# Introducing the Ensemble-Based Dual Entropy and Multiobjective Optimization for Hydrometric Network Design Problems: EnDEMO

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## Abstract

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## 1. Introduction

- (1)
- Evaluate the existing hydrometric networks by applying the entropy-based transinformation analysis;
- (2)
- Identify potential station locations in the study area and define the simulated time series for each station;
- (3)
- Apply the traditional, deterministic, entropy-based hydrometric network design approach to find the optimal networks;
- (4)
- Apply the proposed ensemble-based design approach;
- (5)
- Compare the optimal networks and suggest the locations where the additional monitoring is recommended.

## 2. Study Area and Data Preparation

^{2}. The minimum network guideline by World Meteorological Organization (WMO) [22] recommends 1875 km

^{2}per station for streamflow monitoring in interior plains. This shows that the NCRB currently requires the installing of new monitoring stations to collect more representative information from this large basin. The number of existing stations and the number of catchments associated with each sub-basin are summarized in Table 1. For the streamflow monitoring, catchment topology is one of the important aspects: hence, the outlets of the catchments were selected as potential station locations in this study (see Figure 2). The number of catchments which is equivalent to the number of potential stations is 2693. This study follows the delineation result which was accomplished during the Hydrological Prediction for the Environment (HYPE) modeling process [23], which will be described in Section 4.

## 3. Background

#### 3.1. Information Theory

#### 3.2. Transinformation Analysis

#### 3.3. Dual Entropy and Multiobjective Optimization (DEMO)

## 4. Methodology

- (1)
- Obtain daily observed streamflow time series from the existing stations. In this study, the HYDAT database of Environment and Climate Change Canada (ECCC) and the National Water Information System (NWIS) database of the United States Geological Survey (USGS) were used [23].
- (2)
- Estimate the transinformation (TI) index for each existing station to evaluate the current network.
- (3)
- Draw the TI index maps by spatially interpolating the values from (2).
- (4)
- Select a regionalization method or a hydrologic model to generate estimated (or synthetic) time series data at potential station locations. In this study, simulated time series from HYPE model by Stadnyk and Bajracharya (2019) were used.
- (5)
- Obtain the daily estimated runoff time series data for potential stations from (4).
- (6)
- Run the dual entropy and multiobjective optimization (DEMO) tool.
- (7)
- Determine the optimal networks and analyze the networks by creating maps of the station selection frequency.
- (8)
- Generate ensemble runoff time series to account for the model uncertainty in runoff estimation.
- (9)
- Run ensemble-based DEMO (EnDEMO) with the ensemble time series. Ten ensemble members were applied in this study.
- (10)
- Analyze the optimal networks from the EnDEMO application and draw maps of the station selection frequency.
- (11)
- Compare the DEMO with the EnDEMO results and make recommendations.

#### 4.1. Ensemble-Based DEMO (EnDEMO)

- Daily estimated flows from each subcatchment are aggregated to monthly streamflow time series. The monthly historical streamflows are log-transformed and then normalized by the sample mean and the standard deviation values, so that the resulting standardized log-transformed monthly streamflow time series follow a standard normal distribution.
- Then synthetic ensembles are randomly populated from the standardized log-transformed monthly flows that satisfy its statistics.
- Additional requirements are set when randomly sampling the synthetic ensemble time series. The method of Cholesky matrix decomposition is used to populate monthly synthetic flows that preserve the month-to-month and year-to-year historical autocorrelation matrix of the raw log-transformed streamflow time series.
- The ensemble standardized log-transformed flows are transformed back to real space, ensemble, monthly streamflows by de-standardizing and log-inversing.

- The k-nearest neighbor is estimated for each ensemble month from all estimated monthly values collected from the surrounding subcatchments using the real space Euclidean distance.
- The k-nearest neighbors in each month are ranked from the nearest to the furthest sites.
- Using a Kernel estimator, the probability of selecting a neighbor is estimated for each site that has an estimated daily time series and the closest neighbor is selected accordingly.
- The final step is to proportionally disaggregate the ensemble monthly streamflow to daily time series using the selected neighboring site.

#### 4.2. Hydrological Prediction for the Environment (HYPE) Model

## 5. Results and Discussions

#### 5.1. TI Index Map

- Highly Deficit: TI index of 0.0 to 0.3;
- Deficit: TI index of 0.3 to 0.6;
- Average: TI index of 0.6 to 0.8;
- Above Average: TI index of 0.8 to 1.0.

#### 5.2. DEMO Results

_{2}3652 = 11.83 bits, the range of the joint entropy of the optimal solutions, which is from 9.5 bits to 10.5 bits is appropriate. The number of the selected stations in each optimal network in the Nelson sub-basin ranges from 33 to 190 and total correlation varies from 42.45 bits to 288.14 bits. In general, total correlation is lower in the Pareto-front when the number of the selected stations is lower and vice versa. The shapes of the Pareto-front for other sub-basins were similar that of the Nelson sub-basin, so they are not presented here. Instead, Table 3 shows the number of the optimal networks, ranges of the number of the selected stations in the optimal networks, and their objective values for each sub-basin.

#### 5.3. Uncertainty Considerations using EnDEMO

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 5.**Pareto-front plot from the dual entropy and multiobjective optimization (DEMO) result of the Nelson sub-basin.

**Figure 6.**Spatial distributions of the selected optimal networks and their locations (red circles) in the Pareto-front for the Nelson sub-basin. (

**a**,

**b**) Optimal network which has the maximum joint entropy among the optimal solutions; (

**c**,

**d**) optimal network which has the median joint entropy and total correlation values among the optimal solutions; (

**e**,

**f**) optimal network which has the minimum total correlation among the optimal solutions.

**Figure 7.**Spatial distributions of the selected optimal networks and their locations (red circles) in the Pareto-front for the Saskatchewan sub-basin. (

**a**,

**b**) Optimal network which has the maximum joint entropy among the optimal solutions; (

**c**,

**d**) optimal network which has the median joint entropy and total correlation values among the optimal solutions; (

**e**,

**f**) optimal network which has the minimum total correlation among the optimal solutions.

**Figure 9.**Map of the station selection frequency from ensemble-based dual entropy and multiobjective optimization (EnDEMO) results.

Sub-Basins | Numbers of Existing Stations | Numbers of Potential Stations |
---|---|---|

Assiniboine | 54 | 271 |

Red | 41 | 218 |

Winnipeg | 25 | 377 |

Lake Winnipeg | 14 | 294 |

Saskatchewan | 98 | 695 |

Nelson | 9 | 219 |

Upper Churchill | 23 | 556 |

Lower Churchill | 3 | 61 |

Model Parameters | Parameter Value |
---|---|

Population Size | 3000 |

Maximum Generations | 6000 |

Number of Decision Variables | Number of Potential Stations (N) |

Crossover Operator | Single Point Crossover |

Crossover Probability | 1.0 |

Mutation Operator | Bit String Mutation |

Mutation Probability | 2/N |

Variable Type | Binary |

Sub-Basins | Numbers of Optimal Networks | Numbers of Selected Station Range | Joint Entropy Range (bits) | Total Correlation Range (bits) |
---|---|---|---|---|

Assiniboine | 1343 | 29–219 | 10.57–11.39 | 37.76–227.91 |

Red | 1191 | 19–179 | 9.22–10.78 | 19.03–158.54 |

Winnipeg | 1134 | 34–228 | 10.95–11.47 | 57.31–333.89 |

Lake Winnipeg | 1415 | 60–262 | 10.87–11.63 | 65.66–344.52 |

Saskatchewan | 685 | 22–162 | 11.47–11.82 | 70.80–235.38 |

Nelson | 1205 | 33–190 | 9.52–10.45 | 42.45–288.14 |

Upper Churchill | 1036 | 70–319 | 10.99–11.54 | 82.35–456.01 |

Lower Churchill | 454 | 6–57 | 5.24–7.25 | 5.13–76.48 |

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**MDPI and ACS Style**

Keum, J.; Awol, F.S.; Ursulak, J.; Coulibaly, P. Introducing the Ensemble-Based Dual Entropy and Multiobjective Optimization for Hydrometric Network Design Problems: EnDEMO. *Entropy* **2019**, *21*, 947.
https://doi.org/10.3390/e21100947

**AMA Style**

Keum J, Awol FS, Ursulak J, Coulibaly P. Introducing the Ensemble-Based Dual Entropy and Multiobjective Optimization for Hydrometric Network Design Problems: EnDEMO. *Entropy*. 2019; 21(10):947.
https://doi.org/10.3390/e21100947

**Chicago/Turabian Style**

Keum, Jongho, Frezer Seid Awol, Jacob Ursulak, and Paulin Coulibaly. 2019. "Introducing the Ensemble-Based Dual Entropy and Multiobjective Optimization for Hydrometric Network Design Problems: EnDEMO" *Entropy* 21, no. 10: 947.
https://doi.org/10.3390/e21100947