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

^{1}

^{2}

^{*}

## 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

## References

- Herschy, R.W. Streamflow Measurement; Elsevier: London, UK, 1985. [Google Scholar]
- Simpson, M.R.; Oltman, R.N. Discharge Measurement System Using a Acoustic Doppler Current Profiler with Applications to Large Rivers and Estuaries. In U.S. Geological Survey Water-Supply Paper; 2395; U.S. Geological Survey: Reston, VA, USA, 1993; 32p. [Google Scholar]
- Fujita, I.; Muste, M.; Kruger, A. Large-scale particle image velocimetry for flow analysis in hydraulic applications. J. Hydraul. Res.
**1998**, 36, 397–414. [Google Scholar] [CrossRef] - Tauro, F.; Porfiri, M.; Grimaldi, S. Orienting the camera and firing lasers to enhance large scale particle image velocimetry for streamflow monitoring. Water Resour. Res.
**2014**, 50, 7470–7483. [Google Scholar] [CrossRef] - Costa, J.E.; Spicer, K.R.; Cheng, R.T.; Haeni, F.P.; Melcher, N.B.; Thurman, E.M. Measuring stream discharge by non-contact methods: A proof-of-concept experiment. Geophys. Res. Lett.
**2000**, 27, 553–556. [Google Scholar] [CrossRef] - Welber, M.; Le Coz, J.; Laronne, J.B.; Zolezzi, G.; Zamler, D.; Dramais, G.; Hauet, A.; Salvaro, M. Field assessment of noncontact stream gauging using portable surface velocity radars (SVR). Water Resour. Res.
**2016**, 52, 1108–1126. [Google Scholar] [CrossRef] [Green Version] - Muste, M.; Fujita, I.; Hauet, A. Large-scale particle image velocimetry for measurements in riverine environments. Water Resour. Res.
**2008**, 44, W00D19. [Google Scholar] [CrossRef] [Green Version] - Chiu, C.-L. Velocity distribution in open channel flow. J. Hydraul. Eng.
**1989**, 115, 576–594. [Google Scholar] [CrossRef] - Chiu, C.-L.; Hsu, S.H.; Tung, N.-C. Efficient methods of discharge measurements in rivers and streams based on the probability concept. Hydrol. Process.
**2005**, 19, 3935–3946. [Google Scholar] [CrossRef] - Chiu, C.-L.; Tung, N.-C. Maximum velocity and regularities in open-channel flow. J. Hydraul. Eng.
**2002**, 128, 390–398. [Google Scholar] [CrossRef] - Shannon, C.E. A mathematical theory of communication. Bell Syst. Tech. J.
**1948**, 27, 623–656. [Google Scholar] [CrossRef] - Jaynes, E.T. Information theory and statistical mechanics I. Phys. Rev.
**1957**, 106, 620–630. [Google Scholar] [CrossRef] - Xia, R. Relation between mean and maximum velocities in a natural river. J. Hydraul. Eng. ASCE
**1997**, 123, 123,720–723. [Google Scholar] [CrossRef] - Moramarco, T.; Saltalippi, C.; Singh, V.P. Estimation of mean velocity in natural channel based on Chiu’s velocity distribution equation. J. Hydrol. Eng.
**2004**, 9, 42–50. [Google Scholar] [CrossRef] - Mirauda, D.; Pannone, M.; De Vincenzo, A. An entropic model for the assessment of streamwise velocity dip in wide open channels. Entropy
**2018**, 20, 69. [Google Scholar] [CrossRef] [Green Version] - Stearns, F.P. On the current meter, together with a reason why the maximum velocity of water flowing in open channel is below the surface. Trans. Am. Soc. Civ. Eng.
**1883**, 3, 20–32. [Google Scholar] - Termini, D.; Moramarco, T. Dip phenomenon in high-curved turbulent flows and application of entropy theory. Water
**2018**, 10, 306. [Google Scholar] [CrossRef] [Green Version] - Fulton, J.; Ostrowski, J. Measuring real-time streamflow using emerging technologies: Radar, hydroacoustics, and the probability concepts. J. Hydrol.
**2008**, 357, 1–10. [Google Scholar] [CrossRef] - Moramarco, T.; Barbetta, S.; Tarpanelli, A. From Surface Flow Velocity Measurements to Discharge Assessment by the Entropy Theory. Water
**2017**, 9, 120. [Google Scholar] [CrossRef] [Green Version] - Tsallis, C. Possible generalization of Boltzmann-Gibbs statistics. J. Stat. Phys.
**1988**, 52, 479–487. [Google Scholar] [CrossRef] - Singh, V.P. Introduction to Tsallis Entropy Theory in Water Engineering; CRC Press/Taylor and Francis: Boca Raton, FL, USA, 2016. [Google Scholar]
- Cui, H.; Singh, V.P. One-dimensional velocity distribution in open channels using Tsallis entropy. J. Hydrol. Eng.
**2014**, 19, 290–298. [Google Scholar] [CrossRef] - Cui, H.; Singh, V.P. Two-dimensional velocity distribution in open channels using the Tsallis entropy. J. Hydrol. Eng.
**2013**, 18, 331–339. [Google Scholar] [CrossRef] - Singh, V.P.; Luo, H. Entropy theory for distribution of one-dimensional velocity in open channels. J. Hydrol. Eng.
**2011**, 16, 725–735. [Google Scholar] [CrossRef] - Alvisi, S.; Barbetta, S.; Franchini, M.; Melone, F.; Moramarco, T. Comparing grey formulations of the velocity-area method and entropy method for discharge estimation with uncertainty. J. Hydroinf.
**2014**, 16, 797–811. [Google Scholar] [CrossRef] - Alimenti, F.; Bonafoni, S.; Gallo, E.; Palazzi, V.; Vincenti Gatti, R.; Mezzanotte, P.; Roselli, L.; Zito, D.; Barbetta, S.; Corradini, C.; et al. Non-Contact Measurement of River Surface Velocity and Discharge Estimation with a Low-Cost Doppler Radar Sensor. IEEE Trans. Geosci. Remote Sens.
**2020**, 1–13. [Google Scholar] [CrossRef] - Fulton, J.W.; Mason, C.; Eggleston, J.; Nicotra, M.; Chiu, C.L.; Henneberg, M.; Best, H.; Cederberg, J.; Holnbeck, S.; Lotspeich, R.; et al. Remote Sensing of Surface Velocity and River Discharge Using Radars and the Probability Concept at 10 USGS Streamgages. Remote Sens.
**2020**, 12, 1296. [Google Scholar] [CrossRef] [Green Version] - Moramarco, T.; Barbetta, S.; Bjerklie, D.M.; Fulton, J.W.; Tarpanelli, A. River Bathymetry Estimate and Discharge Assessment from Remote Sensing. Water Resour. Res.
**2019**, 55, 6692–6711. [Google Scholar] [CrossRef] - Moramarco, T.; Singh, V.P. Formulation of the entropy parameter based on hydraulic and geometric characteristics of river cross sections. J. Hydrol. Eng.
**2010**, 15, 852–858. [Google Scholar] [CrossRef]

**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 |

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**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