# A Data Driven Approach to Bioretention Cell Performance: Prediction and Design

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## Abstract

**:**

## 1. Introduction

## 2. Methods and Materials

#### 2.1. Site Description and Experiment Procedure

#### 2.2. Data Collection and Analysis

_{i}) and released (V

_{o}) from the bioretention cell, the corresponding flow rates Q

_{i}and Q

_{o}, and the duration of the two hydrographs, t

_{i}and t

_{o}. The initial (θ

_{i}), peak (θ

_{p}) and final (θ

_{f}) moisture level of the soil media was measured at four depths: 150, 300, 500 and 1000 mm below the surface of the cell. A number of climatic parameters, including air temperature (T), radiation, wind speed and direction, and precipitation were also collected. In addition to these, a number of variables were calculated from the observed data, including peak and center of mass flow rates and the time to peak for both inlet and outlet hydrographs and the delay or lag-time between the peaks and center of mass.

_{o}, all parameters that were directly influenced by V

_{o}were not considered as regressors. This limited the candidates to the following nine parameters: T, V

_{i}, t

_{i}, influent peak flow rate and center-of mass flow rate (labeled Q

_{pi}and Q

_{ci}, respectively) and θ

_{i}at four depths. Pearson’s correlation coefficient and significance tests (estimated with Student’s t-test) were carried out on these parameters to determine which had the highest influence on the dependent variable V

_{o}.

_{o}= volume of effluent (m

^{3}); and V

_{i}= volume of influent (m

^{3}). A value of ΔV less than 100% indicates a decrease in runoff volume between the inlet and outlet, or a lower volume of effluent than influent.

#### 2.3. Model Construction

_{i}are the independent variables, A and B

_{i}are the regression coefficients and n is the total number of regressors used. Different permutations of regressors were used to create six different models, which are summarized in Section 4 below. For each regression analysis, the logarithms of the data were used as inputs into the model to overcome the nonlinearity between the independent and dependent variables. These models were evaluated using the coefficient of determination (R

^{2}), root mean square error (RMSE), mean absolute error (MAE) and the corrected Akaike information criterion (AICc). These error statistics were calculated using the following equations:

## 3. Experimental and Data Analysis Results

_{o}) for the cold climate experiments. The reasons for the differences in the two climatic conditions are discussed in detail in [15].

ID | T | V_{i} | t_{i} | Q_{pi} | Q_{ci} | θ_{i}-1 | θ_{i}-2 | θ_{i}-3 | θ_{i}-4 | V_{o} |
---|---|---|---|---|---|---|---|---|---|---|

(°C) | (L) | (min) | (L/s) | (L/s) | (m^{3}/m^{3}) | (m^{3}/m^{3}) | (m^{3}/m^{3}) | (m^{3}/m^{3}) | (L) | |

S1 | 14 | 8,004 | 116 | 2.62 | 1.11 | 0.230 | 0.249 | 0.282 | 0.402 | 739 |

S2 | 22 | 7,520 | 21 | 13.04 | 4.08 | 0.307 | 0.250 | 0.284 | 0.403 | 162 |

S3 | 20 | 7,736 | 36 | 11.52 | 3.31 | 0.299 | 0.302 | 0.328 | 0.433 | 237 |

S4 | 13 | 7,943 | 31 | 8.83 | 4.73 | 0.313 | 0.308 | 0.324 | 0.421 | 586 |

F1 | 11 | 8,230 | 34 | 11.79 | 4.02 | 0.597 | 0.312 | 0.179 | 0.410 | 394 |

F2 | 9 | 8,120 | 32 | 10.59 | 4.34 | 0.307 | 0.351 | 0.482 | 0.465 | 728 |

F3* | −1 | 4,739 | 28 | 6.22 | 2.33 | 0.459 | 0.358 | 0.399 | 0.513 | 3 |

W1* | 3 | 3,919 | 19 | 6.43 | 3.35 | 0.236 | 0.353 | 0.346 | 0.370 | 0 |

Sp1* | 4 | 3,866 | 21 | 7.03 | 2.23 | 0.236 | 0.353 | 0.346 | 0.370 | 0 |

Sp2 | 9 | 8,601 | 27 | 11.37 | 4.49 | 0.335 | 0.391 | 0.357 | 0.392 | 909 |

Sp3 | 16 | 3,877 | 14 | 9.67 | 3.28 | 0.335 | 0.391 | 0.357 | 0.392 | 5 |

S5 | 12 | 8,395 | 34 | 6.30 | 3.39 | 0.335 | 0.391 | 0.357 | 0.392 | 409 |

S6 | 22 | 7,842 | 36 | 7.45 | 4.28 | 0.181 | 0.286 | 0.278 | 0.314 | 479 |

S7 | 15 | 4,848 | 17 | 6.53 | 4.81 | 0.335 | 0.391 | 0.357 | 0.392 | 0 |

S8 | 17 | 4,857 | 16 | 6.06 | 3.89 | 0.326 | 0.385 | 0.340 | 0.388 | 0 |

S9 | 17 | 3,717 | 21 | 7.28 | 4.81 | 0.389 | 0.434 | 0.395 | 0.429 | 0 |

S10^{+} | 21 | 17,995 | 88 | 8.30 | 1.91 | 0.364 | 0.414 | 0.368 | 0.405 | 3,253 |

F4* | −10 | 8,840 | 61 | 3.35 | 2.33 | 0.417 | 0.460 | 0.376 | 0.368 | 2,282 |

W2* | 5 | 8,398 | 62 | 8.03 | 1.88 | 0.179 | 0.347 | 0.306 | 0.356 | 1,005 |

W3* | 2 | 9,014 | 68 | 5.46 | 1.51 | 0.292 | 0.454 | 0.372 | 0.404 | 1,883 |

W4* | −4 | 8,951 | 69 | 9.70 | 2.89 | 0.130 | 0.189 | 0.267 | 0.362 | 1,293 |

W5* | 5 | 8,581 | 64 | 5.18 | 1.62 | 0.158 | 0.242 | 0.320 | 0.372 | 2,311 |

S11 | 19 | 6,922 | 67 | 4.79 | 1.03 | 0.307 | 0.337 | 0.321 | 0.357 | 424 |

S12 | 10 | 8,868 | 67 | 5.08 | 1.71 | 0.423 | 0.457 | 0.404 | 0.407 | 2,044 |

^{+}Analysis concluded that experiment S10 was an outlier, with respect to the magnitude of V

_{i}and was removed from further calculations; θ

_{i}-1, θ

_{i}-2, θ

_{i}-3 and θ

_{i}-4 refer to soil moisture measured 150, 300, 500 and 1,000 mm below the surface of the bioretention cell.

_{o}and the other nine parameters. Figure 2 shows pair-by-pair scatter plots of V

_{o}and these parameters. The figure illustrates that a single parameter cannot be used to describe all the variance in V

_{o}. Correlation coefficients were calculated for each pair; V

_{o}was significantly (with a p-value < 0.05) correlated with V

_{i}(Pearson’s correlation coefficient r, was calculated to be 0.73) and t

_{i}(r = 0.64). Non-significant (with a p-value > 0.05) correlation was found between V

_{o}and θ

_{i}-2 (the soil moisture measured 300 mm below the surface, with r = 0.14). Based on this, the candidates for regressors for the MLR were reduced from the original nine to V

_{i}, t

_{i}and θ

_{i}-2. Figure 3 shows pair-by-pair scatter plots of these three parameters versus V

_{o}; in this figure the values plotted are the logarithm of the original observations. The linear correlation between these variables is clearer after the transformation. It should be noted that for events F3 to Sp1, Sp3 and S7 to S9, V

_{o}was less than 10 L (in fact, V

_{o}was equal to zero for five of the seven cases, i.e., no effluent drained from the bioretention cell); these values were replaced with values of 1 L to facilitate the log-transformation. As such, experiments producing minimal effluent volume were considered to be of the same population (with the higher effluent volumes arising from a different population due to different generating mechanisms) with a very small variance. Thus, all effluent volumes less than 0.1% of the inlet volume, or 10 L, were assigned a value of 1 L.

**Figure 3.**Pair by pair scatter plots of log-transformed V

_{o}, the two significantly correlated variables, V

_{i}, and t

_{i}, and weakly correlated θ

_{i}-2.

## 4. Model Construction and Validation

_{o}were identified, the data was divided for model construction and validation; 15 data points were used for model construction and 8 were used for validation. The data were split to ensure an equal representation of cold climate events in both sets as well as no significant deviation from the population mean. Table 2 summarizes statistics of the original and the divided data, and identifies the experiment ID that corresponds to each data set.

**Table 2.**Summary of statistics of all experiments, construction and validation data sets for the three regressor candidates and V

_{o}.

Statistic | V_{i} (L) | t_{i} (min) | θ_{i}-2 (m^{3}/m^{3}) | V_{o} (L) |
---|---|---|---|---|

All experiments | ||||

Mean | 7034 | 42 | 0.347 | 691 |

Standard Deviation | 1955 | 25 | 0.073 | 770 |

Experiments used for construction: | ||||

S1, S2, S3, F3, W1, SP2, SP3, S5, S7, S9, F4, W2, W3, S11, S12 | ||||

Mean | 6893 | 44 | 0.371 | 673 |

Standard Deviation | 2050 | 29 | 0.068 | 799 |

Experiments used for validation: | ||||

S4, F1, F2, SP1, S6, S8, W4, W5 | ||||

Mean | 7299 | 38 | 0.303 | 724 |

Standard Deviation | 1866 | 19 | 0.064 | 763 |

_{i}and t

_{i}as the regressors, emerged as the best performing model. Model 2 outperformed the other models based on a collective evaluation of error statistics, rather than leading in a single category. An advantage of using Model 2 to predict bioretention performance is that these two variables are typically readily available from historical records, and thus can be easily used to predict bioretention cell performance, without the need for other detailed site characteristics (e.g., soil type or the degree of imperviousness in upstream catchment area).

**Table 3.**Summary of six multiple linear regression (MLR) model results, including regressors, x

_{i}used, the constant term A, coefficients B

_{i}, and error statistics for both model construction (top row) and model validation (bottom row).

Model # | Input | A | B_{1} | B_{2} | B_{3} | R^{2} | RMSE | MAE | AICc | |
---|---|---|---|---|---|---|---|---|---|---|

1 | x_{1} | log(V_{i}) | −33.94 | 9.39 | - | - | 0.98 | 0.195 | 0.136 | −22.2 |

0.98 | 0.161 | 0.109 | −11.9 | |||||||

2 | x_{1} | log(V_{i}) | −30.60 | 8.14 | 0.91 | - | 0.99 | 0.128 | 0.086 | −28.5 |

x_{2} | log(t_{i}) | 0.98 | 0.178 | 0.145 | −11.1 | |||||

3 | x_{1} | log(V_{i}) | −33.74 | 9.42 | 0.71 | - | 0.98 | 0.191 | 0.123 | −22.5 |

x_{2} | log(θ_{i}-2) | 0.98 | 0.203 | 0.144 | −10.1 | |||||

4 | x_{1} | log(V_{i}) | −30.34 | 8.16 | 0.93 | 0.77 | 0.99 | 0.120 | 0.079 | −29.5 |

x_{2} | log(t_{i}) | |||||||||

x_{3} | log(θ_{i}-2) | 0.97 | 0.221 | 0.164 | −9.41 | |||||

5 | x_{1} | log(t_{i}) | −4.46 | 4.08 | - | - | 0.58 | 0.901 | 0.616 | 0.744 |

0.68 | 0.721 | 0.553 | −0.050 | |||||||

6 | x_{1} | log(t_{i}) | −4.25 | 4.09 | 0.53 | - | 0.59 | 0.888 | 0.600 | 0.526 |

x_{2} | log(θ_{i}-2) | 0.65 | 0.754 | 0.600 | −0.408 |

_{o}. It is important to note this model was designed so that predicted V

_{o}values between 0 L and 10 L were set equal to 1 L to be consistent with the rule used described in Section 3. This assumes that predicted low effluent volumes (between 0 L and 10 L) are essentially the same population as the case when no runoff drains from the bioretention cell. Figure 4 also shows that the observed and predicted values closely follow the 1:1 line, in both the construction and validation data sets.

_{i}ranged from 3720 L to 9100 L, while t

_{i}ranged from 14 min to 116 min. The application of the model is approximately limited to these values. Furthermore, the limit of the predicted values of this model are bounded by V

_{o}= 0 L (100% of inlet runoff captured) and by V

_{o}≤ V

_{i}(maximum outlet runoff cannot be greater than inlet runoff). Therefore, it is important to ensure that the input variables used for prediction do not exceed the upper limits of V

_{i}and t

_{i}.

**Figure 4.**Model 2 results: time series comparison of observed and predicted log(V

_{o}) for (

**a**) construction, and (

**c**) validation data sets; and comparison of predicted and observed data sets for (

**b**) construction and (

**d**) validation data sets.

## 5. Model Application

_{o}using the developed model. An important intermediate step in this process is the conversion of the precipitation depth to V

_{i}. A method that accounts for the size of a bioretention cell and the size and degree of imperviousness of its upstream catchment is used and described below.

**Figure 5.**Summary of historical maximum precipitation depth and corresponding duration for Calgary, collected from Environment Canada for the years 1947–2007.

_{i}from the historical precipitation depths. First, the area of the catchment has to be defined; typical guidelines recommend that a bioretention cell should be sized from 5% to 20% of the catchment area [24,25]. For this research, four different catchment sizes were used, where bioretention cell area (which was equal to 32 m

^{2}for the experiments) was 5%, 10%, 15% and 20% of the total catchment area. The corresponding catchment areas are 160, 213.3, 320 and 640 m

^{2}.

^{2}catchment, if the amount of impervious area in a catchment is 40%, the area of the impervious surfaces is 128 m

^{2}.

^{2}, while the bioretention area (i.e., P) is 32 m

^{2}, meaning the I/P = 128 m

^{2}/32 m

^{2}= 4. The I/P approach is useful since it can combine both factors: the total area, and the imperviousness of the catchment, into a single relationship which can be formulated as:

_{i}= volume of influent (m

^{3}); d = precipitation depth (m); A

_{B}= bioretention cell area (m

^{2}); and I/P = Impervious to Pervious Ratio. Equation (7) takes into account the runoff generated from the impervious area in the catchment and also the precipitation that occurs on the bioretention cell itself. A summary of the I/P calculated for the 16 combinations of catchment size and impervious areas are shown below in Table 4. Though Equation (7) shows that V

_{i}is a function of the area of bioretention cell, A

_{B}, this is not explicitly necessary, and Equation (7) can be rewritten in a more general form as:

_{imp}is the impervious area of the catchment (m

^{2}); P

_{oC}is the size of the bioretention cell as a percentage of the total area of the catchment, e.g., between 5% and 20% (%); and A

_{C}is the area of the catchment that drains for the bioretention cell (m

^{2}).

**Table 4.**Summary of I/P values calculated for 16 different combinations of catchment size and percent impervious area in the catchment.

Percent Impervious Area | Percentage of catchment size | |||
---|---|---|---|---|

5% | 10% | 15% | 20% | |

100% | 20 | 10 | 6.7 | 5 |

80% | 16 | 8 | 5.3 | 4 |

60% | 12 | 6 | 4 | 3 |

40% | 8 | 4 | 2.7 | 2 |

_{i}values were computed for the entire historical data set. Values of V

_{i}that exceeded 9,100 L were excluded from further analysis. These values were then used as input in Model 2. The resultant V

_{o}, determined from the model output, were used to calculate ΔV using Equation (1). The results are plotted in Figure 6 below. The figure shows predicted ΔV values versus the precipitation depth at six different duration intervals. Further, at each duration, the ΔV values are further categorized by I/P values. The solid black lines represent predicted values calculated by the model, whereas the dashed lines are extrapolation of the predicted trends.

_{i}, and also the characteristics of the experimental bioretention cell used to generate the data.

**Figure 6.**Bioretention cell design guide for Calgary: runoff volume reduction as a function of precipitation depth, duration and I/P ratio.

## 6. Conclusions

^{2}(0.991), RMSE (0.128), MAE (0.086) and AIC (−28.8). Historical precipitation depth and duration was then used to predict bioretention cell performance under 16 different catchment characteristics, including varied levels of imperviousness and catchment size. These results were collated and extrapolated to create a novel design tool, shown in Figure 6. This figure was then used to demonstrate how future bioretention cell design size can be estimated if performance targets are known. An example also showed how bioretention cell performance can be estimated if the size and characteristics of the catchment are known. Though the methodology was applied to only one case study, Calgary, it can be applied in any region where the relevant data is available.

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**MDPI and ACS Style**

Khan, U.T.; Valeo, C.; Chu, A.; He, J.
A Data Driven Approach to Bioretention Cell Performance: Prediction and Design. *Water* **2013**, *5*, 13-28.
https://doi.org/10.3390/w5010013

**AMA Style**

Khan UT, Valeo C, Chu A, He J.
A Data Driven Approach to Bioretention Cell Performance: Prediction and Design. *Water*. 2013; 5(1):13-28.
https://doi.org/10.3390/w5010013

**Chicago/Turabian Style**

Khan, Usman T., Caterina Valeo, Angus Chu, and Jianxun He.
2013. "A Data Driven Approach to Bioretention Cell Performance: Prediction and Design" *Water* 5, no. 1: 13-28.
https://doi.org/10.3390/w5010013