# Discharge Estimation Using Tsallis and Shannon Entropy Theory in Natural Channels

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction:

## 2. Method

#### 2.1. Tsallis Entropy-Based Non-Contact Discharge Estimation.

#### 2.2. Shannon Entropy-Based Non-Contact Discharge Estimation

#### 2.3. Cross-Sectional Mean Flow Velocity and Discharge Estimation

- Identify the location of the y-axis (vertical where the maxum velocity is recorded) through historical records.
- Measure multiple point velocities along this vertical, including the surface water velocity on this identified y-axis.
- Tabulate the pairs of cross-sectional mean and maximum velocities for different flood events.
- Estimate Tsallis’ G value using Equation (2) and Shannon’s M value using Equation (5).
- Estimate the maximum velocity from the surface flow velocity measurements using Equation (3) for Tsallis’ theory and Equation (6) for Shannon’s theory, and compare these with the observed maximum velocity of that event.
- Estimate the cross-sectional mean velocity using Equation (2) for Tsallis’ theory and Equation (5) for Shannon’s theory.
- Determine the cross-sectional flow area corresponding to the recorded water surface level.
- Estimate the discharge using the estimated cross-sectional mean velocity obtained in step (8) and the cross-sectional area using Q = AV.

## 3. Study Area

## 4. Results and Discussion

#### 4.1. Non-Contact Discharge Assessment Using Tsallis Entropy

^{2}= 0.98 for both the gauge stations) with the observed values. Thefore, the maximum flow velocities estimated by Equation (3) for all the events of both gauge stations were used to assess the cross-sectional mean flow velocity using the Tsallis entropy-based linear relationship in Equation (2).

#### 4.2. Non-Contact Discharge Assessment Using Shannon Entropy

_{max}using Equation (3). The vertical velocity profile at any location of the flow section of a flow event can be estimated using Equation (1) by using the estimated value of ${U}_{max}$ instead of the observed value of ${U}_{max}$, such as that depicted by Equation (3) by leveraging the observed surface velocity and the dip. The estimated entropy constants were G = 4.02 for the Pontelagsucro station of the Po River and 3.52 for the Ponte Nuovo station of the Tiber River.

## 5. Conclusions

- -
- Non-contact monitoring techniques based on the use of surface flow velocity measurements at river gauge stations by employing surface velocity radar (SVR) and large scale particle image velocimetry (LSPIV) are a valuable alternative approach to the traditional discharge estimation methods. These approaches eliminate the drawbacks of using the traditional methods for monitoring high flow conditions, which prove to be inefficient and subject to accuracy problems, as well as pose safety problems for the operators during high flow conditions. By sampling the maximum surface flow velocity at the y-axis and applying entropy theory, one can accurately estimate the river discharge, which makes the non-contact technology highly appealing for river monitoring. It is worth noting that the uncertainty analysis of entropy-based methods using velocity measurements provide a variation on estimates that, as shown by Alvisi [25], does not exceed 10% on average for high flows. It is also worth noting that recent studies showed that entropy-based models can be applied for any flow conditions using both ground measurements [26,27] and satellite observations [28], and this is of considerable interest for new satellite missions, such as SWOT (Surface Water and Ocean Topography- NASA) and Sentinel (European Space Agency).
- -
- Tsallis entropy theory provided similar performance to the one based on Shannon entropy theory when estimating the cross-sectional mean flow velocity and the velocity profile distribution at the y-axis.
- -
- It was shown that the measure of the surface flow velocity along the y-axis allowed us to efficiently estimate the maximum velocity for which the mean flow velocity can be accurately assessed, regardless of the type of entropy approach applied. The proposed method can be easily replicable for any river site and this finding provides a considerable benefit when using the non-contact techniques for monitoring discharge during any flow conditions, and in particular, during high flow. This is linked to the fact that the key variable U
_{max}can be easily monitored during high flow and the entropy parameter characterizing the slope of the linear entropy relationship does not depend on the hydraulic gradient, which influences the dynamics of flooding. Indeed, as shown by Moramarco and Singh [29], the entropy parameter is linked to the ratio between the geometric and hydraulic characteristics of a river site, which remains constant during a flood. - -
- Finally, the analysis of velocity profiles at the y-axis showed that by using the observed dip values, both Tsallis and Shannon entropy theories could be used to study the secondary currents when dip phenomena occur. This aspect will be investigated in detail in terms of a two-dimensional velocity distribution when secondary currents occur by using the velocity dataset referring to the gauged river stations with different geometric and hydraulic characteristics and including Indian rivers.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**The relation between the cross-sectional mean flow velocity U

_{m}and the maximum velocity U

_{max}for (

**a**) Pontelagoscuro station and (

**b**) Ponte Nuovo station.

**Figure 2.**Comparison of pertinent estimated variables with the corresponding observed ones using Tsallis entropy for Ponteslagscuro station (

**a**,

**c**,

**e**) and Ponte Nuovo (

**b**,

**d**,

**f**).

**Figure 3.**Comparison of the pertinent estimated variables with the corresponding observed ones using Shannon entropy for Ponteslagscuro station (

**a**,

**c**,

**e**) and Ponte Nuovo (

**b**,

**d**,

**f**).

**Figure 4.**The relative percentage error in estimating the cross-sectional mean velocity using Tsallis and Shannon entropies: (

**a**) Pontelagscuro and (

**b**) Ponte Nuovo.

**Figure 5.**Typical velocity profiles estimated using Tsallis and Shannon entropies, and a comparison with the observed velocity points (depth (y-axis) represents the vertical where ${U}_{max}$ occurs) for the gauging stations at (

**a**) Pontelagscuro and (

**b**) Ponte Nuovo.

**Table 1.**Flow data set details: ${N}_{e}$—number of events considered, ${N}_{v}$—total number of verticals (from the ${N}_{e}$ events), Q—measured discharge, D—dep, A—area.

River | Station | ${\mathit{N}}_{\mathit{e}}$ | ${\mathit{N}}_{\mathit{v}}$ | Q (m^{3}/s) | D (m) | A (m^{2}) | Period |
---|---|---|---|---|---|---|---|

Po | Pontelagoscuro | 48 | 595 | 316–5026 | 5.41–15.46 | 913–2833 | 1984–1992 |

Tiber | Ponte Nuovo | 22 | 186 | 2.65–506 | 0.91–6.07 | 25.48–278.16 | 1985–2000 |

**Table 2.**Estimated percentage of errors in estimating the mean flow velocity using Tsallis and Shannon entropy and using only the observed surface velocities.

Metrics | Pontelagscuro | Ponte Nuovo | ||
---|---|---|---|---|

Tsallis | Shannon | Tsallis | Shannon | |

Mean (%) | 5.59 | 5.59 | 7.55 | 7.58 |

Standard Deviation (%) | 6.95 | 6.95 | 8.79 | 8.49 |

NSE * | 0.99 | 0.99 | 0.99 | 0.99 |

${R}^{2}$ | 0.98 | 0.98 | 0.99 | 0.99 |

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**MDPI and ACS Style**

Vyas, J.K.; Perumal, M.; Moramarco, T. Discharge Estimation Using Tsallis and Shannon Entropy Theory in Natural Channels. *Water* **2020**, *12*, 1786.
https://doi.org/10.3390/w12061786

**AMA Style**

Vyas JK, Perumal M, Moramarco T. Discharge Estimation Using Tsallis and Shannon Entropy Theory in Natural Channels. *Water*. 2020; 12(6):1786.
https://doi.org/10.3390/w12061786

**Chicago/Turabian Style**

Vyas, Jitendra Kumar, Muthiah Perumal, and Tommaso Moramarco. 2020. "Discharge Estimation Using Tsallis and Shannon Entropy Theory in Natural Channels" *Water* 12, no. 6: 1786.
https://doi.org/10.3390/w12061786