# A Self-Contained and Automated Method for Flood Hazard Maps Prediction in Urban Areas

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Operational Method for Flooded Areas Forecasting

#### 2.1. Hydronet Algorithm

#### 2.1.1. Large-Scale Topographic Data Correction (Step 1)

#### Computation of the “Upper” Solution

#### Computation of the “Lower” Solution and of the Final One

**A**the subset of the basin cells where the “upper” solution is strictly greater than the “lower” solution. Using good quality DEM data, the size of

**A**, where zu

_{i}or zl

_{i}is not equal to ${z}_{i}^{*}$, is of the order of 1% of all the cells or even less. Moreover, these cells are clustered inside many small areas disconnected from each other. Say

**A**

_{j}the j

^{th}sub-set of the cells of

**A**connected to each other. The final solution is found by minimizing the square difference between the elevations measured in each sub-set

**A**

_{j}and the elevations obtained as a linear combination of the “upper” and “lower” solutions in the same sub-set. The weight of the linear combination is the same for all the cells of the sub-set; therefore, the properties of both the “lower” and “upper” solutions are saved in the final one, named zc. See a detailed description of the algorithm in Appendix C.

^{6}cells, the stationarity is achieved in about 10 min (including I/O operations) using a single core of Intel(R) i7-4770 CPU 3.4 GHz.

#### 2.1.2. Automatic Generation of the Hydrographic Network and of the TIN Hydraulic Computational Mesh (Steps 2–4)

^{*}(Step 1) is the basis for the computation of the hydrographic network. This network can be thought of as the ensemble of the segments connecting the centers of cells i, iout(i). These segments can only have four possible directions—the x-direction, the y-direction, and the two diagonal directions. The use of all the i–iout(i) segments would lead to a bad representation of the basin and, most importantly, it would not help the generation of a TIN as support for the runoff propagation model. In order to reduce the complexity of the hydrographic network, we need to order all the cells inside the basin according to their drainage area. Our final hydrographic network is the network resulting from the i–iout(i) segments connecting all the cells characterized by a drainage area/basin area ratio greater than a minimum value, named threshold ratio Sill.

#### 2.1.3. Computation of the Final Node Elevation (Step 5)

#### 2.2. Runoff Model

^{2}) is the extension of the Thiessen polygon and d (hours) is the duration of every single storm. In the implemented model, the measured rainfall intensity is multiplied by the ARF factor.

#### 2.3. Routing Model

## 3. Case Study and Model Setup

^{2}. In the downstream part of the basin, the urban area of Pistrino, on the left of the Sovara stream, is subject to flooding phenomena during intense rainfall events.

^{−2}. In the case of correction by the “filling up” methods only, the same error is equal to 8.17 × 10

^{−2}, that is almost twice the one obtained by the Hydronet. Figure 9 shows how the correction by the “filling up” methods can provide a significant local overestimation of the bottom elevation inside channels, with an artificial flood prediction in the neighboring areas. In the same figure, observe that along a single channel of the case study, the “filling up” method raises the bottom elevation up to 2 m, for a maximum length of about 900 m. On the contrary, the maximum correction carried out in the same channel by the Hydronet is restricted to 0.75 m.

## 4. Results

^{3}/s (as calculated by our next simulations), the flow overtops the inlet culvert, inundates the Pistrino urban area, and a large part of it leaves the basin through the left boundary of the catchment, along a very flat area, without passing through the outlet gauged section.

^{3}), while the peak discharge is used to calibrate the Manning roughness n (Figure 14). For the hydraulic analysis, zero water depth is assigned as the initial condition. The calibrated parameters are CN = 84, λ = 0.01, and n = 0.050 s/m

^{1/3}. The observation that an initial abstraction ratio λ equal to 0.20, as suggested in [26], provides a strong delay of the simulated storm hydrograph with respect to the observed one, which is missing in the computed optimal one. The optimal CN parameter is also much higher than the regional values suggested in the literature [26]. This can be justified by the fully distributed structure of the proposed model, which simulates the water storage effect during the flow routing. The same effect is embedded in the CN parameters when the Curve Number method is applied for the discharge computation. In this first calibration event, the errors in the volume and peak discharge estimation are both close to 1.1%, and the Nash–Sutcliffe efficiency (NSE) is equal to 0.882.

^{1/3}, CN = 84, and λ = 0.01, calibrated for the 2012 event, have been used to simulate the 2016 event. The observation that the CN parameter is a function of the ground saturation level and, is used in the context of nowcasting, should be corrected according to the actual saturation level. The errors between computed and observed hydrographs otherwise grow up to 14.5% and 10.2%, respectively, for the volume and for the peak discharge, but the shape of the computed hydrograph is quite similar to the measured one. The NSE found in this validation event is equal to 0.776. The relatively small impact of the CN parameter is confirmed by its optimization for the 2016 flood event. In this case, the optimal value of CN equal to 77 allows to drop the errors up to 0.3% and 4.7%, respectively for the volume and for the peak discharge but leads to anNSE equal to 0.753, similar to the previous one. The results obtained for the 2016 event are shown in Figure 16. In both events, the differences between the measured and the computed discharges are likely to depend mainly on the actual heterogeneity of the rainfall distribution, which has been accounted for in the model only through the ARF reduction of the mean value over each polytope.

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A. Mathematical Statement of the Constrained Minimization (CM) Problem and Computation of the “Upper” Auxiliary Solution

_{c}− 1

_{i}is the corrected one, ic(i,m) is the index of the m

^{th}cell surrounding cell i in any of the eight available directions, ε is a minimum slope (parameter), c

_{i,m}is the distance between the center of the two cells, i and ic(i,m), Nc is the total number of cells, as well as the index of the outlet cell. The elevation of the basin outlet cell is not subject to then constraint (A2). We call the solution of problem (A1) and (A2) an optimal solution.

_{i}cell elevations equal to Z, except the basin outlet one. Z is the maximum cell elevation in the basin, and the elevation zu

_{Nc}of the basin outlet cell always remains the measured one.

_{i,m}is the index of the cell neighbor to cell i in the m

^{th}direction, c

_{i,m}is the distance between the centers of cells i and ic

_{i,m}. The eight possible directions are Nord, Nord-Est, Est, Sud-Est, Sud, Sud-West, West, Nord-West (Figure A1). If the cell is a boundary cell, one or more directions will be missing. Updating the residue with:

## Appendix B. Computation of the “Lower” Auxiliary Solution

_{T}the number of the leaf cells. Say T(m) the cell index of the m

^{th}leaf cell. The “lower” solution zl is initialized by setting all the zl(i) cell elevations equal to Z, except the leaf cells, where we set zl(i) = z

^{*}(i). This also initializes an auxiliary vector I(i) = Inv(i).

_{c}(the outlet cell index) and I(N

_{c}) = 1 stop; If iout(i) ≠ N

_{c}and I(N

_{c}) = 1 set i= iout(i) and go back to operator (1B); If iout(i) ≠ N

_{c}and I(N

_{c}) > 1 set k = k + 1 and start again from i = T(k). See the flow chart of the “lower” solution computation in Figure A3.

## Appendix C. Optimal Combination of the “Upper” and the “Lower” Solutions

^{th}branch and Iz(i) the branch index associated to cell i. Assume Iz(i) = 0 if zu

_{i}= zl

_{i}. Call Ar(j) the cell index of the root of the j

^{th}branch, such that Iz(Iout(Ar(j))) = 0. Say A(j,k) the cell index of the k

^{th}element of the j

^{th}branch. Search for the optimal combination of the “upper” and “lower” solutions in each branch, given by:

## Appendix D. Correction of the TIN Node Elevations

^{th}connected node, nste(i) is the number of nodes connected to node i and N

_{n}is the total number of nodes, including the node at the center of the DEM outlet cell, which is assumed to have index N

_{n}.

## Appendix E. Culvert Modelling

^{th}time step. Mixed edges take the piezometric gradient of the channel edge connected to their culvert section and conveyance, which is a fraction c of the conveyance of the same channel edge. Because the conveyance per unit width is well-known to be almost proportional to the power 5/3 of the water depth, the fraction c

_{s,i}of the mixed edge linking culvert section s with the i

^{th}2D node is computed at the end of the previous time step as:

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**Figure 2.**The eight neighboring cells surrounding a generic cell i inside the basin and the corresponding directions. The possible location of the cell iout(i) is also shown.

**Figure 8.**Sovara stream basin—hydrographic network computed with different values of the Sill threshold ratio.

**Figure 9.**Comparison of the Digital Elevation Model correction between the Hydronet and the “filling up” method.

**Figure 13.**(

**a**) Hyetographs and (

**b**) total rainfall depths observed during the two investigated flood events.

**Figure 14.**November 2012 flood event: comparison between observed rainfall, observed and computed discharge hydrographs at the hydrometric section of Pistrino (basin outlet).

**Figure 16.**The November 2016 flood event: a comparison between the observed rainfall, and the observed and computed discharge hydrographs at the hydrometric section of Pistrino (basin outlet).

Vector Name | Description |
---|---|

zcm | Optimal solution of the Constrained Minimum (CM) problem, never found |

zu | “Upper” auxiliary solution of the CM problem |

zl | “Lower” auxiliary solution of the CM problem |

zc | Sub-optimal solution of the CM problem, linear combination of zu and zl |

zp | TIN node elevations, interpolated from zc elevation in the cell centers and from other point elevations |

z | Corrected final elevation in the TIN nodes |

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**MDPI and ACS Style**

Sinagra, M.; Nasello, C.; Tucciarelli, T.; Barbetta, S.; Massari, C.; Moramarco, T.
A Self-Contained and Automated Method for Flood Hazard Maps Prediction in Urban Areas. *Water* **2020**, *12*, 1266.
https://doi.org/10.3390/w12051266

**AMA Style**

Sinagra M, Nasello C, Tucciarelli T, Barbetta S, Massari C, Moramarco T.
A Self-Contained and Automated Method for Flood Hazard Maps Prediction in Urban Areas. *Water*. 2020; 12(5):1266.
https://doi.org/10.3390/w12051266

**Chicago/Turabian Style**

Sinagra, Marco, Carmelo Nasello, Tullio Tucciarelli, Silvia Barbetta, Christian Massari, and Tommaso Moramarco.
2020. "A Self-Contained and Automated Method for Flood Hazard Maps Prediction in Urban Areas" *Water* 12, no. 5: 1266.
https://doi.org/10.3390/w12051266