*Water* **2021**, *13*, 3128

**Figure 5.** Hyetographs of 20 observed rainstorm events used in the model development and application [18].

#### **4. Results and Discussion**

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

Before carrying out the rainfall-induced inundation, a great number of the gridded rainstorms should be reproduced in advance. Thus, in this study, using the SG\_GSTR method with the above statistical moments of gridded rainfall characteristics, 1000 simulations can be obtained. After that, the SOBEK 1D-2D hydrodynamic model is employed to carry out the inundation simulation with 20 m × 20 m DEM of Nankan River watershed provided by Water Resource Agency in Taiwan (see Figure 5). Figure 6 presents the river-channel internet and computation nodes in the SOBEK 1D-2D dynamic model set up according to the geographical and hydrological data in the study area. Within the Nankan River watershed, a mesh composed of 335 × 520 computation grids whose the spatial resolution is 20 m × 20 m is adopted. The above topographic, hydrologic, and hydraulic features used in Nankan River SOBEK model are listed in Table 3. Finally, using the SOBEK model for the Nankan River watershed with 1000 simulations of the gridded rainstorm events, the resulting 2D inundation simulations, including the gridded inundation depths and corresponding flooding area, could be accordingly obtained.

**Figure 6.** 2D SOBEK model for the study area (Nankan River watershed) (note: the circle is the rainfall-runoff computing node for each sub-basin).

Thereby, this study implements the 2D inundation simulation by taking into account the spatiotemporal uncertainty in the rain field to achieve the goal of providing detailed 2D inundation information with high spatial and temporal resolution as the training datasets for the proposed SM\_EID\_IOT model.

#### *4.2. Identification of Potential Locations of Roadside IoT Sensors*

On the basis of the results from the simulated grid-based inundation depths within the study area, a 2D inundation simulation with high spatial and temporal resolution can be used to determine the locations of the roadside water-level sensors. In detail, the road-side water-level sensors can be set up at the locations in association with high flooding frequency calculated from the above 2D inundation simulations for the correction of flooding forecasts [30]. Under the consideration of IoT quality, the most appropriate locations of roadside water-level sensors can be accordingly determined and named IoT sensors.

To quantify the flooding risk at all grids, this study utilizes the following equation to calculate the flooding probability (*P<sup>f</sup> h<sup>f</sup>* > 0 ) within the study area, the Nankan River watershed (see Figure 4):

$$\begin{array}{l}P\_f\left(h\_f>0\right)=\frac{\sum\_{i=1}^{N\_{GRS}}\left(I\_f\left(h\_f^i\right)\right)}{N\_{GRS}}\\I\_f\left(h\_f^i\right)=1,\text{ if }\max\left(h\_f\right)>0\\I\_f\left(h\_f^i\right)=0,\text{ if }\max\left(h\_f\right)=0\end{array} \tag{10}$$

where *I<sup>f</sup> h i f* is the flooding indicator and *max hf* stands for the maximum gridded inundation. In Equation (10), if the maximum of gridded inundation depth is greater than zero, the corresponding flooding probability *I<sup>f</sup> h i f* is equal to one; otherwise, it is equal to 0. Thus, in reference to Figure 7, it can be seen that inundation mainly takes place in the four regions. Apparently, the first 46 grids associated with high flooding probabilities (approximately 0.7) can be identified in the two locations marked with a red line, near the locations (TWD97X:280200.2, TWD97Y:2765356.7) and (TWD97X:281523.6, TWD97Y:2761435.5), respectively; they can be treated as the potential inundated grids. Therefore, among the aforementioned potential inundated spots, the locations of desired roadside IoT sensors can be determined.

**Figure 7.** Flooding risk map and locations of potential inundated spots within the study area (the Nankan River watershed).

In conclusion, the quantification of flood risk at all grids within the study area can be carried out using a large number of inundation simulations by the hydraulic numerical model with the generated grid-based rainstorm events through the proposed SM\_GSTR model. Additionally, the resulting big data of rainfall-inundation simulations are advantageous to the identification of roadside inundation-depth sensors, which can be used in the real-time error correction of flood forecasts [1].

As the Nankan River watershed lacks practical roadside IoT sensors, the grid with high flooding risk, the TWD97 coordination of which is (TWD97X: 281523.6, TWD97Y: 2761435.5), is selected as the potential IoT sensor. As a result, the corresponding simulations of rainfall-induced inundation, consisting of the inundation depths and associated areal average rainfalls, are employed in the development and validation of the proposed SM\_EID\_IOT model.

#### *4.3. Determination of Critical Resolutions in Time and Space*

In this study, the correlation and sensitivity analysis for the inundation depth at the IoT sensor of interest (TWD97X: 281523.6, TWD97Y: 2761435.5) is carried out to detect the appropriate critical value of the spatial and temporal resolutions. In detail, the critical distance to the IoT sensor for calculating the areal average rainfalls and the forward time steps from the current time for selecting the observed inundation depths and areal average rainfalls are required for deriving Equation (9) within the proposed SM\_EID\_IOT model.

#### 4.3.1. Critical Resolution in Space

To quantify the fitness of the areal average rainfall for various critical distances to the inundation depth, Pearson correlation coefficients (*ρ*) are adopted, as shown in Figure 8, presenting that most correlation coefficients gradually increase/decrease with the critical distances, i.e., the critical spatial resolution *L<sup>c</sup>* used in Equation (9). For example, regarding the 5th, 500th, and 1000th simulation cases, the correction coefficient generally declines from the critical distance of 1.5 km to 4 km critical distance; it then remains constant. Contrarily, in the case of the remaining simulation cases, the correlation coefficient remains constant.

**Figure 8.** Relationship between the inundation depth and associated areal average of rainfall calculated using the simulated gridded rainfall with the specific distances from the IoT sensor of interest.

To quantify and assess the effect of critical distances regarding the calculation of areal average rainfall on the estimated inundation depth, the statistical properties of Pearson coefficients (i.e., mean value µ and standard deviation σ) calculated from 1000 simulation cases are computed as shown in Figure 9, where the mean value of the correlation coefficient *ρ* declines with the critical distance from 1.5 km (*ρ* = 0.655) km to 3 km (*ρ* = 0.645), and the correlation coefficient then reaches its constant value (*ρ* = 0.645); further, its standard deviations for various critical distances approximate 0.69, except for the 2 km critical distance (*σ* = 0.688).

To sum up, the above results from the correlation analysis reveals that, in spite of the 2 km distance having the smallest variation, the areal average rainfall calculated from the precipitations at the grids, the distance of which to the IoT sensor is less than or equal to 3 km, exhibits a stable variation in terms of the correlation coefficient. In conclusion, the critical resolution in space is assigned as 3 km for the calculation of the areal average rainfalls.

**Figure 9.** Correlation coefficients and associated statistical properties of inundation depths with the areal average rainfall calculated using the simulated gridded rainfall with specific distances from the IoT sensor of interest.

#### 4.3.2. Critical Resolution in Time

Generally speaking, the water level at the current time is markedly related to the areal average rainfall and inundation depths at the previous time steps. Thus, the temporal resolution of the areal average rainfall and inundation depths in terms of the correlations within the specific forward time steps, i.e., critical temporal resolution *T<sup>c</sup>* hour in the standardized regression equation, possibly affect the estimation of the inundation depths at the current time step for the IoT sensors. The standardized regression equation is as follows (Speed and Yu, 1993):

$$\frac{\mathcal{Y} - \overline{\mathcal{Y}}}{\mathcal{S}\_Y} = \sum\_{i=1}^{n} \beta\_i \frac{X\_i - \hat{X}\_i}{\mathcal{S}\_{X\_i}} \tag{11}$$

where *Y* and *X* are the model output and inputs; *Y* and *X*ˆ account for the mean of the model output and inputs, respectively; *S<sup>Y</sup>* and *S<sup>X</sup>* separately represent the corresponding standard deviation; and *β<sup>i</sup>* denotes the regression coefficient that is inversely related to the model outputs; otherwise, the model parameter is proportional to the model output in the case of the associated *β<sup>i</sup>* being positive. Note that the standard regression coefficient *β<sup>i</sup>* accounts for the sensitivities of the *i*th model parameter to the model outputs, meaning that a larger absolute value indicates that the change in the ith model parameter more significantly impacts the estimation of the model outputs. Moreover, the model parameter is associated with the negative *β<sup>i</sup>* . Accordingly, the above specific forward period *k* hours should be treated as the sensible factors, which can be determined based on the regression coefficients *β* (i.e., the sensitivity coefficient) of the standard regression Equation (11).

In this study, using 1000 simulations of the rainfall-induced inundation, the inundation depths at the six particular time steps—0.3, 0.5, 0.6, 0.7,0.8, and 0.9 times the duration—and the inundation depths as well as the areal average rainfall at the forward 6 h are used in the establishment of the standard regression equation; their resulting regression coefficients could be obtained as shown in Figure 10. By observing Figure 10, the average of absolute sensitivity coefficients regarding the areal average rainfall and inundation depths at the forward 1–6 h, it is known that, with respect to the inundation depth, the average of the absolute sensitivity coefficients of the forward period from *T<sup>c</sup>* = 1 h to *T<sup>c</sup>* = 3 h ranges between 0.563 and 0.43, which are obviously greater than the coefficients regarding the forward *T<sup>c</sup>* = 4 h to *T<sup>c</sup>* = 6 h (about 0.029–0.127). Furthermore, in the case of the areal

average rainfall, the absolute sensitivity coefficients corresponding to the forward 3 h change on average from 0.123 to 0.426, which are significantly greater than the coefficients at the forward 4–6 h (approximately 0.015–0.048).

**Figure 10.** The averages of absolute sensitivity coefficients regarding the areal average rainfall and inundation depths at the various forward time steps.

To sum up the above results, the inundation depths at a particular time step are strongly and significantly related to the areal average of rainfall and inundation depths at the forward *T<sup>c</sup>* = 3 h. Therefore, in referring to Equation (9), the relationship of the estimated/forecasted inundation depth regarding the lead time (*t* + 1 h) at a specific IoT sensor ˆ*h t*+1 *IOT* with the observation of the areal average rainfall at the lead time (*t* + 1 h) and current time (t hour) as well as the forward 2 h and the inundation depths at the forward 3 h (i.e., *t*, *t* − 1 and *t* − 2 h) can be written as follows:

$$
\hat{h}\_{\rm IOT}^{t+1} = f\_{\rm ANN-GA-MTF} \left( \overline{\mathbb{R}}\_{\rm IOT'}^{t+1} \overline{\mathbb{R}}\_{\rm IOT'}^{t} \overline{\mathbb{R}}\_{\rm IOT'}^{t-1} \overline{\mathbb{R}}\_{\rm IOT'}^{t-2} h\_{\rm IOT'}^{t} h\_{\rm IOT'}^{t-1} h\_{\rm IOT}^{t-2} \right) \tag{12}
$$

where *R t IOT* is the rainfall forecast at the lead time (*t* + 1 h); *R t IOT*, *R t*−1 *IOT*, and *R t*−2 *IOT* are the areal average rainfalls calculated using the gridded rainfall at the current time (*t* hour) and forward 2 h (*t* − 1 and *t* − 2 h) within the distance of 3 km to the location of the IoT sensor; and *h t IOT*, *h t*−1 *IOT*, and *h t*−2 *IOT* represent the observed inundation depths at the current time (*t* hour) and forward 2 h (*t* − 1 and *t* − 2 h).

#### *4.4. Development of the Proposed SM\_EID\_IOT Model*

In referring to the framework of the model development, the proposed SM\_EID\_IOT model for estimating the inundation depths at the lead time regarding the IoT sensor is developed based on the ANN-derived ANN\_GA-SA\_MTF model by taking into account the uncertainty factors, including the areal average rainfall at the lead time and forward 3 h and the inundation depths at the forward 3 h. Furthermore, according to the induction to the ANN\_GA-SA\_MTF model, the initial conditions regarding the parameters should be given in advance, including the number of the hidden layers, the total number of neurons used, and the candidate transfer functions, as listed in Table 2. It is well-known that the three-layer network structure, comprising one input layer, one output layer, and one hidden layer, is commonly adopted in hydrological/hydraulic modeling (e.g., Wu et al., 2021); thus, the hidden layer used in the derivation of the SM\_EID\_IOT model is derived on the basis of the three-layer ANN-based model (i.e., the ANN\_GA-SA\_MTF model).

Moreover, in general, the number of neurons in the entire ANN network structure can be set up by means of a variety of formulae listed in Table 1 in accordance with the number of model inputs and outputs. Figure 11 indicates the resulting number of neurons from the different equations, ranging from 3 neurons to 127 neurons. Accordingly, their average number of 8 neurons is employed in the model development. In addition, the statistical properties, the mean and standard deviation, of the ANN weights are assigned as 0 and 1, respectively. As for the remaining parameters, their initial conditions can be referred to in Table 4.

**Figure 11.** Summary for the estimation of the number of hidden neurons via various methods.



Using the parameter definition shown in Table 5, the SM\_EID\_IOT model can be developed by training the ANN\_GA-SA\_MTF model with 650 simulations of the training datasets, extracted from 1000 rainfall-induced inundation simulations obtained in Sections 4.1 and 4.2. Table 5 summarizes the results from the parameter calibrations under consideration of the transfer function TF1 (Sigmoid function). Furthermore, according to Figure 12, the inundation-depth estimates are the weighted average using the model estimates resulting from a variety of transfer functions with the corresponding weights, as

shown in Figure 13, in which TF2 (Tanh function) and TF5 (ReLU function) are associated with the maximum and minimum weights, 0.10045 and 0.0996, respectively, indicating that the various transfer functions significantly contribute to different degrees to the estimation of the inundation depths at the IoT sensors. Consequently, it is necessary to take into account the effect of uncertainty in the formulation of the transfer functions on the estimation of model outputs via the proposed SM\_EID\_IOT model.

**Table 5.** Summary for the calibrated parameter of the SM\_EID\_IOT model in the study area (the Nankan River Watershed).


**Figure 12.** Summary of the weights of the transfer function for calculating the weighted average of inundation-depth estimates.

#### *4.5. Model Validation*

To demonstrate the reliability and accuracy of the resulting inundation-depth estimates from the proposed SM\_EID\_IOT model, a simulated rainfall-induced inundation event, i.e., the 825th simulated rainstorm event (see Figure 13) is adopted as the validated one, where the duration and average rainfall intensity regarding the validated event are 57 h and 3.7 mm/h, respectively; moreover, in the validated water-level hyetograph, the two peaks of the inundation depths are 0.12 m and 0.1 m at the 30th and 40th hours, respectively. As the Center Weather Bureau (CWB) in Taiwan can provide the gridded rainfall forecasts at lead times of 3 h, the model verification focuses on the evaluation in comparison with the inundation-depths estimates at the 1, 2, and 3 h lead times, namely, *t* + 1, *t* + 2, and *t* + 3 h, respectively (*t* = current time).

**Figure 13.** Relationship between the areal average rainfall and inundation depths regarding the 825th validated rainfall-induced flood event.

#### 4.5.1. Reliability Quantification of Inundation-Depth Estimates

Accordingly, as for the 460th simulation case, the resulting inundation-depth estimates and associated 95% confidence intervals at the IoT sensor can be obtained from the SM\_EID\_IOT model as compared with the validated ones, as shown in Figure 14. It can be seen that the temporal change in the inundation-depth estimates at the three lead times resembles variation regarding the areal average rainfall in accordance with the high correlation. Moreover, at the 1 h lead time, the estimated inundation depths mostly lie around the 95% confidence interval, except for the 30th–31th and 40th–41th hour, where the inundation-depth estimates are about 0.119 m and 0.09 m, exceeding the upper bounds of 0.075 m and 0.085 m, respectively. Similar results can be found for the inundationdepth forecasts at the 2 h and 3 h lead times. The above results imply that the proposed SM\_EID\_IOT model can produce the inundation-depth estimate with high likelihood of approaching the true values (i.e., observations).

As can be seen in Figure 14, in spite of the proposed SM\_EID\_IOT model possibly producing reasonable inundation-depth estimates, the inundation-depth estimates at the 1 h lead time are underestimated as compared with the validated data at the 30th–32th hours regarding the 825th simulated event. Hence, to evaluate the accuracy of the inundationdepth forecasts at the various lead times, the performance in comparison with the estimated inundation depths and validated ones at the 31th–51th hours is carried out in terms of the root mean error (RMSE) and correlation coefficients, as shown in Figure 15. According to the results from Figure 15, the RMSE increases with the lead time; however, the correlation coefficient declines with the lead time. For example, although the estimations exhibit an obvious difference from the validation data in association with a large root mean error square (RMSE) (about 0.01 m), the corresponding correlation coefficient approaches 0.3, meaning the change in the average rainfall in time is close to that regarding the inundation depth; similar conclusions are also made based on the results from the 2 h and 3 h lead time. The above difference between the estimation and validations might be caused by the uncertainties in the observation and model parameters.

**Figure 14.** Comparison among the validated, estimated, and corrected inundation depths as well as the quantified 95% confidence intervals for the validated rainfall-induced flood event by the proposed SM\_EID\_IOT model.

**Figure 15.** Summary of the performance indices of the inundation-depth estimates and comparison with the validation datasets at various lead times.

#### 4.5.2. Real-Time Correction of Inundation-Depth Estimates

To facilitate the accuracy of the results from the proposed SM\_EID\_IOT model, the inundation-depth estimates are supposed to be adjusted based on the real-time measured data. The real-time error correction method is developed via the time series approach and Kalmen filtering equation (RTEC\_TS&KF) [30]. Figures 16 and 17 show the comparison between the validated, estimated, and corrected inundation depths, respectively, as well as the corresponding performance indices, respectively, indicating that the RMSE value increases with the lead time; in contrast, the correlation coefficient markedly decreases with the lead time. In spite of the RMSE values for the corrected inundation-depth estimates significantly increasing with the lead time, from 0.007 m to 0.014 m, on average, they are less than those for the forecasts (0.012 m). Moreover, with respect to the consistency wof the validations, the correlation coefficients for both inundation-depth estimations and corrections generally decrease with the lead time, 0.28–0.26 (estimations) and 0.74–0.06 (corrections), respectively. For illustration, the correlation coefficients for the corrections at the 1 h lead time approximate 0.7, obviously greater than that from the estimates (about 0.28). In particular, at the 3 h lead time with the worst correlation, −0.25 (estimations) and 0.07 (corrections), the corrections have better consistency with the validated datasets than the estimations. The above results reveal that the corrected inundation-depth forecasts exhibit better agreement with the observations than the underestimated/overestimated inundation-depth forecasts, with the marked errors even for the long lead times being able to be immediately adjusted.

**Figure 16.** *Cont.*

**Figure 16.** Comparison among the validated inundation depths and the corrected as well as corrected ones by the proposed SM\_EID\_IOT model during the validated rainfall-induced flood event.

**Figure 17.** *Cont.*

**Figure 17.** Summary of the performance indices of the inundation-depth estimates and the corresponding corrections for the validated rainfall-induced flood event.

In summary, the accuracy of the resulting inundation-depth estimates at various lead times (hours) from the proposed SM\_EID\_MTF with a reasonable reliability can be effectively improved based on the difference between the observations and estimates at the previous time steps during a rainfall-induced flooding event.

#### **5. Conclusions**

This study intends to propose a stochastic ANN-derived model for the estimation of the inundation depths at the roadside water-level sensors set up through the Internet of Things (IoT) (named the SM\_EID\_IOT). The proposed SM\_EID\_IOT is developed on the basis of the artificial neural network model ANN\_GS-SA\_MTF (Wu et al., 2021), in which the associated parameters are calibrated by means of the modified genetic algorithm (GA-SA) [30] under the consideration of multiple transfer functions. A basin located in northern Taiwan, the Nankan River watershed, is selected as the study area and the associated grid-based precipitation data regarding 20 historical rainstorms provided by Center Weather Bureau in Taiwan are utilized to reproduce 1000 simulations of the rainfallinduced inundations via as the training dataset for the development of the proposed SM\_EID\_IOT model.

According to the results from the correlation and sensitivity analysis, the inundation depths at the IoT sensor for the forward periods of 3 h (i.e., critical temporal resolution) and the corresponding precipitations at the neighboring grids within the specific distance of 3 km (i.e., critical spatial resolution) to the IoT sensor should be regarded as the uncertainty factors for the resulting inundation-depth estimate (i.e., model inputs) from the proposed SM\_EID\_IOT model. Additionally, the results from the model demonstration indicate that the validated inundation depths at the lead times of 3 h are almost located within the quantified 95% confidence intervals by the proposed SM\_EID\_IOT model, revealing that the proposed SM\_EID\_IOT can provide the inundation-depth estimates at the lead times of 3 h with a high likelihood of approaching the validated datasets. Furthermore, the corrected inundation-depth estimates by the real-time error correction method RTEC\_TS&KF method integrated within the proposed SM\_EID\_IOT model could effectively improve the accuracy of the inundation-depth estimates by 50%; thereby, the estimate exhibits a good match with the validated datasets under a better temporal correlation (i.e., correlation coefficient approaching 0.8). Consequently, the proposed SM\_EID\_IOT model is capable of estimating more accurate inundation depths at the IoT sensors of interest with high reliability.

In addition to the estimation of the inundation depths at the particular locations at which the roadside water-level IoT sensors are set up, the rainfall-induced flooding map is necessarily delineated in order to primarily estimate the possible inundation area under conditions of the flood-rated indices, such as the flash-food potential index (FFPI)

and flooding potential index (FPI) [48], and rainfall-rated variables, such as the radar precipitation [49,50]. As a result of the flooding map being composed of the gridded inundation depths, the inundation depths at the ungauged locations should be quantified; by doing so, future work would improve the application of the framework and detailed concepts of developing the proposed SM\_EID\_IOT model in the derivation of stochastic ANN-derived modeling (i.e., the ANN\_GA-SA MTF model) for the inundation-depth estimates at the ungauged locations within the flood-prone zones.

**Author Contributions:** S.-J.W. Conceptualization, methodology, investigation, validation, original draft preparation, and writing—review and editing; C.-T.H. Resources and software; C.-H.C. Resources. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research received no external funding.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** This paper was written based on the results from the several researches made by authors which were not mentioned in any reports.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**


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