# Towards Informed Water Resources Planning and Management

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

#### 1.1. Management under Stationary Conditions

#### 1.2. Management under Non-Stationary Conditions

## 2. Basic Concepts

#### 2.1. Decisions under Uncertainty

#### 2.2. The Mathematical Representation of Knowledge

#### 2.3. Deterministic versus Probabilistic Forecasts

## 3. Probabilistic Predictions

#### 3.1. Short Term Probabilistic Forecasts

#### 3.2. Medium Term Probabilistic Forecasts

#### 3.3. Long-Term Probabilistic Climate Projections

## 4. Attracting the Interest of Decision Makers

- inappropriate definition of predictive uncertainty;
- misunderstanding of the meaning of predictive uncertainty and of its role in decision-making;
- unclear role and use of epistemic uncertainty (such as parameter uncertainty), which is often confused with predictive uncertainty;
- incorrect use of ensembles in the assessment of predictive uncertainty;
- misunderstanding of the mechanism and of the advantages for using predictive uncertainty in the Bayesian decision-making process.

^{3}while environmental, social, and economic losses rapidly increase when the volume overtops the upper operational limit of 600 Mm

^{3}. The utility function in Figure 10, expressed in monetary terms (€), is generally set up in cooperation with the decision maker to reflect his or her subjective views and risk propensity.

^{3}and standard deviation 80 Mm

^{3}. As can be visually noticed from Figure 10, the integral of the product between the predictive density, represented by the thin, grey, bell-shaped curve, and the utility function gives a large expected loss of about 1 million $\u20ac$. By releasing water from the reservoir, although there would be loss of precious water volume, expected losses could be dramatically reduced. Releasing water is equivalent to shifting downwards the predictive density by the released quantity. The situation of Figure 10 after releasing 350 Mm

^{3}shows that the updated predictive density, represented by the black solid line bell shaped curve, is shifted downwards and the expected losses ($~30\text{}\u20ac$) become practically null.

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Losses by the Aswan reservoir accrued over one year, depending on the management strategy (reprinted from [3]). The thicker solid line (No Forecast) represents the results of management based on the historical monthly average Nile inflows; the dotted line represents the results of management based on the hypothetical knowledge of future inputs (Perfect Forecast); the thinner continuous line represents the results of a management based on imperfect knowledge of future inputs provided by a simple AR (1) (Uncertain Forecast) forecasting model, while accounting at each time step for the predictive probability distribution in estimating the volume to be released. Note that the information produced, even by a very simple model such as the AR(1), leads to significantly loss reduction (over 65%) by approaching the lower limit of losses obtainable with a perfect forecast, namely the retrospective perfect knowledge of future reservoir inflows [3].

**Figure 2.**The monthly non-stationary water levels of the Nile at Roda’s Nilometer (in cm) during the period 1871–1971 considered as a weakly stationary stochastic process. We identified two 20-year sub-periods. Within each period, the stochastic process could be thought of as weakly stationary and ergodic, while the process over the 100-year period was clearly non-stationary.

**Figure 3.**Comparison between the expected losses estimated (

**a**) deterministically, according to the expected value of the volume forecast, and (

**b**) probabilistically, by integrating the product between losses and their predictive probability of occurrence.

**Figure 4.**In pane (

**a**), the representation of uncertainty, or rather our knowledge, in the form of probability density allows its informative use in decision-making schemes. In (

**b**), where the expected value surrounded by the limits of $\pm \text{}1$ standard errors are plotted, only the information on the dispersion of the observations is provided, which gives a measure of the uncertainty, but does not allow us to use it in the decision-making phase.

**Figure 5.**The mathematical representation of knowledge: (

**a**) perfect ignorance; (

**b**) incomplete knowledge; (

**c**) perfect knowledge.

**Figure 6.**Uncertainty in the evolution of a chaotic physical system (Real World) and its modeled representation (Virtual World of Models). In green the evolution from the current state to the actual future value (solid line) and to possible alternative future states (green zone). In red, the evolution of the current state to the expected value of the future state (solid line) while the evolution of the model’s predictions from the present to the future (orange zones), is not necessarily coinciding with the real future states (redrawn from [4]).

**Figure 7.**Baseline window and two predictive windows for the prost-processed CMIP5 RCP 4.5 temperature projections, river Po basin, Northern Italy. Seasonal mean observed temperature (blue), unprocessed ensemble output (pink), ensemble mean (light red) and post-processed predictive mean (flash red), spring (MAM). The grey-shaded areas indicate the 50% and 95% credible intervals (redrawn from [4]).

**Figure 8.**Baseline window and two predictive windows for the prost-processed CMIP5 RCP 8.5 precipitation projections, river Po basin, Northern Italy. The precipitation is given as average over a 1° × 1° reference cell centered in 10.08° E and 45.03° N. Seasonal mean observed precipitation (blue), unprocessed ensemble output (brown), ensemble mean (light red) and post-processed predictive mean (flash red), spring (MAM). The grey-shaded areas indicate the 50% and 95% credible intervals.

**Figure 9.**Probability density functions of observed and projected precipitation means (in mm) for a 1° × 1° reference cell centered in 8.08° E and 45.32° N, river Po basin, Northern Italy. Observations (blue dashed), post-processed control period (blue), prognostic window 2035–2065 (red) and 2070–2100 (green) projections, for the summer quarter June, July, August (JJA).

**Figure 10.**A simplified example of how a probabilistic forecast (thin black bell-shaped predictive density) can be used to derive appropriate decisions for reservoir releases. For a given probabilistic prediction (grey bell-shaped solid line), the expected loss, namely, the integral of the product of the density times the utility function of Equation (1) (solid thick black curve), is rather large. By releasing water, thus, reducing the volume in the reservoir, the probabilistic forecast of the cumulated volume is shifted downwards and is represented by the solid line bell shaped curve. As can be noticed, the expected utility value now becomes negligible. The appropriate amount to be released will then be found by comparing the expected utility function value to the cost of lack of future water availability (redrawn from [2]).

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Reggiani, P.; Talbi, A.; Todini, E. Towards Informed Water Resources Planning and Management. *Hydrology* **2022**, *9*, 136.
https://doi.org/10.3390/hydrology9080136

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Reggiani P, Talbi A, Todini E. Towards Informed Water Resources Planning and Management. *Hydrology*. 2022; 9(8):136.
https://doi.org/10.3390/hydrology9080136

**Chicago/Turabian Style**

Reggiani, Paolo, Amal Talbi, and Ezio Todini. 2022. "Towards Informed Water Resources Planning and Management" *Hydrology* 9, no. 8: 136.
https://doi.org/10.3390/hydrology9080136