeoffs, but as a scientist/citizen, I express those opinions in others ways, such as in the voting booth.

Consider a specific example. At the local and regional level, many scientists volunteer their services on community environmental affairs boards. I have done this in the past in Durham, NC. If, as a member of the Durham Environmental Affairs Board (EAB), I am asked to assist the Durham County Commissioners with their decision concerning allowable land use to protect a water supply reservoir, how should my scientific assessment be expressed? Remember, as a member of the Durham EAB, I am serving as a technical expert. In my role as a technical expert, I am in a good position to use analysis and scientific assessment to make "if-then" statements. Examples of this are: "If Durham adopts land use strategy A, then based on my modeling and scientific analysis, X is predicted to happen to water quality" and "If Durham adopts strategy B, then Y is predicted to happen" and so on. The elected County Commissioners then take the scientific assessment from the EAB, along with technical assessments concerning other relevant attributes, and decide. Decisions by the County Commissioners should reflect community values, tradeoffs, and preferences. If they do not, then the voters have the opportunity to express their dissatisfaction with the Commissioners in the next election.

In summary, the community preferences and values are expressed in the decision by the Commissioners. Scientists provide technical assessments that may require interpretation and explanation. However, the scientific input should not be expressed as a management recommendation, and thus take decision making authority from those who have decision making responsibility.

What does this discussion mean, or imply, about standard setting (e.g., establishing nutrient standards)? Very simply, standard setting is decision making; it should be based on the same principles as outlined above. Thus, if scientists recommend specific water quality water quality standards, then they are making decisions, and in doing so they are expressing their values and preferences. Here, too, the proper role of the scientist is one of assessment, not recommendation and not decision making. For phosphorus water quality standards, scientific assessment might be expressed as "if the growing season total phosphorus standard is set at 0.040mg/l, then algal bloom conditions are predicted for 10 percent of the waterbodies in the state and recreational conditions are expected to be ...; if instead it is set at 0.030mg/l, then ..."  Similarly, an economic assessment might be provided to indicate the expected costs to achieve the standard. All important assessments like these are then provided to the appropriate decision maker(s) who must employ citizen values, preferences, and regulatory mandates to make the necessary tradeoffs and establish the standard.

With respect to my current project on the “adequacy” of water quality data to support decision making, stating that data are “adequate” for decision making is a value judgment that should not be made by scientists. In this situation, the scientist should assess the uncertainty in the data in terms that are understandable to stakeholders and decision makers. This allows decision makers to determine when data are adequate for their needs.

As scientists, we may truly believe that we know the best actions for water quality management. But, are all scientists going to be as enlightened as we are? Further, we may encounter decision makers who want us to recommend an action, and in effect make the decision for them. In those situations, will all scientists have the personal integrity that we have? We should resist these tempting opportunities, and instead work toward better-informed decision makers and citizens. In the long run, democracy and government accountability will benefit and good decisions will follow.

Assessing Water Quality Standards Compliance – A Bayesian Approach

If a water quality management plan is developed and implemented to meet a water quality standard, monitoring is usually the basis for assessing compliance and determining if any management modifications are needed. Yet, we know that lags in implementation of plans, lags in pollutant concentration change and/or in biotic response, measurement uncertainty, and natural variability all may lead to errors in inferences based on measurements. This has led some in the water quality modeling community to recommend the use of models to assess progress. However, all water quality models have prediction uncertainties, some of which can be quite large. So, which assessment is more reliable – the model forecast or the monitoring data?

We believe that both assessments can, and should, be used to evaluate compliance and the adequacy of management actions. That is, even though the model is just that – a model – and even though it will always yield uncertain predictions, it has value in forecasting impacts (otherwise we would not be using it to develop the management plan). Likewise, lags, natural variability, and measurement uncertainty do not prevent useful inferences to be derived from measurements. Qian and Reckhow (2007) present a Bayesian approach for pooling pre-implementation model forecasts with post-implementation measurements to assess compliance with the relevant water quality standard. In its simplest form, this Bayesian approach involves a variance-weighted combination of the model forecast and the post-implementation monitoring data. 

Bayes Theorem lies at the heart of Bayesian inference; it is based on the use of probability to express knowledge and the combining of probabilities to characterize the advancement of knowledge. The simple, logical expression of Bayes Theorem stipulates that, when combining information, the resultant (or posterior) probability is proportional to the product of the probability reflecting à priori knowledge (the prior probability) and the probability representing newly acquired data/knowledge (the sample information, or likelihood function). Expressed more formally, Bayes Theorem states that the posterior probability for “y” conditional on experimental outcome “x” (written p(y|x)) is proportional to the probability of y before the experiment (written p(y)) times the probabilistic outcome of the experiment (written p(x|y)):                   

                          (1)

Here, we are interested in whether chlorophyll concentrations in the Neuse River Estuary will be in compliance with North Carolina’s water quality standard of 40 μg/l chlorophyll, following implementation of management actions to reduce nitrogen input to the estuary.  Many states in the US require that the probability of exceeding a water quality standard must be less than 10%, so compliance with the chlorophyll standard is said to be achieved if there is less than a 10% chance (probability) of chlorophyll exceeding 40 μg/l. 

For the Neuse River Estuary, two models were developed to assess the impact of management actions to reduce nitrogen loading to the Neuse River Estuary.  One model is SPARROW (SPAtially Referenced Regressions On Watershed attributes); the SPARROW model was used to predict nitrogen loading to the estuary, and a second model (NeuBERN; a Bayes network) was developed to predict the chlorophyll a concentrations in the estuary, based on the SPARROW nitrogen load predictions.  As a result of application of these models, a plan for nitrogen load reduction was developed that was expected to achieve compliance with the chlorophyll standard. Once the nitrogen load reduction plan was implemented, monitoring of chlorophyll in the estuary was initiated to help assess compliance.

As stated above, the pre-implementation predictions from the model were augmented with post-implementation chlorophyll a measurements from the estuary to better assess the probability of compliance.  Qian and Reckhow (2007) illustrated the methods for combining model predictions and monitoring data using the Neuse River data as an example.  The basis of their work is the repeated application of Bayes Theorem (Equation 1) as a vehicle to combine information from different sources. Prediction of chlorophyll concentration based on the linked SPARROW-NeuBERN model output constitutes the prior probability distribution of chlorophyll concentration in the estuary. Monitoring data for chlorophyll obtained after implementation of the management plan provides the likelihood function.  The resulting posterior distribution (recall Equation 1) represents the combined information from model output and monitoring data. 

Bayes Theorem was applied in a sequential manner on an annual basis beginning with data collected in 1992. When data from the next year (1993) became available, the posterior distribution developed in the previous time step (1992) became the prior distribution for assessing compliance during the next time step (1993).  This iterative Bayesian updating process, sequentially-presented in Figure 1 and summarized in Figure 2, represents a natural mechanism for information accumulation which can be effective in assessing changes in water quality status.

In Figure 1, the solid bell-shaped curve is the prior distribution estimated for each year, the dashed bell-shaped curve is the posterior distribution, and the vertical line is the North Carolina chlorophyll standard of 40 μg/l. The posterior distribution represents a probability-weighted average of prior and monitoring data.   It is evident in the graph on the upper left of Figure 1 that our model-predicted chlorophyll a concentration distribution in the Neuse River Estuary differed from the 1992 observed chlorophyll concentrations (displayed in logarithmic scale in Figure 1), which are represented by the histogram.  However, as the Bayesian updating analysis proceeds through the 1990s, the updating of each year’s prior with new data causes the prior and posterior probabilities to gradually merge. The composite analysis presented in Figure 2 summarizes the gradual convergence of sequentially updated posterior distributions to the distribution represented by the data histogram.

Figure 1. Sequential updating of chlorophyll a concentration distributions in the Neuse River Estuary are presented in natural logarithm scale.  Each panel represents one year. The solid bell-shaped lines are the prior distributions, the dashed bell-shaped lines are the posterior distributions, and the histograms are the annual monitoring data. The North Carolina chlorophyll water quality standard is shown by the vertical line segment.

Figure 2. The sequentially updated posterior distributions are shown to converge to the distribution represented by the data histogram.

 In summary, the use of Bayes Theorem yields a model/data consensus on water quality in the Neuse River Estuary. Natural variability (exacerbated by the hurricanes that often strike this area of North Carolina) causes the chlorophyll observations (Figure 1) to fluctuate from year to year. The initial modeling effort, along with Bayesian updating brings stability to the assessment. As a result, we can be confident that chlorophyll achieved compliance with the water quality standard.

Qian, S., and K.H. Reckhow. 2007. Combining model results and monitoring data for water quality assessment. Environmental Science and Technology.41:5008-5013.

http://www.academia.edu/4172762/Combining_Model_Results_and_Monitoring_Data_for_Water_Quality_Assessment