# From Surface Flow Velocity Measurements to Discharge Assessment by the Entropy Theory

^{*}

## Abstract

**:**

_{surf}, is proposed. The approach, based on the entropy theory, involves the actual location of maximum flow velocity, u

_{max}, which may occur below the water surface (dip phenomena), affecting the shape of velocity profile. The method identifies the two-dimensional velocity distribution in the cross-sectional flow area, just sampling u

_{surf}and applying an iterative procedure to estimate both the dip and u

_{max}. Five gage sites, for which a large velocity dataset is available, are used as a case study. Results show that the method is accurate in simulating the depth-averaged velocities along the verticals and the mean flow velocity with an error, on average, lower than 12% and 6%, respectively. The comparison with the velocity index method for the estimation of the mean flow velocity using the measured u

_{surf}, demonstrates that the method proposed here is more accurate mainly for rivers with a lower aspect ratio where secondary currents are expected. Moreover, the dip assessment is found more representative of the actual location of maximum flow velocity with respect to the one estimated by a different entropy approach. In terms of discharge, the errors do not exceed 3% for high floods, showing the good potentiality of the method to be used for the monitoring of these events.

## 1. Introduction

_{surf}, measured at a location (vertical) into depth-averaged velocity, u

_{vert}, considering the different hydraulic and geometric characteristics of river sites which affect the velocity profile shapes [11]. In the case of maximum flow velocity occurs on the water surface, the two-parameters power law velocity distribution developed by Dingman [12] could be applied to this end.

_{surf}, by a velocity index, k = u

_{vert}/u

_{surf}, equal to 0.85 (e.g., [7] for SVR and [9] for LSPIV). However, the possible occurrence of the dip phenomenon could make the assumption ‘weak’ for create flows where a monotonous velocity distribution does not take place. Consequently, the velocity index value might be not representative of the two-dimensional spatial velocity distribution anymore, leading to failure of the assessment of discharge.

_{max}occurs below the water surface, is the main issue. The dip may significantly affect the estimation of the depth-averaged velocity from which, through the velocity area method, the mean flow can be assessed and, hence, the discharge.

## 2. Method

_{max}, occurs, holds for all verticals in flow area, yielding:

_{maxv}(x

_{i}) is the maximum velocity along the ith vertical, x

_{i}is the position of the i

^{th}sampled vertical from the left bank, M is the entropic parameter, h(x

_{i}) is the dip, i.e., the depth of u

_{maxv}(x

_{i}) below the water surface, D(x

_{i}) the vertical depth, y is the distance of the velocity point from the bed, and N

_{v}is the number of verticals sampled across the river section. The entropic parameter M, which is a characteristic of the section [20], can be easily estimated through the pairs (u

_{m}, u

_{max}) of the available velocity dataset at a gauge site by using the linear entropic relation [14]:

_{m}is the mean flow velocity and u

_{max}is the maximum flow velocity in flow area.

_{maxv}occurs below the water surface, it could be estimated for each vertical as a function of surface velocity, ${u}_{surf}$, such as proposed by [22]:

#### 2.1. Dip Estimate

_{m}and u

_{max}can be estimated using the velocity-area method [23], and, hence, $\Phi \left(M\right)$ can be computed by Equation (2) [19]. However, the uncertainty due to the presence of secondary currents lead us to modify Equation (4) in:

_{obs}) can be easily estimated by the observed pairs (u

_{m}, u

_{max}). For ungauged sites, the M parameter can be even estimated by expressing its value in terms of hydraulic and geometric characteristics such as proposed by [21] and/or following [20], who identified a simple way to estimate M from the maximum surface velocity typically located near the middle of the channel [24].

_{i}) that can be used to estimate $\delta \left({x}_{i}\right).$ This implies the interest in comparing the two approaches. However, it is worth noting that our target is not to estimate the effectiveness of these relationships in the dip assessment but rather to analyze whether, if given a distribution of dip in flow area, the proposed method is able to provide an accurate estimate of velocity, and considering that Equation (4) was tested with laboratory and field data, it is sufficient for our purpose. However, as stressed above, a comparison with Equation (7) is addressed as well and shown afterwards.

#### 2.2. The Velocity Index

_{vert}, estimated by velocity points along verticals sampled by using, e.g., current meter across the cross-sectional flow area. In this case, SVR is used for discharge monitoring, the measure of velocity is limited to the surface, and there is the need to turn the surface velocity u

_{surf}, into u

_{vert}. For that, an approach widely applied is to consider the velocity index, k, defined as:

_{vert}/u

_{surf}

_{vert}, in the case of secondary currents taking place.

## 3. Case Study

^{2}), Ponte Felcino (1970 km

^{2}), and Ponte Nuovo (4100 km

^{2}) along the Tiber River and Rosciano (1950 km

^{2}) on the Chiascio River. One section is on the Po River, i.e., Pontelagoscuro (70,000 km

^{2}).

## 4. Results and Discussion

#### 4.1. The Velocity Index Analysis

_{vert}, is estimated starting from the measured surface velocity, u

_{surf}, i.e., u

_{vert}(x

_{i},D

_{i}) = k u

_{surf}(x

_{i},D

_{i}), and compared with the depth-averaged velocity assessed by the sampled velocity points along each vertical.

_{mean}and k

_{variance}as a function of (W/D) could be of support in understanding the variability of the velocity index and, hence, to evaluate how much the depth-averaged velocity can vary at each vertical sampled in correspondence to the surface velocity measurement. However, more gage sites are necessary to identify a robust relationship that can be leveraged in addressing the uncertainty in the k estimation.

#### 4.2. Comparison of Depth-Averaged Velocities

#### 4.3. Comparison of Mean Flow Velocity and Discharge

_{maxv}values observed and computed by the entropy method for each vertical is found good with the coefficient of determination, R

^{2}, varying from 0.84 for the narrower sites (Santa Lucia and Ponte Felcino) to 0.94 for the wider ones. Moreover, by way of example, Figure 8 shows for the highest flood in the dataset of the Ponte Nuovo gauged site, the comparison between the velocity profile at four different verticals computed by the entropy method, and the velocity points sampled by current meter. It is worth noting that the velocity profiles are obtained just by sampling u

_{surf}, and no information is used in terms of dip. It is noticeable how the velocity profile shape computed by Equation (1) is able to embed the observed points and, particularly, u

_{maxv}. Similar performances are obtained for the velocity dataset of the other investigated gage sites.

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 1.**Schematic representation of dip estimation procedure using measured surface velocity, u

_{surf}(for symbols see text).

**Figure 2.**Box-plot of the values of the velocity index, k, as a function of the aspect ratios, W/D, for the five gage sites based on the velocity dataset. The line corresponding to the value of 0.85 is also shown along with the name of the gage sites.

**Figure 3.**Mean and variance of the velocity index, k, for the selected gage sites as a function of the aspect ratio (W/D).

**Figure 4.**Box-plot of absolute percentage errors ($\u03f5$) in estimating the depth-averaged velocity using the entropy approach (Entr) and the velocity index (Vel Ind).

**Figure 5.**Absolute percentage error in estimating the depth-averaged velocity using the dip assessment by the iterative procedure proposed here (Entr Iter) and Chiu’s formula (Entr Chiu) for (

**a**) the Rosciano and (

**b**) the Pontelagoscuro gage sites.

**Figure 6.**Box-plot of the absolute percentage error ($\u03f5$) in estimating the cross-sectional mean flow velocity using the Entropy approach (Entr) and the velocity index (Vel Ind).

**Figure 7.**Comparison between observed discharge (Obs) and discharge computed by the Entropy approach (Entr) and the velocity index (Vel Ind) for (

**a**) Santa Lucia; (

**b**) Ponte Nuovo; and (

**c**) Pontelagoscuro.

**Figure 8.**Velocity profiles estimated by entropy method (Equations (1) and (5)) plotted against velocity points sampled by current meter along four verticals during the highest flood of the velocity dataset at the Ponte Nuovo gage site. x is the distance from the left bank, while the y-axis represents the vertical where u

_{max}occurs.

**Table 1.**Flow dataset characteristics: N

_{m}= number of velocity measurements considered, N

_{v}= total number of verticals (from the N

_{m}measurements), $\Phi \left({M}_{obs}\right)$ = observed entropy parameter, Q = measured discharge, D = average flow depth, W = average channel width. The period of sampling is also shown.

River | Site | N_{m} | N_{v} | $\mathit{\Phi}\left({\mathit{M}}_{\mathit{o}\mathit{b}\mathit{s}}\right)$ | Q (m^{3}/s) | D (m) | W (m) | W/D | Period |
---|---|---|---|---|---|---|---|---|---|

Tiber | Santa Lucia | 16 | 93 | 0.66 | 30–185 | 2.54 | 22 | 8.79 | 1987–2008 |

Tiber | Ponte Felcino | 8 | 78 | 0.60 | 28–411 | 3.22 | 38 | 11.66 | 1990–2003 |

Tiber | Ponte Nuovo | 12 | 77 | 0.66 | 10–540 | 3.98 | 50 | 12.30 | 1986–2005 |

Chiascio | Rosciano | 8 | 67 | 0.60 | 20–378 | 2.74 | 35 | 12.90 | 1982–2002 |

Po | Pontelagoscuro | 8 | 89 | 0.68 | 380–4000 | 6.32 | 270 | 43.63 | 1987–1992 |

**Table 2.**Mean and standard deviation of the absolute percentage error in estimating the depth-averaged velocity using the dip assessment by the iterative procedure proposed here (Entr Iter) and Equation (7) (Entr Chiu) for the investigated gage sites.

Gage Site | Entr Iter | Entr Chiu | ||
---|---|---|---|---|

Mean (%) | Standard Deviation (%) | Mean (%) | Standard Deviation (%) | |

Santa Lucia | 9.8 | 8.8 | 12.5 | 9.1 |

Ponte Felcino | 11 | 9.6 | 19 | 11.1 |

Ponte Nuovo | 8.6 | 7.9 | 15.4 | 10.2 |

Rosciano | 8.9 | 8.1 | 15.1 | 11.4 |

Pontelagoscuro | 7.2 | 6.9 | 8.9 | 7.8 |

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**MDPI and ACS Style**

Moramarco, T.; Barbetta, S.; Tarpanelli, A.
From Surface Flow Velocity Measurements to Discharge Assessment by the Entropy Theory. *Water* **2017**, *9*, 120.
https://doi.org/10.3390/w9020120

**AMA Style**

Moramarco T, Barbetta S, Tarpanelli A.
From Surface Flow Velocity Measurements to Discharge Assessment by the Entropy Theory. *Water*. 2017; 9(2):120.
https://doi.org/10.3390/w9020120

**Chicago/Turabian Style**

Moramarco, Tommaso, Silvia Barbetta, and Angelica Tarpanelli.
2017. "From Surface Flow Velocity Measurements to Discharge Assessment by the Entropy Theory" *Water* 9, no. 2: 120.
https://doi.org/10.3390/w9020120