A new fuzzy neural network method to predict minimum dissolved oxygen (DO) concentration in a highly urbanised riverine environment (in Calgary, Canada) is proposed. The method uses abiotic factors (non-living, physical and chemical attributes) as inputs to the model, since the physical mechanisms governing DO in the river are largely unknown. A new two-step method to construct fuzzy numbers using observations is proposed. Then an existing fuzzy neural network is modified to account for fuzzy number inputs and also uses possibility theory based intervals to train the network. Results demonstrate that the method is particularly well suited to predicting low DO events in the Bow River. Model performance is compared with a fuzzy neural network with crisp inputs, as well as with a traditional neural network. Model output and a defuzzification technique are used to estimate the risk of low DO so that water resource managers can implement strategies to prevent the occurrence of low DO.

The City of Calgary is a major economic hub in western Canada. With a rapidly growing population, currently estimated in excess of 1 million, the city is undergoing expansion and urbanisation to accommodate the changes. The Bow River is a relatively small river that flows through the city and provides approximately 60 % of the residents with potable water (Khan and Valeo, 2015, 2016). In addition to this, water is diverted from within the city for irrigation, is used as a source for commercial and recreational fisheries, and is the source of drinking water for communities downstream of Calgary (Robinson et al., 2009; Bow River Basin Council, 2015). This highlights the importance of the Bow River, not just as a source of potable water, but also as a major economic resource.

However, urbanisation has the potential to reduce the health of the Bow River, which is fast approaching its assimilative capacity and is one of the most regulated rivers in Alberta (Bow River Basin Council, 2015). Three wastewater treatment plants (shown in Fig. 1) and numerous stormwater outfalls discharge their effluent into the river and are considered to be a major cause of water quality degradation in the river (He et al., 2015). This highlights some of the major impacts on the Bow River from the surrounding urban area. A number of municipal and provincial programmes are in place to reduce the loading of nutrients and sediments into the river such as the Total Loadings Management Plan and the Bow River Phosphorus Management Plan (Neupane et al., 2014) as well as modelling efforts – namely the Bow River Water Quality Model (Neupane et al., 2014; Golder Associates Ltd., 2004) – to predict the impact of different water management programmes on the water quality.

An aerial view of the City of Calgary, Canada, showing the locations
of

One of the major concerns is that low dissolved oxygen (DO) concentration has occurred on a number of occasions over the last decade in the Bow River within the city limits. DO is an indicator of overall health of the aquatic ecosystem (Dorfman and Jacoby, 1972; Hall, 1984; Canadian Council of Ministers of the Environment, 1999; Kannel et al., 2007; Khan and Valeo, 2014a, 2015), and low DO – which can be caused by a number of different factors (Pogue and Anderson, 1995; Hauer and Hill, 2007; He et al., 2011; Wen et al., 2013) – can impact various organisms in the water body. While the impact of long-term effects of low DO are largely unknown, acute events can have devastating effects on aquatic ecosystems (Adams et al., 2013). Thus, maintaining a suitably high DO concentration, and water quality in general, is of utmost importance to the City of Calgary and downstream stakeholders, particularly as the city is being challenged to meet its water quality targets (Robinson et al., 2009).

A number of recent studies have examined the DO in the Bow River and the factors that impact its concentration. Iwanyshyn et al. (2008) found the diurnal variation in DO and nutrient (nitrate and phosphate) concentration were highly correlated, suggesting that biogeochemical processes (photosynthesis and respiration of aquatic vegetation) had a dominant impact on nutrient concentration rather than wastewater treatment effluent. Further, Robinson et al. (2009) found that the DO fluctuations in the river were primarily due to periphyton rather than macrophyte biogeochemical processes. In both studies, the seasonality of DO, nutrients, and biological concentration, and external factors (e.g. flood events) were demonstrative of the complexity in understanding river processes in an urban area, and that consideration of various inputs, outputs and their interaction is important to fully understand the system. He et al. (2011) found that seasonal variations in DO in the Bow River could be explained by a combination of abiotic factors (such as climatic and hydrometric conditions), as well as by biotic factors. The study found that while photosynthesis and respiration of biota are the main drivers of DO fluctuation, the role of nutrients was ambiguous. Neupane et al. (2014) found that organic materials and nutrients from point and non-point sources influence DO concentration in the river. The likelihood of low DO was highest downstream of wastewater treatment plants and that non-point sources have a significant impact in the open-water season. Using a physically based model, Neupane et al. (2014) predicted low DO concentration more frequently in the future in the Bow River owing to higher phosphorus concentration in the water, as well as climate change impacts.

A major issue of modelling DO in the Bow River is that rapid urbanisation within the watershed has resulted in substantial changes to land-use characteristics, sediment and nutrient loads, and to other factors that govern DO. Major flood events (like those in 2005 and 2013) completely alter the aquatic ecosystem, while new wastewater treatment plants (e.g. the Pine Creek wastewater treatment plant) added in response to the growing population further increases the stress downstream. These types of changes in a watershed increase the complexity of the system: the interaction of numerous factors over a relatively small area and across different temporal scales means that DO trends and variability in urban areas are more difficult to model and evaluating water quality in urban riverine environments is a difficult task (Hall, 1984; Niemczynowicz, 1999).

The implication of this is that the simplistic representation described in
conceptual, physically based models is not suitable for complex systems,
i.e. where the underlying physical mechanisms behind the factors that govern
DO are still not clearly understood, in a rapidly changing urban environment.
Physically based models require the parameterisation of several different
variables which may be unavailable, expensive and time consuming
(Antanasijević et al., 2014; Wen et al., 2013; Khan et al., 2013). In
addition to this, the increase in complexity in an urban system
proportionally increases the uncertainty in the system. This uncertainty can
arise as a result of vaguely known relationships among all the factors that
influence DO, in addition to the inherent randomness in the system (Deng et
al., 2011). The rapid changes in an urban area render the system

In this research, we propose a new method to predict DO concentration in the
Bow River using a data-driven approach, as opposed to a physically based
method that uses

Possibility theory is an information theory that is an extension of fuzzy set
theory for representing uncertain, vague or imprecise information (Zadeh,
1978). Fuzzy

Data-driven models, such as neural networks, regression based techniques, fuzzy rule based systems, and genetic programming, have seen widespread use in hydrology, including DO prediction in rivers (Shrestha and Solomatine, 2008; Solomatine et al., 2008; Elshorbagy et al., 2010). Wen et al. (2013) used artificial neural networks (ANNs) to predict DO in a river in China using ion concentration as the predictors. Antanasijević et al. (2014) used ANNs to predict DO in a river in Serbia using a Monte Carlo approach to quantify the uncertainty in model predictions and temperature as a predictor. Chang et al. (2015) also used ANNs coupled with hydrological factors (such as precipitation and discharge) to predict DO in a river in Taiwan. Singh et al. (2009) used water quality parameters to predict DO and BOD in a river in India. Other studies (e.g. Heddam, 2014; Ay and Kisi, 2011) have used regression to predict DO in rivers using water temperature or electrical conductivity, amongst others, as inputs. In general, these studies have demonstrated that there is a need and demand for less complex DO models, have led to an increase in the popularity of data-driven models (Antanasijević et al., 2014), and shown that the performance of these types of models is suitable. Recent research into predicting DO concentration in the Bow River in Calgary using abiotic factors (these are non-living, physical and chemical attributes) as inputs have shown promising results (He et al., 2011; Khan et al., 2013; Khan and Valeo, 2015). The advantage of using readily available data (i.e. the abiotic inputs) in these studies is that if a suitable relationship between these factors and DO can be found, changing the factors (e.g. increasing the discharge rate downstream of a treatment plant) can potentially reduce the risk of low DO.

While fuzzy set theory based applications, particularly applications using
fuzzy

Given the importance of DO concentration as an indicator of overall aquatic ecosystem health, there is a need to accurately model and predict DO in urban riverine environments, like that in Calgary, Canada. In this research a new data-driven method is proposed that attempts to address the issues that plague numerical modelling of DO concentration in the Bow River. The FNN method proposed by Alvisi and Franchini (2011) is adapted and extended in two critical ways. The existing method uses crisp (i.e. non-fuzzy) inputs and outputs to train the network, producing a set of fuzzy number weights and biases, and fuzzy outputs. The method is adapted to be able to handle fuzzy number inputs to produce fuzzy weights and biases, and fuzzy outputs. The advantage is that the uncertainties in the input observations are also captured within the model structure. To do this, a new method of creating fuzzy numbers from observations is presented based on a probability–possibility transformation. Second, the existing training algorithm is based on capturing a predetermined set of observations (e.g. 100, 95 or 90 %) within the fuzzy outputs. The selection of the predetermined set of observations in the original study was an arbitrary selection. A new method that exploits the relationship between possibility theory and probability theory is defined to create a more objective method of training the FNN. A consequence of this is that the resulting fuzzy number outputs from the model can then be directly used for risk analysis, specifically to quantify the risk of low DO concentration. This information is extremely valuable for managing water resources in the face of uncertainty. The impact of using fuzzy inputs and the new training criteria is evaluated by comparing results to the existing FNN method (by Alvisi and Franchini, 2011) as well as with a traditional, crisp ANN.

Following previous research for this river, two abiotic inputs (daily mean
water temperature,

The Bow River is 645 km long and averages a 0.4 % slope over its length
(Bow River Basin Council, 2015) from its headwaters at Bow Lake in the Rocky
Mountains to its confluence with the Oldman River in southern Alberta, Canada
(Robinson et al., 2009; Environment Canada, 2015). The river is supplied by
snowmelt from the Rocky Mountains, rainfall and discharge from groundwater.
The City of Calgary is located within the Bow River Basin and the river has
an average annual discharge of 90 m

A summary of low DO events in the Bow River between 2004 and 2012 and the corresponding minimum acceptable DO concentration guidelines.

The City of Calgary routinely samples a variety of water quality parameters along the Bow River to measure the impacts of urbanisation, particularly from three wastewater treatment plants and numerous stormwater runoff outfalls that discharge into the river. DO concentration measured upstream of the city is generally high throughout the year, with little diurnal variation (He et al., 2011; Khan et al., 2013; Khan and Valeo, 2015). The DO concentration downstream of the city is lower and experiences much higher diurnal fluctuation. The three wastewater treatment plants are located upstream of this monitoring site, and are thought to be responsible, along with other impacts of urbanisation, for the degradation of water quality (He et al., 2015).

For this research, 9 years of DO concentration data were collected from one of the downstream stations from 2004 to 2012. The monitoring station was located at Pine Creek and sampled water quality data every 30 min (from 2004 to 2005) and every 15 min (from 2006 to 2007). The station was then moved to Stier's Ranch and sampled data every hour (in 2008) and every 15 min (2009 to 2011). The monitoring site was moved further downstream to its current location (at Highwood) in 2012 where it samples every 15 min. During this period a number of low DO events have been observed in the river and are summarised below in Table 1 corresponding to different water quality guidelines.

Note that even though daily minimum DO was observed to be below
5 mg L

A YSI sonde is used to monitor DO and

Daily mean flow rate,

Fuzzy sets were proposed by Zadeh (1965) in order to express imprecision in
complex systems, and can be described as a generalisation of classical set
theory (Khan and Valeo, 2015). In classical set theory, an element

Fuzzy numbers express uncertain or imprecise

Traditional representation of a fuzzy numbers has been using symmetrical,
linear membership functions, typically denoted as triangular fuzzy numbers.
The reason for selecting this type of membership function has to do with its
simplicity: given that a fuzzy number must, by definition, be convex and
normal, a minimum of three elements are needed to define a fuzzy number (two
elements at

However, recent research (Khan et al., 2013; Khan and Valeo, 2014a, b,
2015, 2016) has shown that such a simplistic representation may not be
appropriate for hydrological data, which are often skewed and non-linear.
This issue is further highlighted if the probability–possibility framework
mentioned above is used: it implies that for a triangular membership
function, the fuzzy number bounded by the support [8 12] mg L

Multiple frameworks exist to transform a probability distribution to a possibility distribution and vice versa; a comparison of different conceptual approaches is provided in Klir and Parvais (1992), Oussalah (2000), Jaquin (2010), Mauris (2013) and Dubois and Prade (2015). However, a major issue of implementing fuzzy number based methods in hydrology is that there is no consistent, transparent and objective method to convert observations (e.g. time series data) into fuzzy numbers, or generally speaking to construct the membership function associated with fuzzy values (Abrahart et al., 2012; Dubois and Prade, 1993; Civanlar and Trussel, 1986).

A popular method (Dubois et al., 1993, 2004) converts a probability
distribution to a possibility distribution by relating the area under a
probability density function to the membership level (Zhang, 2009). In this
framework, the possibility is viewed as the upper envelope of the family of
probability measures (Jacquin, 2010; Ferrero et al., 2013; Betrie et al.,
2014). There are two important considerations for this transformation, first
it guarantees that something must be possible before it is probable; hence,
the degree of possibility cannot be less than the degree or probability –
this is known as the consistency principle (Zadeh, 1965). Second is order
preservation, which means if the possibility of

For

However, a major drawback of this transformation is that it theoretically requires a full description of the probability density function, or in the finite case, the probability associated with each element of the fuzzy number, the probability mass function. For many hydrological applications this might not be possible because the hourly time series data may not adequately fit the mould of a known class of probability density functions, or one distribution amongst many alternatives may have to be selected based on best-fit. This best-fit function may not be universal, e.g. data from one 24 h period may be best described by one class or family of probability density function, while the next day by a completely different class of density function. This means working with multiple classes of distribution functions for one application, which can be cumbersome. Also, given that each day may only have 24 data points (or fewer on days with missed samples), it is difficult to select one particular function.

In previous research by Khan and Valeo (2015), a new approach to create a
fuzzy number based on observations was developed. This process used a
histogram based approach to estimate the probability mass function of the
observations, and then Eq. (1) was used to estimate the membership function
of the fuzzy number. To create the histogram, the bin size was selected based
on the extrema observations for a given day and the number of the
observations. A linear interpolation scheme was then used to calculate the
fuzzy number at five predefined membership levels. This method has a few
shortcomings, namely that the bin-size selection was arbitrarily selected
based on the magnitude and number of observations, which does not necessarily
result in the optimum bin size. This lack of optimality means that the
resulting histogram may either be too smooth so as not to capture the
variability between membership levels, or too rough and uneven so that the
underlying shape of the membership function is difficult to discern. This is
a common issue with histogram selection in many applications (Shimazaki and
Shinomoto, 2007). Secondly, the aforementioned transformation used by Khan
and Valeo (2015) only allows one element to have

In this research, a two-step procedure is proposed to create fuzzy numbers on
the inputs (i.e.

Shimazaki and Shinomoto (2007) proposed a method to find the optimum bin size
of a histogram when the underlying distribution of the data is unknown. The
basic premise of the method is that the optimum bin size (

Using this property, the optimum bin size can be found as follows. Let

Partitioning positions are defined as the first and last points that define a
bin. The most common way of defining a partitioning position is to centre it
on some value

For this research this bin-size optimisation algorithm is implemented to
determine the optimum histogram for the two input variables,

Once the

This process is then repeated for the next

First, the

Once all the

In creating this discretised empirical membership function this way (rather
than assuming a shape of the function) means that this function best
reflects the possibility distribution of the observed data. However, it also
means that all fuzzy numbers created using this method are not guaranteed to
be defined at the same

Artificial neural networks (ANNs) are a type of data-driven model that are defined as a massively parallel distributed information processing system (Elshorbagy et al., 2010; Wen et al., 2013). ANN models have been widely used in hydrology when the complexity of the physical systems is high owing partially to an incomplete understanding of the underlying process and the lack of availability of necessary data (He et al., 2011; Kasiviswanathan et al., 2013). Further, ANNs arguably require fewer data and do not require an explicit mathematical description of the underlying physical process (Antanasijević et al., 2014), making them a simpler and practical alternative to traditional modelling techniques.

A multilayer perceptron (MLP) is a type of feedforward ANN and is one of the most commonly used in hydrology (Maier et al., 2010). A trained MLP network can be used as a universal approximator with only one hidden layer (Hornik et al., 1989). This means that models are relatively simple to develop and theoretically have the capacity to approximate any linear or non-linear mapping (ASCE Task Committee on Application of Artificial Neural Networks in Hydrology, 2000; Elshorbagy et al., 2010; Napolitano et al., 2011; Kasiviswanathan et al., 2013). Further, the popularity of MLPs has meant that subsequent research has continued to use MLPs (He and Valeo, 2009; Napolitano et al., 2011) and thus form a reference for the basis of comparing ANN performance (Alvisi and Franchini, 2011).

In the simplest case, an MLP consists of an input layer, a hidden layer, and an output layer as shown in Fig. 2. Each layer consists of a number of neurons (or nodes) that each receive a signal, and on the basis of the strength of the signal, emit an output. Thus, the final output layer is the synthesis and transformation of all the input signals from both the input and the hidden layer (He and Valeo, 2009).

An example of a three-layer multilayer perceptron feed-forward ANN, with two input neurons, the hidden-layer neurons, and one output neuron. WIH are the weights between the input and hidden layers, WHO are the weights between the hidden and output layers, BH are the biases in the hidden layer, and BO is the bias in the output layer.

The number of neurons in the input (

The values of all the weights and biases in the MLP are calculated by
training the network by minimising the error – typically mean squared error
(

Most ANNs have a deterministic structure without a quantification of the uncertainty corresponding to the predictions (Alvisi and Franchini, 2012; Kasiviswanathan and Sudheer, 2013). This means that users of these models may have excessive confidence in the forecasted values and misinterpret the applicability of the results (Alvisi and Franchini, 2011). This lack of uncertainty quantification is one reason for the limited appeal of ANN among water resource managers (Abrahart et al., 2012; Maier et al., 2010). Without this characterisation, the results produced by these models have limited value (Kasiviswanathan and Sudheer, 2013).

In this research, two methods are proposed to quantify the uncertainty in MLP
modelling to predict DO in the Bow River. First, the uncertainty in the input
data (daily mean water temperature and daily mean flow rate) is represented
through the use of fuzzy numbers. These fuzzy numbers are created using the
probability–possibility transformation discussed in the previous section.
Second, the

Alvisi and Franchini (2011) proposed a method to create a FNN, where the
weights and biases, and by extension the output, of the neural network are
fuzzy numbers rather than crisp (non-fuzzy) numbers. These fuzzy numbers
quantify the total uncertainty of the calibrated parameters. Most fuzzy set
theory based applications of ANN in hydrology have used fuzzy logic, e.g.
the widely used Adaptive Neuro-Fuzzy Inference System, where automated
IF-THEN rules are used to create

In their FNN, the MLP model presented in Eqs. (5) and (6) is modified to
predict an interval rather than a single value for the weights, biases and
output, corresponding to an

In this research, this method is modified in two ways. First, the inputs

The second modification of the original algorithm is related to the
selection of the percent of data included in the predicted interval (

In the adopted possibility–probability framework, the interval [

Selected values for

Note that for practical purposes,

For this research a three layer, feedforward MLP architecture was selected
to model minimum daily DO (the output) using fuzzified daily flowrate (

This FNN optimisation algorithm was implemented in MATLAB (version 2015a).
First, the built-in MATLAB Neural Network Toolbox was used to estimate the
value of weights and biases using the midpoint of the interval at

Risk analyses for complex systems is challenging for a number of reasons,
including an insufficient understanding of the failure mechanisms (Deng et
al., 2011). The use of imprecise information (e.g. fuzzy numbers) is an
effective method of conducting a risk analysis (Deng et al., 2011). However,
communicating uncertainty is an important, yet difficult task, and many
different frameworks exist to do so; water quality indices (Sadiq et al.,
2007; Van Steenbergen et al., 2012) are one example. Since water resource
managers often prefer to use probabilistic measures (rather than
possibilistic ones), it is important to convert the possibility of low DO to
a comparable probability for effective communication of risk analysis. Note
that the linguistic parameters (e.g. “most likely”) that are often used to
convey risk or uncertainty (Van Steenbergen et al., 2012) have a probability
based meaning – in this case “most likely” is a measure of

In this research, a

The bin-size optimisation and the probability–possibility transformation
algorithms were applied to the collected

Sample results of probability–possibility transformation for flow
rate,

Figure 3 shows sample results of converting hourly

The example in the first row illustrates cases where the bin-size
optimisation algorithm calculates an optimum bin size, corresponding to the
minimum cost function

The second row in Fig. 3 shows the results for 20 August 2009, where the
optimum bin size was found to be 4 times higher than the original bin size
(

Since the objective of the bin-size algorithm was to reduce the error between
the histogram created using

Similar results can be seen in the third row in Fig. 3, where the optimised
bin size is 4.5 times greater than the original bin size
(

The fourth row shows a different phenomenon, where instead of smoothing out
the original membership function, the combined bin-size optimisation and
transformation algorithm creates a membership function with more specificity.
In this case

Sample results of probability–possibility transformation for water
temperature,

The last example for

Figure 4 shows similar results for the five water temperature examples, where
the

A major difference between the

Thus, without using the bin-size optimisation algorithm, there is a risk that the resulting membership functions will be too vague and will not represent the information that can be gained from the observations. It is worth nothing that for these three examples, if linear interpolation is used on the original membership function, the resulting interpolated fuzzy numbers will all have equal intervals (due to the trapezoidal shape), transferring no useful information to the final fuzzy number.

Overall, the above examples illustrate the advantages of using the coupled methods of bin-size optimisation and probability–possibility transformation to create fuzzy numbers for the FNN application. The applicability of this method is not necessarily restricted to this application and can be applied whenever there is a need to construct fuzzy numbers from observed data. The utility of the first component, bin-size optimisation to estimate the density function is that in cases where either not enough information is available to define a probability distribution, or if the data do not follow the mould of a known density function, or if assumptions about the class of distribution cannot be made, the optimum bin size can be calculated to define an empirical distribution for the probability–possibility transformation. The advantage of the second component, the algorithm to construct the possibility distribution (i.e. the membership function of the fuzzy number) is that it provides a consistent, transparent and objective method to convert observations (e.g. time series data) into fuzzy numbers – which has been cited as a major hurdle in implementing fuzzy number based applications in the literature (Abrahart et al., 2010; Dubois and Prade, 1993; Civanlar and Trussel, 1986). A noteworthy component of this algorithm is that the fuzzy numbers do not reduce to the simple, triangular shaped functions that are widely used, but rather the functions better represent the information from the observations.

The

Once the observations of the abiotic input parameters (^{®}

The

The results of the optimisation component of the algorithm are summarised in
Table 4, which shows the percentage of data (

Percentage of data captured within each

A sample of the fuzzy weights and biases produced through the optimisation
are shown in Fig. 5. Note that the membership functions are assumed to be
piecewise linear (following similar assumptions made in Alvisi and Franchini,
2011; Khan et al., 2013; Khan and Valeo, 2015), i.e. that the intervals at
each membership levels can be joined to create a fuzzy number. This can be
confirmed by the fact that each of the weights and biases are

Sample plots of the produced membership functions for the weights and biases of the fuzzy neural network for both the proposed and existing methods.

The figure demonstrates that enough

A comparison of the predicted and observed minimum DO at the

Connecting this back to the results in Table 4, these two particular weights
and biases show why the percentage of data calculated at

Tables 3 and 4 and Fig. 5 demonstrate the overall success of the proposed approach to calibrate an FNN model as compared to a crisp ANN, as well as an FNN that uses crisp inputs. The optimisation algorithm is defined based on the principles of possibility theory (i.e. defining the amount of data to include in each interval) and is a transparent, repeatable and objective (not arbitrary) method to create the fuzzy numbers for the FNN model.

The observed versus crisp predictions (black dots) and fuzzy predictions at

Figure 6 also demonstrates the benefit of the FNN approach as compared to the
crisp ANN approach with respect to predicting low DO (i.e. when DO is less
than 6.5 mg L

A comparison of the observed and predicted minimum DO trends for 2004 (top panel) and 2006 (bottom panel).

A comparison of the observed and predicted minimum DO trends for 3 sample years: 2007 (top panel) and 2010 (bottom panel).

Trend plots of observed minimum DO and predicted fuzzy minimum DO for the years 2004, 2006, 2007 and 2010 are illustrated in Figs. 7 and 8. These results are shown only for the proposed method for clarity; the difference between the existing method (using crisp inputs) and the proposed method (using fuzzy inputs) is discussed later. These years were selected due to the high number of low DO occurrences in each year (as listed in Table 1), and highlight the utility of the proposed method to predict minimum DO using abiotic factors in the absence of a complete understanding of the physical mechanisms that govern DO in the Bow River. Note that for each year, 50 % of the data are training data, 25 % are validation data and 25 % are testing data. However, for clarity this difference is not individually highlighted for each data point in these figures.

Zoomed-in views of the trend plots for 4 sample years corresponding to important periods with low DO occurrences.

In Figs. 7 and 8, the predicted minimum DO at equivalent membership levels
(e.g. 0

In each of the years shown, the majority of the observations tend to fall
within the

The trend plot for 2004 shows that observed DO decreases rapidly from late
June to late July, followed by a few days of missing data and near-zero
observations, before increasing to higher concentrations. Details of this
trend are shown in Fig. 9 which shows magnified versions of important periods
for each year. The reason for this rapid decrease in 2004 is unclear and may
be related to problems with the real-time monitoring device which was in its
first year of operation that year. However, it demonstrates that the efficacy
of data-driven methods is dependent on the quality of the data. Since the
proposed method was calibrated to capture

The time series plot for 2006 shows that all the observations fall within
the predicted intervals, and that the predicted trend generally follows the
observed trend. The majority of the 25 low DO events (

In contrast to the results from 2004 and 2006, the majority of observations
are captured at higher membership levels (i.e. greater than

The trend plot for 2010 is shown in Fig. 8, and it is clear that all
observations fall within the

Figure 9 shows details of a low DO (

The analysis of the trend plots for these four sample years shows that the
proposed FNN method is extremely versatile in capturing the observed daily
minimum DO in the Bow River using

Figure 10 shows a comparison of the predicted minimum DO trends from both the
proposed FNN method (solid black line) and the existing FNN method (dashed
black line), along with the observed data (circles) for each membership
level for the 2009 data. These figures show that despite the use of more
data for the inputs (i.e. fuzzy numbers versus crisp numbers), both methods
are optimised to show similar results (due to the optimisation algorithm
requiring a specific amount of data being captured at each level). This
shows that the optimisation algorithm developed in this manuscript for fuzzy
inputs successfully mimics the original algorithm developed by Alvisi and
Franchini (2011) that only used crisp inputs. Thus, when modelling a complex
system, such as the minimum daily DO in the Bow River, the uncertainty in
the inputs can also be quantified and propagated through the data-driven
model, by using the proposed method. This is a major advantage over the
original model (Alvisi and Franchini, 2011) that only allowed crisp inputs
to be used. Note that as per Table 4, both methods approximately capture the
same amount of data at each interval, however as Fig. 10 indicates this does
not necessarily mean that the predicted intervals are exactly the same for
both methods. Both methods predict unique intervals, with the overall result
being that

Comparison of predicted trends of the proposed (solid black line) and existing (dashed black line) methods shown for 2009 for each membership level. Observations are shown as black circles.

A comparison of average annual interval widths of predicted fuzzy numbers using the proposed and existing FNN methods for four selected membership levels.

Figure 11 compares with width of predicted intervals at four selected
membership levels for both methods, generally showing mixed results. As
discussed above the existing method does not predict an interval for

Sample plots of low DO events and the corresponding risk of low DO
calculated using a possibility–probability transformation for the
5 mg L

The utility of the FNN method is illustrated through an analysis of the
ability of the proposed model to predict low DO events, and then a
possibility–probability transformation is used to assess the risk of these
low DO events. The number of occasions when observed DO was below any of the
three guidelines used for this research are summarised in Table 1. The FNN
model was cable to capture 100 % of all low DO events (i.e. below 5, or
6.5 or 9.5 mg L

Once all the low DO events were identified, the inverse transformation
(defuzzification) described in Sect. 2.4 was used to estimate the
probability of low DO. The primary reason for converting from possibility to
probability is to improve the communication of the risk of low DO. For each
low DO event (i.e. at 5, or 6 or 9.5 mg L

For the first case,

For the 184 cases where DO was observed to be less than 6.5 mg L

For the last, most conservative case, the probability of predicting DO to be
less than 9.5 mg L

It is worth noting here that the proposed FNN model was designed to only
include data from the April to October each year, corresponding to the
ice-free period (as defined in Sect. 2.1). This implies that the analysis
has been conducted on the time period that is most critical or susceptible
to low DO. Thus, as the proposed FNN model predicts, there is possibility of
low DO on most days (as shown in the trend plots in Figs. 7 and 8). However,
the consistency principle (Zadeh, 1978) implies that an event must be
possible before it is probable. Thus, a possibility to predict low DO does
not imply that it will occur with a high probability. In fact, nearly all
the possibility of low DO events occurs at low membership levels (i.e.

The predicted membership functions of minimum DO for nine examples are shown
Fig. 12, along with the observed minimum DO (the vertical dashed line).
Three samples are shown for each low DO guideline: 5, 6.5, or 9.5 mg L

For the 5 mg L

The examples for the 6.5 mg L

The last row in Fig. 12 show sample low DO results for the 9.5 mg L

These examples are meant to illustrate the potential utility of the
data-driven and abiotic input parameter DO model, which can be used to assess
the risk of low DO. Given that it is a data-driven approach, the model can be
continually updated as more data are available, further refining the
predictions. Various combinations of input values can be used to predict
fuzzy minimum DO and defuzzification technique can be used to

A new method to predict DO concentration in an urbanised watershed is proposed. Given the lack of understanding of the physical system that governs DO concentration in the Bow River (in Calgary, Canada), a data-driven approach using fuzzy numbers is proposed to account for the uncertainty. Further, the model uses abiotic (non-living, physical and chemical attributes) factors as inputs to the model. Specifically, water temperature and flow rate were selected, which are routinely monitored, and thus a large data set is available.

The data-driven approach proposed is a modification of an existing fuzzy neural network method that quantifies the total uncertainty in the model by using fuzzy number weights and biases. The proposed model refines the exiting model by (i) using possibility theory based intervals to calibrate the neural network (rather than arbitrarily selecting confidence intervals), and (ii) using fuzzy number inputs rather than crisp inputs. This research also proposes a new two-step method to construct these fuzzy number inputs using observations. First a bin-size optimising algorithm is used to find the optimum histogram (as an estimate of the underlying but unknown probably density function of the observations). Then a probability–possibility transformation is used to determine the shape of the fuzzy number membership function.

The results demonstrate that the network training algorithm proposed can be successfully implemented. Model results demonstrate that low DO events are better captured by the fuzzy network as compared to a non-fuzzy network. A defuzzification technique is then used to calculate the risk of low DO events. Generally speaking, the method demonstrates that a data-driven approach using abiotic inputs is a feasible method for predicting minimum daily DO. Results from this research can be implemented by water resource managers to assess conditions that lead to and quantify the risk of low DO.

The data used in this research may be obtained from the City of Calgary and Environment Canada.

Usman T. Khan conducted all aspects of this research and wrote the manuscript. Caterina Valeo assisted in manuscript preparation and revisions.

The authors would like to thank S. Alvisi from the Università degli Studi di Ferrara for providing the MATLAB code for the original Fuzzy Neural Network model. The authors would also like to thank: the Natural Sciences and Engineering Research Council of Canada; the Ministry of Advanced Education, Innovation and Technology – Government of British Columbia; and the University of Victoria, for funding this research. The authors are grateful for the City of Calgary and Environment Canada for providing the data used in this research. Lastly, the authors would like to acknowledge the comments of two anonymous reviewers, whose feedback greatly helped improve this manuscript. Edited by: D. Solomatine