# Uncertainty Analysis for Image-Based Streamflow Measurement: The Influence of Ground Control Points

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## Abstract

**:**

## 1. Introduction

## 2. Study Site and Measuring Instruments

#### 2.1. Description of Study Site

#### 2.2. Measuring Instruments

## 3. Methods: LSPIV Measurement and Uncertainty Assessment

#### 3.1. LSPIV Measurement

#### 3.1.1. Collinearity Equations

_{i}, y

_{i}, z

_{i}) corresponds to a point in the image space P(u

_{i}, v

_{i}) and then converges to the perspective center C behind the image. Hence, the perspective center point C, the image point P, and the object point O construct a collinear line.

_{x}, C

_{y}, and C

_{z}), the perspective center in the image space coordinates (C

_{u}, C

_{v}, and f), and three rotation angles (azimuth angle θ, roll angle β, and tilt angle τ). Figure 4 depicts the relationship of the three rotation angles between the object and image spaces. It indicates that X, Y, and Z are the three coordinate axes in the object space, while U, V, and F are the three coordinate axes in the image space. The azimuth angle θ denotes the angle between the F-axis direction (in the image space) and the Y-axis direction (in the object space); the roll angle β expresses the angle between the V-axis direction and the Z-axis direction; and the tilt angle τ represents the angle between the F-axis direction and the Z-axis direction. Since the coordinate systems in the object image spaces are different, a transformation/conversion needs to be performed through the rotation angle matrix. The collinearity equations and rotation coefficients can be expressed as

_{i}denotes the distance from any image point to the center of the image; k

_{1}and k

_{2}are the coefficients of radial distortion; p

_{1}and p

_{2}express the coefficients of tangential distortion; U

_{dif}/V

_{dif}represents the difference between the perspective center C

_{u}/C

_{v}and any image point u

_{i}/v

_{i}projecting in the u/v coordinate system (the image space); f denotes the distance from the perspective center to the image or the equivalent focal length; X

_{dif}/Y

_{dif}/Z

_{dif}expresses the difference between the perspective center C

_{x}/C

_{y}/C

_{z}and any object point x

_{i}/y

_{i}/z

_{i}projecting in the x/y/z coordinate system (the object space). The parameters of the rotation angle matrix (M

_{1}~M

_{9}) are composed of the azimuth angle θ, the roll angle β, and the tilt angle τ [47], i.e.,

_{1}= −cosτ cosθ − sinτ cosβ sinθ;

_{2}= cosτ sinθ − sinτ cosβ cosθ;

_{3}= −sinτ sinβ;

_{4}= sinτ cosθ − cosτ cosβ sinθ;

_{5}= −sinτ sinθ − cosτ cosβ cosθ;

_{6}= −cosτ sinβ;

_{7}= −sinβ sinθ;

_{8}= −sinβ cosθ;

_{9}= cosβ.

#### 3.1.2. Image Matching

#### 3.1.3. Surface Velocity and River Discharge

#### 3.2. Uncertainty Assessment: Monte Carlo Simulations

## 4. Results and Discussion

#### 4.1. Streamflow Measurement Using LSPIV

_{u}= 812 pixels and C

_{v}= 617 pixels. The focal length f was 0.018 m for the far-field camera, and it ranged from 0.012 m to 0.017 m for the near-field camera. In the object space, the coordinates C

_{x}, C

_{y}, and C

_{z}ranged from −0.393 m to 0.373 m, −1.749 m to 0.41 m, and 0.778 m to 1.744 m, respectively. Both the azimuth angle θ and roll angle β were close to 180° (from 168.15° to 189.17°). The tilt angle τ of the near-field camera (between 110.72° and 115.46°) was larger than that of the far-field camera (96.63° to 99.28°), giving an ROI close to the right bank for the near-field camera (Figure 8c) and an ROI near the left bank for the far-field camera (Figure 8d).

^{2}= 0.55 and a p-value ~10

^{−8}is statistically significant under a 95% confidence interval. Furthermore, in terms of accuracy, the mean absolute error (MAE) and root mean square error (RMSE) of the averaged surface velocity are summarized in Table 2. The values of MAE and RMSE ranged from 0.097 m/s to 0.154 m/s and from 0.107 m/s to 0.191 m/s, respectively. Overall, the LSPIV method demonstrated strong reliability in measuring the river surface velocity. Among the four experiments, the best performance was obtained on 3 May 2020. For the remaining three dates, the performances with slight differences were also satisfactory.

#### 4.2. Uncertainty in GCPs and Camera Parameters

_{u}and C

_{v}are integers (i.e., the pixels in the image space), forming several groups in a vertical line for the 5000 GCP samples. The probability distributions considered in this study included the Gumbel, Weibull, beta, normal, and log Pearson type III functions. Two statistical indices, i.e., the standard error SE and correlation coefficient R, were used to determine the optimal probability function for each parameter. All the probability density functions have good correlations, with R > 0.95. The values of the standard errors are listed in Table A1 in the Appendix A. According to their fitness, the parameters and their distribution functions are summarized as follows:

- Near-field camera: C
_{u}(normal), C_{v}(normal), f (log Pearson type III), C_{x}(beta), C_{y}(beta), C_{z}(log Pearson type III), θ (normal), β (normal), and τ (log Pearson type III); - Far-field camera: C
_{u}(log Pearson type III), C_{v}(normal), f (log Pearson type III), C_{x}(log Pearson type III), C_{y}(Weibull), C_{z}(log Pearson type III), θ (normal), β (normal), and τ (log Pearson type III).

#### 4.3. Uncertainty in Streamflow Measurement: GCP Measurement Times

#### 4.4. Uncertainty in Streamflow Measurement: GCP Measurement Accuracy

^{3}/s to 5.7 m

^{3}/s for the case of SE = 30 mm. While the mean discharge Q

_{mean}was 4.82 m

^{3}/s, the mode of discharges with an occurrence probability of 24% was about 4.62 m

^{3}/s (Figure 17d). Note that the mean of the surface velocities was potentially influenced by extreme values. The median (or mode) would be a more appropriate way to represent the surface velocities for comparison. The uncertainty in the discharge caused by the GCP measurement accuracy can be expressed as half of the normalized confidence interval, i.e., Q

_{diff}= (Q

_{97.5%}− Q

_{2.5%})/Q

_{mean}/2 = 20.7%, where Q

_{97.5%}and Q

_{2.5%}denote discharges with cumulative probabilities of 97.5% and 2.5%, respectively. In the case of SE = 3 mm, the measured discharges were distributed mainly in a range between 4.5 m

^{3}/s and 4.7 m

^{3}/s (with occurrence probabilities of 29% and 26% or a total of 55% in the distribution), returning a mean discharge of 4.59 m

^{3}/s and a normalized half confidence interval of 10.7% (Figure 17f). As the accuracy of the GCP measurements increased (SE = 30 mm, 10 mm, and 3 mm), the uncertainty in the LSPIV streamflow measurements (Q

_{diff}=20.7%, 12.8%, and 10.7%) decreased, returning median discharges (Q

_{median}= 4.62 m

^{3}/s, 4.6 m

^{3}/s, and 4.59 m

^{3}/s) closer to those (4.59 m

^{3}/s) obtained from the flow meter.

_{diff}was about 12% when the number of GCPs reached 19. Additionally, Schweitzer and Cowen [54] demonstrated that an accurate GCP-based georeferencing method was able to reduce uncertainty in the streamflow measurement by a factor of five or more in comparison to the direct method. Overall, both previous and current works clearly imply that a high-precision instrument for GCP measurement is necessary.

#### 4.5. Limitations and Future Work

## 5. Conclusions

_{diff}= 20.7%, 12.8%, and 10.7%), returning median discharges (Q

_{median}= 4.62 m

^{3}/s, 4.60 m

^{3}/s, and 4.59 m

^{3}/s) closer to those (4.59 m

^{3}/s) obtained from the flow meter.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A

**Table A1.**Standard errors of various probability distributions for the parameters of the near-field and far-field cameras.

Parameter | Near-Field Camera | Far-Field Camera | ||||||||
---|---|---|---|---|---|---|---|---|---|---|

Beta | Gamma | Normal | Log Pearson III | Weibull | Beta | Gamma | Normal | Log Pearson III | Weibull | |

C_{u} (Pixels) | 0.5064 | 0.4716 | 0.4454 | 0.4964 | 0.7246 | 0.4924 | 0.6789 | 0.4223 | 0.4151 | 0.7273 |

C_{v} (Pixels) | 0.5715 | 0.5750 | 0.5064 | 0.5201 | 0.6487 | 0.5609 | 0.5482 | 0.4903 | 0.5005 | 0.6654 |

f (m) | 6 × 10^{−5} | 12 × 10^{−5} | 15 × 10^{−5} | 5 × 10^{−5} | 35 × 10^{−5} | 15 × 10^{−5} | 27 × 10^{−5} | 45 × 10^{−5} | 11 × 10^{−5} | 76 × 10^{−5} |

C_{x} (m) | 0.0013 | 0.0129 | 0.0042 | 0.0014 | 0.0059 | 0.0035 | 0.0042 | 0.0035 | 0.0022 | 0.018 |

C_{y} (m) | 0.0134 | 0.0334 | 0.0306 | 0.0151 | 0.0269 | 0.0475 | 0.1521 | 0.1255 | 0.1131 | 0.0473 |

C_{z} (m) | 0.0030 | 0.0072 | 0.0123 | 0.0027 | 0.0295 | 0.0034 | 0.0100 | 0.0130 | 0.0032 | 0.0320 |

θ (Degrees) | 0.0289 | 0.0533 | 0.0167 | 0.0209 | 0.1611 | 0.0731 | 0.0659 | 0.0400 | 0.0460 | 0.2977 |

β (Degrees) | 0.0460 | 0.0718 | 0.0195 | 0.2213 | 0.2115 | 0.0392 | 0.0266 | 0.0218 | 0.0256 | 0.1393 |

τ (Degrees) | 0.0178 | 0.1120 | 0.0189 | 0.0176 | 0.1726 | 0.0670 | 0.0680 | 0.0714 | 0.0341 | 0.3639 |

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**Figure 1.**Location map of the study site. (

**a**–

**c**) study area; (

**d**) image of study site; (

**e**) river cross-sectional profile.

**Figure 2.**Instruments for measurement: (

**a**) camera, lens, and protective shell; (

**b**) swivel base; and (

**c**) hand-held propeller digital flow meter.

**Figure 3.**Measurement procedure for using image-based surface velocimetry and a propeller digital flow meter.

**Figure 4.**Schematic diagram of the rotation angles (i.e., θ, β, and τ for azimuth, roll, and tilt, respectively) among the collinear parameters.

**Figure 6.**Flow chart for the measurement of surface velocity using LSPIV and the estimation of river discharge.

**Figure 7.**The procedure for determining the optimal probability distribution function in the frequency analysis.

**Figure 9.**Surface velocity fields and contours measured using LSPIV at (

**a**,

**e**) 9:00 on 3 May 2020, (

**b**,

**f**) 16:00 on 26 July 2020, (

**c**,

**g**) 11:00 on 1 November 2020, and (

**d**,

**h**) 14:00 on 3 December 2020.

**Figure 10.**Comparison of surface velocities measured by LSPIV and the flow meter on (

**a**) 3 May 2020, 10 a.m., (

**b**) 26 July 2020, 12 p.m., (

**c**) 1 November 2020, 11 a.m., and (

**d**) 3 December 2020, 4 p.m.

**Figure 11.**Comparison of surface velocities measured by LSPIV and the flow meter on (

**a**) 3 May 2020, (

**b**) 26 July 2020, (

**c**) 1 November 2020, and (

**d**) 3 December 2020.

**Figure 13.**Comparison of the samples and various probability distribution functions for the nine parameters of the near-field camera.

**Figure 14.**Comparison of the samples and various probability distribution functions for the nine parameters of the far-field camera.

**Figure 15.**Uncertainty analysis of the averaged surface velocities from LSPIV based on Monte Carlo simulations: (

**a**) one, (

**b**) three, and (

**c**) five GCP measurements. Note that the red dots represent the average surface velocity measured by the flow meter; the light blue area denotes the 95% (i.e., from 2.5% to 97.5%) confidence interval; and the dark blue area expresses the 50% (i.e., from 25% to 75%) confidence interval.

**Figure 16.**Box plot for the averaged surface velocities from LSPIV with different GCP measurement times.

**Figure 17.**Uncertainty analysis of the averaged surface velocities from LSPIV and the estimated river discharges based on Monte Carlo simulations: (

**a**,

**d**) SE = 30 mm, (

**b**,

**e**) SE = 10 mm, and (

**c**,

**f**) SE = 3 mm in GCPs. Note that the red dots represent the average surface velocity measured by the flow meter; the light blue area denotes the 95% (i.e., from 2.5% to 97.5%) confidence interval; and the dark blue area expresses the 50% (i.e., from 25% to 75%) confidence interval.

**Table 1.**Parameters of the near-field (NF) and far-field (FF) cameras in the four field experiments.

Date/Camera | C_{u}(Pixels) | C_{v}(Pixels) | f (m) | C_{x}(m) | C_{y}(m) | C_{z}(m) | θ (Degrees) | β (Degrees) | τ (Degrees) | |
---|---|---|---|---|---|---|---|---|---|---|

3 May 2020 | FF | 812 | 617 | 0.018 | −0.16 | −1.595 | 1.648 | 178.83 | 184.16 | 99.28 |

NF | 812 | 617 | 0.015 | −0.261 | −0.629 | 1.064 | 168.15 | 186.42 | 111.77 | |

26 July 2020 | FF | 812 | 617 | 0.018 | −0.189 | −1.696 | 1.605 | 178.65 | 184.09 | 97.38 |

NF | 812 | 617 | 0.014 | 0.070 | 0.410 | 0.778 | 175.92 | 178.02 | 110.72 | |

1 November 2020 | FF | 812 | 617 | 0.018 | −0.393 | −1.675 | 1.648 | 178.26 | 189.17 | 96.63 |

NF | 812 | 617 | 0.017 | −0.145 | −0.634 | 1.133 | 160.76 | 184.64 | 115.46 | |

3 December 2020 | FF | 812 | 617 | 0.018 | 0.373 | −1.749 | 1.744 | 178.86 | 178.25 | 98.76 |

NF | 812 | 617 | 0.012 | 0.017 | −0.080 | 1.132 | 176.70 | 184.84 | 112.26 |

**Table 2.**The averaged surface velocities and two statistical indices (MAE and RMSE) between the results measured by LSPIV (V

_{LSPIV}) and the flow meter (V

_{FM}).

Date | 3 May | 26 July | 1 November | 3 December |
---|---|---|---|---|

Max water depth (m) | 0.66 | 0.79 | 0.62 | 0.85 |

VFM (m/s) | 0.528 | 0.750 | 0.554 | 0.829 |

VLSPIV (m/s) | 0.485 | 0.665 | 0.492 | 0.758 |

MAE (m/s) | 0.097 | 0.154 | 0.104 | 0.098 |

RMSE (m/s) | 0.107 | 0.191 | 0.111 | 0.110 |

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## Share and Cite

**MDPI and ACS Style**

Liu, W.-C.; Huang, W.-C.; Young, C.-C. Uncertainty Analysis for Image-Based Streamflow Measurement: The Influence of Ground Control Points. *Water* **2023**, *15*, 123.
https://doi.org/10.3390/w15010123

**AMA Style**

Liu W-C, Huang W-C, Young C-C. Uncertainty Analysis for Image-Based Streamflow Measurement: The Influence of Ground Control Points. *Water*. 2023; 15(1):123.
https://doi.org/10.3390/w15010123

**Chicago/Turabian Style**

Liu, Wen-Cheng, Wei-Che Huang, and Chih-Chieh Young. 2023. "Uncertainty Analysis for Image-Based Streamflow Measurement: The Influence of Ground Control Points" *Water* 15, no. 1: 123.
https://doi.org/10.3390/w15010123