# On the Uncertainty of the Image Velocimetry Method Parameters

^{*}

## Abstract

**:**

## 1. Introduction

_{i}v

_{i}, i = 1 … n

_{i}is the area of the ith segment of the measured cross-section, v

_{i}is the average velocity of the midsection of the ith segment of the cross-section, and n is the number of segments (typically, no fewer than 10 and at least 25 for wide rivers [3]). Conventionally, the velocity is sampled manually with a current meter either at 60% of depth below the free surface (for shallow depths 0.76 m), or at 20% and 80% or at 20%, 60%, and 80% of depth, weighted appropriately to estimate the mean velocity v

_{i}over the depth ([3] (5.4.5)). It is evident that such campaigns are tedious, costly and at high flows, particularly under flooding conditions, prohibitively dangerous and thus practically impossible.

## 2. Materials and Methods

_{i}= Φ

_{TR}(i/(n + 1); a, b, c), i = 1 … n

_{i}is the ith value from the total n values produced by the random number generator, Φ

_{TR}is the inverse cumulative of the triangular distribution function, and, a, b, and c (with a < c < b) are the parameters of the triangular distribution.

^{3}sets of parameter values were formed employing the Cartesian product on the three sets of which each one has n generated parameter values. Free-LSPIV was run for each one of these n

^{3}sets. This produced n

^{3}velocity profiles along the cross-section from which the lower limit of the 90% confidence interval, the mean value, and the upper limit were derived.

^{3}runs) and n = 5 (5

^{3}runs). The results of these simulations were compared to evaluate the sufficiency of a limited number of simulations to successfully capture the influence of uncertainty of parameters on the results.

## 3. Results

#### 3.1. Reliability of Monte Carlo Simulations with Low Number of Iterations

_{i}} × {x2

_{i}} × {x3

_{i}}, i = 1 … n

_{i}are the values of the contrast adjustment obtained by Equation (2) for (a, b, c) = (0.3, 0.9, 0.7), x2

_{i}are the values of the minimum acceptable cross-correlation obtained by Equation (2) for (a, b, c) = (0.3, 0.9, 0.6), and x3

_{i}are the values the IA size obtained by Equation (2) for (a, b, c) = (0.5 L, 2.0 L, 1.0 L).

^{3}and 20

^{3}runs was evaluated employing the Nash Sutcliffe Efficiency (NSE) coefficient. The results are shown in Table 2.

#### 3.2. The Impact of the Mean IA Size on the Confidence Intervals

#### 3.3. Mean Velocity of Monte Carlo Simulations as Estimator of Cross-Section Velocity Profile

_{i}(the average velocity of the midsection of the ith segment) from the corresponding surface velocities [11]. The surface velocities of the single run LSPIV were obtained from Pearce et al. [20]. The reference discharge was estimated from the surface velocities of the ADCP measurements. The reason for doing so (instead of directly using the discharge estimated by the ADCP method) was to exclude the influence of the error of the estimation of v

_{i}(from the surface velocity) from the comparisons.

## 4. Discussion

^{3}and 5

^{3}runs displayed in Figure 2, Figure 3 and Figure 4 and in Table 2 suggest that the use of Equation (2) for generating random numbers helped reliably estimate the statistical profiles even with a limited number of simulations. This is very important taking into account the considerable CPU time required by image velocimetry methods.

- Identify the parameters of the method and their plausible maximum/minimum and most likely values.
- Employ Equation (2) to generate sets of random values for the parameters. A small number of values (e.g., n = 5) for each parameter is sufficient.
- Start with a low value for mean IA size (e.g., L × L = 16 × 16 pixels) and produce a small number (e.g., n = 5) of IA sizes employing Equation (2) with (a, b, c) = (½L, 2L, L).
- Run Monte Carlo simulations and obtain the mean and the lower and upper limits.
- Increase L by a factor between 1 and 2. The confidence interval width will initially decrease with L. When the confidence interval stops decreasing with L, go to the next step, otherwise repeat steps 3 and 4.
- Identify the Monte Carlo simulation with the lowest confidence interval width and use the mean velocity profile of this simulation as the best estimation of the surface velocities.

## 5. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A

**Figure A1.**The effects of the Free-LSPIV pre-processing on a video frame: original greyscale (

**a**), background removal (

**b**), contrast adjustment with parameter 0.8 (

**c**), contrast adjustment with parameter 0.7 (

**d**), contrast adjustment with parameter 0.6 (

**e**).

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**Figure 2.**Mean velocity profile and confidence intervals of Monte Carlo simulations with 20

^{3}(

**a**) and 5

^{3}(

**b**) runs, and L = 16.

**Figure 3.**Mean velocity profile and confidence intervals of Monte Carlo simulations with 20

^{3}(

**a**) and 5

^{3}(

**b**) runs, and L = 32.

**Figure 4.**Mean velocity profile and confidence intervals of Monte Carlo simulations with 20

^{3}(

**a**) and 5

^{3}(

**b**) runs, and L = 64.

**Figure 5.**The impact of the mean IA size (L × L) on the accuracy of the estimated velocities and the 90% confidence interval width for Video A and the cross-sections S1 (

**a**) and S2 (

**b**).

**Figure 6.**The impact of the mean IA size (L × L) on the accuracy of the estimated velocities and the 90% confidence interval width for Video B and the cross-sections S1 (

**a**) and S2 (

**b**).

L (px) | 20^{3} Runs (Sec) | 5^{3} Runs (Sec) |
---|---|---|

64 | 88,718 | 1235 |

32 | 19,360 | 323 |

16 | 5208 | 86 |

L (px) | Lower Limit | Mean | Upper Limit |
---|---|---|---|

64 | 0.99809 | 0.99977 | 0.99405 |

32 | 0.95002 | 0.98379 | 0.87227 |

16 | 0.97966 | 0.99298 | 0.95515 |

Monte Carlo Mean | LSPIV Single Run | |
---|---|---|

Video A S1 | −10.34% | −22.27% |

Video A S2 | 3.79% | −7.21% |

Video B S1 | 3.79% | −1.71% |

Video B S2 | 16.42% | 9.56% |

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**MDPI and ACS Style**

Rozos, E.; Dimitriadis, P.; Mazi, K.; Lykoudis, S.; Koussis, A.
On the Uncertainty of the Image Velocimetry Method Parameters. *Hydrology* **2020**, *7*, 65.
https://doi.org/10.3390/hydrology7030065

**AMA Style**

Rozos E, Dimitriadis P, Mazi K, Lykoudis S, Koussis A.
On the Uncertainty of the Image Velocimetry Method Parameters. *Hydrology*. 2020; 7(3):65.
https://doi.org/10.3390/hydrology7030065

**Chicago/Turabian Style**

Rozos, Evangelos, Panayiotis Dimitriadis, Katerina Mazi, Spyridon Lykoudis, and Antonis Koussis.
2020. "On the Uncertainty of the Image Velocimetry Method Parameters" *Hydrology* 7, no. 3: 65.
https://doi.org/10.3390/hydrology7030065