# Modeling the 2D Inundation Simulation Based on the ANN-Derived Model with Real-Time Measurements at Roadside IoT Sensors

^{1}

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## Abstract

**:**

^{2}) of over 0.8; and it also can delineate the flooding zone to quantify the corresponding area in high reliability in terms of the precision ratio of about 0.7.

## 1. Introduction

## 2. Methodology

#### 2.1. Model Concept

#### 2.2. Simulation of Rainfall-Induced Inundation Events

#### 2.3. Identification of the Virtual IoT (VIOT) Grids

#### 2.4. Artificial Neural Network Model Associated with Multiple Transfer Functions

_{i}stand for the observed inundation depth at the ith IoT sensor and associated distance to the virtual IoT grids (VIOT), respectively; the equations of the estimated inundation depths at the VIOT grids with the observations at the roadside IoT sensors are then established based on the ANN_GA-SA_MTF model; namely, the ANN_GA-SA_MTF model derived can be regarded as an equation for adjusting the estimated spatial averages of the inundation depths at the specific VIOT grids (${\overline{h}}_{IDW,VIOT1}^{t}$) to be the corresponding inundation-depth estimates (${\widehat{h}}_{EST,VIOT1}^{t}$) as follows:

#### 2.5. Integration with Real-Time Correction Approach

#### 2.6. Model Framework

#### 2.6.1. Conceptual Model

**Step 1**: Collect the gridded hyetographs of historical rainstorm events within the study area and extract their gridded characteristics, i.e., rainfall duration, gridded rainfall depth, the areal average of the cumulative dimensionless rainfall, and the associated bias;

**Step 2**: Generate a significant number of rainfall fields with high spatiotemporal resolutions comprised of the simulated gridded rainfall characteristics by the SM_GSTR model with the statistical properties of gridded rainfall characteristics extracted at Step [1].

**Step 3**: Perform the 2D inundation simulation using the SOBEK model with the numerous gridded rainstorms simulated at Step 2 to obtain the simulations of inundation depths at all the grids, including the VIOT grids and IoT sensors.

**Step 4:**Recognize the inundated grids defined as the virtual (VIOT) grids associated with the probabilities of the corresponding nonzero inundation depths at all the grids from a great number of rainfall-induced rainfall flood events.

**Step 5:**Extract inundation-depth estimates regarding the specific time steps during the simulated flood events at the VIOT grids and IoT sensors regarding the particular time steps.

**Step 6:**Calculate the spatial average of inundation-depth estimates obtained in Step 5 at the roadside IoT sensors through Equation (6) as the model inputs of the ANN_GA-SA_MTF at the VIOT grids, where the corresponding simulations of the inundation depth are treated as the model outputs.

**Step 7**: With the model inputs and outputs summarized in Step 6, training the ANN_GA-SA_MTF model regarding the VIOT grids to determine the associated ANN-related coefficients as the parameters of the SM_EID_VIOT model.

#### 2.6.2. Actual Model

**Step 1**: Collect the observed inundation depths during the rainfall-induced flood events at the IoT sensors using the inverse distance method to calculate the areal average at the VIOT grids of interest, as shown in Figure 4.

**Step 2**: Obtain the resulting inundation-depth estimates at the VIOT grids within the study area from the proposed SM_EID_VIOT model.

**Step 3**: Compute the averages of the inundation-depth estimates at the four VIOT grids around the IoT sensors of interest as the corresponding estimations via Equation (9).

**Step 4**: Carry out the real-time correction for the resulting inundation-depth estimates at all the VIOT grids from the proposed SM_EID_VIOT model coupled with the RTEC_2DIS method based on the bias of the inundation-depth estimates in comparison to the measurements at the IoT sensors obtained in Step 4.

**Step 5:**Delineate the flooding zone according to the corrected inundation-depth estimates at all VIOT grids and summarize the number of the inundated VIOT grids to quantify the flooding area. The framework for the model development and application regarding the proposed SM_EID_VIOT can be referred to in Figure 4.

## 3. Study Area and Data

^{2}and 58.3 km, respectively.

## 4. Results and Discussions

#### 4.1. Simulation of Rainfall-Induced Inundation

#### 4.1.1. Extraction of Gridded Rainstorms

#### 4.1.2. Simulation of Rainfall-Induced Inundation Events

#### 4.2. Identification of Ungauged Locations as VIOT Grids

#### 4.3. Training of ANN-Derived Model

#### 4.4. Model Validation

#### 4.4.1. Extraction of Validation Data

^{2}) and then gradually decreases to 2 km

^{2}.

^{2}) (i.e., the square of the coefficient of correlation) which are commonly utilized in the evaluation of the ANN-derived models [24,26,27]. Furthermore, to evaluate the accuracy of the proposed SM_EID_VIOT in the quantification of the flooding area composed of the inundation-depth estimates, three types of performance indices, including the precision index, recall-index, and F1-index, are addressed as follows [3]:

_{IG_EST}and N

_{IG_VAL}serve as the number of the grids regarded as the inundated ones based on the associated nonzero inundation depths estimated by the proposed SM_EID_VIOT model and SOBEK model as the validated data, respectively, and N

_{IG_EST_VAL}denotes the number of the inundated grids identified both by the proposed SM_EID_VIOT model and SOBEK model. In referring to Equations (12)–(14), a high precision index means that the grids with the most of the nonzero inundation depths estimated by the proposed SM_EID_VIOT model can be regarded as the inundated ones by the SOBEK model, revealing that the proposed SM_EID_VIOT model can provide the practically inundated grids with high reliability, while a great recall-index value indicates that the inundated grids identified by the proposed SM_EID_VIOT model can also be treated as the inundated ones by the SOBEK model, implying that the proposed SM_EID_VIOT model can capture the practically inundated grids with high likelihood. Eventually, by substituting the precision- and recall-index calculated in Equation (14), the F1-index can be obtained, the high value of which implies that the resulting flooding regions from the proposed SM_EID_VIOT model have excellent agreement with the validated data.

#### 4.4.2. Evaluation of the Inundation-Depth Estimates

^{2}) correlation coefficient of 0.8 (see Table 6). Moreover, in the case of the comparison regarding the inundation-depth hydrograph, as shown in Figure 20, the estimated hydrographs at the specific VIOT grids approach the validated data with a small RMSE value on average, less than 0.002 m. Moreover, the temporal change in the inundation depths estimated by the proposed SM_EID_VIOT model significantly resembles validated data by a high coefficient of determination (R

^{2}), over 0.98.

#### 4.4.3. Assessment of the Flooding Area

^{2}) can be found in Table 7.

^{2}), the temporal change in the flooding-area estimates is similar to the validated data with a high correlation coefficient of 0.65 (see Table 7).

## 5. Conclusions

^{2}) and over 0.8, respectively, meaning that the inundation-depth estimates at the ungauged grids close to the roadside IoT sensors are significantly correlated with the validated data at the roadside IoT sensors. Nevertheless, the proposed SM_EID_VIOT model could possibly capture the completed flooding zones with some difficulty due to a low recall index of about 0.4, but the resulting inundated grids recognized by the proposed SM_EID_VIOT model are classified as the actual flooding locations with a high precision ratio of approximately 0.7. This reveals that the proposed SM_EID_VIOT model can delineate the flooding zones and quantify the corresponding flooding area in agreement with the validation datasets regarding the spatial variation.

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**The graphic framework of the development and application regarding the proposed SM_EID_VIOT model.

**Figure 2.**Graphical process of extracting the gridded rainfall characteristics from observed photographs of rainstorm events [22].

**Figure 3.**The brief graphical framework for estimating the inundation depths at the VIOT spots via the ANN_GA-SA_MTF model within the SM_EID_VIOT model.

**Figure 5.**Location of Miaoli County (note: blue circles are the radar-precipitation grid, and red points are the roadside IoT sensors).

**Figure 7.**Summary of the gridded rainfall characteristics from 50 historical rainstorms in Miaoli County.

**Figure 10.**Graphical illustration of the flooding area within Miaoli County drawn based on the maximum of the resulting inundation depths.

**Figure 14.**Summary of the transfer function weights for calculating the weighted average of inundation-depth estimates.

**Figure 15.**The SOBEK model simulated inundation-depth hydrographs at the IoT sensors for the 921st simulation case as the validated data.

**Figure 16.**Maximum and average of the simulated inundation depths at the VIOT grids in the SOBEK model for the 921st simulation case.

**Figure 19.**Comparison of the average and maximum of the inundation depths at the VIOT grids by the proposed SM_EID_VIOT model with the validated data. (

**a**) Average inundation depth. (

**b**) Maximum inundation depth.

**Figure 20.**Comparison of the resulting inundation-depth hydrograph at the specific VIOT grids from the proposed SM_EID_VIOT model with the validated data.

**Figure 21.**Comparison of the estimated flooding regions via the proposed SM_EID_VIOT model with results from the SOBEK model at the specific time steps.

**Figure 22.**Comparison of flooding-area hydrograph quantified estimated via the proposed SM_EID_VIOT model with the results from the SOBEK as validated data.

**Figure 23.**Summary of the performance indices of the estimated flooding region by the proposed SM_EID_VIOT model.

**Table 1.**The formula for estimating the number of hidden neurons [1].

No of Formula | Formula |
---|---|

1 | ${N}_{HN}=\left(\sqrt{1+8\times {N}_{IP}}-1\right)/2$ |

2 | ${N}_{HN}={N}_{IP}-1$ |

3 | ${N}_{HN}=\raisebox{1ex}{$2\times {N}_{IP}$}\!\left/ \!\raisebox{-1ex}{${N}_{IP}$}\right.$+1 |

4 | ${N}_{HN}=\sqrt{{N}_{IP}\times {N}_{OP}}$ |

5 | ${N}_{HN}={2}^{{N}_{IP}}-1$ |

6 | ${N}_{HN}=\raisebox{1ex}{$\left[4\times {({N}_{IP})}^{2}+3\right]$}\!\left/ \!\raisebox{-1ex}{$\left[{({N}_{IP})}^{2}-8\right]$}\right.$ |

**Table 2.**Transform functions commonly used [1].

Transfer Function | Formula | Derivative | |
---|---|---|---|

TF1 | Logistic(soft step, Sigmoid) | $\mathrm{f}\left(\mathrm{x}\right)=\frac{1}{1+{e}^{-\propto x}}$ | $\text{}\mathrm{f}\text{}\prime \left(\mathrm{x}\right)=\mathrm{f}\left(\mathrm{x}\right)\left(1-\mathrm{f}\left(\mathrm{x}\right)\right)$ |

TF2 | Tanh | $\mathrm{f}\left(\mathrm{x}\right)=\mathrm{tanh}\left(\mathrm{x}\right)=\frac{2}{1+{e}^{-2\propto x}}-1$ | $\text{}\mathrm{f}\text{}\prime \left(\mathrm{x}\right)=1-f{\left(x\right)}^{2}$ |

TF3 | Arctan | $\mathrm{f}\left(\mathrm{x}\right)=ta{n}^{-1}\left(\propto x\right)$ | $\text{}\mathrm{f}\text{}\prime \left(\mathrm{x}\right)=\frac{1}{{\left(\propto x\right)}^{2}+1}$ |

TF4 | Identity | $\mathrm{f}\left(\mathrm{x}\right)=\propto $ x | $\mathrm{f}\prime \left(\mathrm{x}\right)=\propto $ |

TF5 | Rectified linear unit (ReLU) | $\mathrm{f}\left(\mathrm{x}\right)=\{\begin{array}{c}0forx0\\ xforx\ge 0\end{array}$ | $\text{}\mathrm{f}\text{}\prime \left(\mathrm{x}\right)=\{\begin{array}{c}0forx0\\ 1forx\ge 0\end{array}$ |

TF6 | Parameteric rectified linear unit (PReLU, leaky ReLU) | $\mathrm{f}\left(\mathrm{x}\right)=\{\begin{array}{c}\propto xforx0\\ xforx\ge 0\end{array}$ | $\text{}\mathrm{f}\text{}\prime \left(\mathrm{x}\right)=\{\begin{array}{c}\propto forx0\\ 1forx\ge 0\end{array}$ |

TF7 | Exponential linear unit(ELU) | $\mathrm{f}\left(\mathrm{x}\right)=\{\begin{array}{c}\propto \left({e}^{x}-1\right)forx0\\ xforx\ge 0\end{array}$ | $\text{}\mathrm{f}\text{}\prime \left(\mathrm{x}\right)=\{\begin{array}{c}f\left(x\right)+\propto forx0\\ 1forx\ge 0\end{array}$ |

TF8 | Inverse abs (IA) | $\mathrm{y}\left(\mathrm{x}\right)\frac{x}{1+\left|\propto x\right|}$ | $\text{}\mathrm{y}\text{}\prime \left(\mathrm{a}\right)=\frac{1}{{\left(1+\left|a\propto x\right|\right)}^{2}}$ |

TF9 | Rootsig (RS) | $\mathrm{y}\left(\mathrm{x}\right)=\frac{\propto x}{1+\sqrt{1+{\left(\propto x\right)}^{2}}}$ | $\mathrm{y}\prime \left(\mathrm{x}\right)=\frac{1}{\left(1+\sqrt{1+{\left(\propto x\right)}^{2}}\right)\sqrt{1+a{\left(\propto x\right)}^{2}}}$ |

TF10 | Sech function (SF) | $\mathrm{y}\left(\mathrm{x}\right)=\frac{2}{\mathrm{exp}\left(\propto x\right)+\mathrm{exp}\left(-\propto x\right)}$ | $\text{}\mathrm{y}\text{}\prime \left(\mathrm{x}\right)=-\mathrm{y}\left(\mathrm{x}\right)\mathrm{tan}\mathrm{h}\left(\propto x\right)$ |

Function | Facilities | Number |
---|---|---|

Hydraulic analysis | Sub-basins | 4731 |

Cross-sections | 9838 | |

Gates | 62 | |

Bridges | 9018 | |

Sewer | 68.6 km | |

Maintenance of sewer system | 1382 | |

Hydrological analysis | Rainfall-runoff node | 4097 |

Parameters | Definition | ||
---|---|---|---|

Transfer functions used | TF1-TF10 | ||

Input factors | Resulting areal average of Inundation depth from IoT sensors | ${\overline{h}}_{IDW,VIOT1}^{t}$ | |

Output factor | Inundation depth at VIOT grids | ${\widehat{h}}_{EST,VIOT1}^{t}$ | |

Number of hidden levels | 1 | ||

Number of neurons | 3 | ||

Calibration of parameters of transfer function | Number of optimizations | 10 | |

$\mathrm{Weights}\text{}\mathrm{of}\text{}\mathrm{neurons}\text{}({\mathsf{\omega}}_{HL})$ | Mean | 1 | |

Standard deviation | 3 | ||

$\mathrm{Bias}\text{}\mathrm{of}\text{}\mathrm{function}\text{}({\mathsf{\theta}}_{TF})$ | Mean | 0 | |

Standard deviation | 1 | ||

$\mathrm{Adjusting}\text{}\mathrm{factor}\text{}({\propto}_{TF})$ | Mean | 1 | |

Standard deviation | 0.005 |

**Table 5.**Summary for the appropriate calibrated parameters of the ANN_GA-SA_MTF model at the 500th VIOT grid.

Transfer Function | No of Optimization | Adjust Factor $\left({\propto}_{\mathit{T}\mathit{F}}\right)$ | 1.00113 | |||||
---|---|---|---|---|---|---|---|---|

1 | OPT1 | Weights of $\mathrm{neurons}\text{}{\omega}_{HL}$ | The 1st hidden layer | Input factors | ||||

1 | Bias | |||||||

Neuron | 1 | 2.27251 | −1.14272 | |||||

2 | 0.58986 | −2.71961 | ||||||

3 | 4.56974 | −4.28293 | ||||||

Output layer | The 1st hidden layer | |||||||

1 | 2 | 3 | Bias | |||||

Input factor | 1 | −0.87316 | 1.04999 | 0.08291 | −0.21039 |

**Table 6.**Performance indices of the resulting inundation-depth estimates from the proposed SM_EID_VIOT model.

Performance Index | Inundation Depth | ||||

Average | Maximum | VIOT1676 | VIOT6655 | VIOT2978 | |

Root mean square error RMSE (m) | 0.103 | 0.015 | 0.002 | 0.000 | 0.000 |

Coefficient of determination (R^{2}) | 0.891 | 0.703 | 0.993 | 1.000 | 1.000 |

Performance Index | Precision Index | Recall Index | F1 | RMSE (km^{2}) | R^{2} | |
---|---|---|---|---|---|---|

Statistical properties | Mean | 0.669 | 0.364 | 0.461 | 0.161 | 0.65 |

Standard deviation | 0.209 | 0.220 | 0.208 | |||

Maximum | 1.000 | 1.000 | 1.000 | |||

Minimum | 0.000 | 0.000 | 0.000 |

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## Share and Cite

**MDPI and ACS Style**

Wu, S.-J.; Hsu, C.-T.; Shen, J.-C.; Chang, C.-H.
Modeling the 2D Inundation Simulation Based on the ANN-Derived Model with Real-Time Measurements at Roadside IoT Sensors. *Water* **2022**, *14*, 2189.
https://doi.org/10.3390/w14142189

**AMA Style**

Wu S-J, Hsu C-T, Shen J-C, Chang C-H.
Modeling the 2D Inundation Simulation Based on the ANN-Derived Model with Real-Time Measurements at Roadside IoT Sensors. *Water*. 2022; 14(14):2189.
https://doi.org/10.3390/w14142189

**Chicago/Turabian Style**

Wu, Shiang-Jen, Chih-Tsu Hsu, Jhih-Cyuan Shen, and Che-Hao Chang.
2022. "Modeling the 2D Inundation Simulation Based on the ANN-Derived Model with Real-Time Measurements at Roadside IoT Sensors" *Water* 14, no. 14: 2189.
https://doi.org/10.3390/w14142189