# Accounting for Economic Factors in Socio-Hydrology: Optimization under Uncertainty and Climate Change

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## Abstract

**:**

## 1. Introduction

- In the original paper the costs of each regime are not fully accounted for. Losses from floods include not only population changes, but also loss of physical assets and costs associated with levee construction. In our model these costs are included.
- In [1] the model for the technosociety case is myopic. Society reacts only after an extreme event without considering the probabilities (frequency and intensity) of future extreme events. Our model makes use of the calculated probabilities in infrastructure decision-making.
- The modeling in [1] is non-stochastic—the sequence of flood events is pre-determined over the century, with 35 extreme events occurring over a 200-year period. A natural extension, therefore, is to explore how the results are affected when the sequence and intensity of extreme events are stochastic in nature requiring optimization under uncertainty to find optimal levee heightening strategies. This is done in our model.
- In the framework with stochastic extreme events, we build an objective decision function based on the net present value. This function allows us to obtain expected present values under uncertainty, depending on the adaptation strategy and the model´s stochastic parameters with two risk factors: frequency and intensity of extreme flood events.
- Furthermore, we include a model to calibrate the stochastic parameters based on historical information. Using this net-present-value framework, optimal strategies are explored, e.g., an optimal levee heightening strategy, which could not be done in the original model.

## 2. Methodology

#### 2.1. Land Services

#### 2.2. Losses in the Floodplain from Floods

#### 2.3. Costs of Raising and Replacing Flood Defenses

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#### 2.4. Optimization Methods

#### 2.4.1. Optimization Methods for a Deterministic Characterization of Flood Events

#### 2.4.2. Optimization Model for a Stochastic Characterization of Flood Events

^{2}= 0.845) and for the generalized Pareto distribution (χ

^{2}= 0.580). Since the goodness of fit test statistics indicate the distance between the data and the fitted distributions, the generalized Pareto distribution with the lower χ

^{2}statistic value is the better fitting model and therefore the one used in the following sections

#### 2.4.3. Optimization Model for Non-Stationarity Arising from Climate Change

## 3. Results and Discussion

#### 3.1. Optimization Results for a Deterministic Characterization of Flood Events

#### 3.2. Optimization Results for a Stochastic Characterization of Flood Events

#### 3.3. Optimization Results for Non-Stationarity Arising from Climate Change

## 4. Conclusions

- The better management option (‘technosociety’ or ‘greensociety’) depends on the initial value of the land services the area provides and the evolution of those services over time. We have plausible cases of high initial land values where the ‘technosociety’ option is better and of low initial land values where the ‘greensociety’ is better.
- The paper shows that a critical parameter for the values under the ‘technosociety’ in determining the values of the flood plain is the safety factor $({\epsilon}_{T})$ (i.e., the proportional raising of defenses when a flooding events occurs). For the deterministic case considered in the original paper, we obtain an optimal safety factor that is significantly higher than in the original paper.
- Whereas [1] consider only myopic behavior toward flood events, our model calculates the value of land services and other costs under uncertainty of extreme event occurrence and intensity. The paper shows the important differences in results when optimizing either under certainty or uncertainty. Our approach allows the economically optimal levee heightening strategy to be calculated under uncertainty.
- We show how climate change affects the optimal response to floods in a socio-hydrology framework. It turns out that the greater the expected impacts from such change the higher the levees but, surprisingly, the optimal safety factors do not always increase.

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A

${\mathit{\epsilon}}_{\mathit{T}}$ | Society |
---|---|

1.1 | Technosociety |

0.0 | Greensociety |

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**Figure 1.**Schematic for river in flood plain and dam: i) The river runs along the side of a square representing the flood plain with area A. ii) Dam geometry with crest b, height H and slope 1:s.

**Figure 5.**Histogram of the objective function at ${\mathsf{\epsilon}}_{\mathrm{T}}$ = 1.1 for the technosociety/high value case with υ = 0.

**Figure 7.**Histogram of the objective function at ${\mathsf{\epsilon}}_{\mathrm{T}}$ = 1.1 for the technosociety/low value case with υ = 0.

**Figure 9.**Objective function for the high value case depending on ${\mathit{\epsilon}}_{\mathit{T}}$ when υ = 0.

**Figure 10.**Objective function for the low value case depending on ${\mathit{\epsilon}}_{\mathit{T}}$ when υ = 0.

Variable | Unit | High Value | Low Value | Basis |
---|---|---|---|---|

Initial Population | No. | 25,000 | 1000 | Assumed |

Area (A) | Hectares | 100,000 | 10,000 | Assumed |

Residential Use Share (μ) | % | 40 | 40 | Common for such use |

Pop. Density (D) | Pop/Ha | 0.25 | 0.10 | Calculated |

Dwelling Density (δ) | Dwelling/Ha. | 0.10 | 0.04 | Calculated |

FAR | Ratio | 0.0025 | 0.0010 | Plausible values |

Value of Land | USD/Ha. | 100 | 100 | Assumed |

Value per Dwelling (B) | USD | 100,000 | 4683 | Plausible range for low and high income countries |

Value of all Land with Dwellings | USD Mn. | 1004.0 | 2.3 | Calculated |

**Table 2.**Parameters of the model as set in [1].

Parameters | Description | Domain | Value |
---|---|---|---|

${\alpha}_{H}$ | Parameter related to relationship between flood water levels to relative damage | Hydrology | 10 m |

${\xi}_{H}$ | Proportion of flood level enhancement due to flood levees | Hydrology | 0.2 |

${\alpha}_{D}$ | Ratio of preparedness/awareness | Demography | 5 |

${\epsilon}_{T}$ | Safety factor for levee heightening | Technology | 1.1 |

${\kappa}_{T}$ | Protection level decay rate | Technology | 2 × 10^{−5} year^{−1} |

${\mu}_{S}$ | Memory loss rate | Society | 0.06 year^{−1} |

Events (k) | Probability (%) |
---|---|

0 | 83.946 |

1 | 14.690 |

2 | 1.285 |

3 | 0.075 |

>3 | 0.003 |

Total | 100.00 |

Parameter Value | 95% Confidence Interval | |
---|---|---|

Shape ($k)$ | −0.466 | −0.761 to −0.172 |

Scale ($\upsilon )$ | 9.812 | 6.476 to 14.868 |

Technosociety | Greensociety | ||
---|---|---|---|

Case of υ = 0 | Case of υ = 1 | ||

High Value | 1719 | 1927 | 5158 |

Low Value | 61 | 127 | 31 |

**Table 6.**Expected net present value (NPV) (million) for the different management regimes in the stochastic case.

Generalized Pareto | High Land Value Scenario | Low Land Value Scenario | ||
---|---|---|---|---|

Technosociety | Greensociety | Technosociety | Greensociety | |

υ = 0 | 993 | 181 | ||

υ = 1 | 1290 | 3947 | 294 | 25 |

Generalized Pareto | High Land Value | Low Land Value |
---|---|---|

υ = 0 | 1.53 | 0.00 |

υ = 1 | 1.64 | 0.00 |

**Table 8.**Impact of climate change for the high value case. Upper section: multipliers for frequency and intensity changes for the 100 and 200-year time horizon respectively. Lower section: optimal levee heightening strategy (for two cases: υ = 0: only extension required after event, υ = 1: rebuilding and extension required after event).

${\mathit{\alpha}}_{\mathit{\lambda}}={\mathit{\alpha}}_{\mathit{\sigma}}$ | 0.000 | 0.001 | 0.002 | 0.003 | 0.004 | 0.005 |
---|---|---|---|---|---|---|

${e}^{{\alpha}_{\lambda}t}$ | ||||||

100 years | 1.00 | 1.11 | 1.22 | 1.35 | 1.49 | 1.65 |

200 years | 1.00 | 1.22 | 1.49 | 1.82 | 2.23 | 2.72 |

Safety Factor for levee heightening ${\mathit{\epsilon}}_{\mathit{T}}$ | ||||||

υ = 0 | 1.53 | 1.50 | 1.45 | 1.35 | 1.27 | 1.17 |

υ = 1 | 1.64 | 1.67 | 1.70 | 1.71 | 1.65 | 1.64 |

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**MDPI and ACS Style**

Abadie, L.M.; Markandya, A.; Neumann, M.B.
Accounting for Economic Factors in Socio-Hydrology: Optimization under Uncertainty and Climate Change. *Water* **2019**, *11*, 2073.
https://doi.org/10.3390/w11102073

**AMA Style**

Abadie LM, Markandya A, Neumann MB.
Accounting for Economic Factors in Socio-Hydrology: Optimization under Uncertainty and Climate Change. *Water*. 2019; 11(10):2073.
https://doi.org/10.3390/w11102073

**Chicago/Turabian Style**

Abadie, Luis M., Anil Markandya, and Marc B. Neumann.
2019. "Accounting for Economic Factors in Socio-Hydrology: Optimization under Uncertainty and Climate Change" *Water* 11, no. 10: 2073.
https://doi.org/10.3390/w11102073