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
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 North
Carolina
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.
