Evaluation of Discharge Measurement Uncertainty of a Surface Image Velocimeter

Abstract

This study aims to develop a framework for evaluating the uncertainty of a surface image velocimeter (SIV) based on the Guide to the Expression of Uncertainty in Measurement (GUM) standard. To achieve this, the uncertainty factors of the SIV were thoroughly reviewed and categorized into those that can be directly incorporated into the functional equation for surface velocity calculation and those that cannot. Factors that can be included in the velocity calculation equation primarily involve image displacement measurement and the accurate determination of the time interval between successive stationary images. Conversely, parameters and image quality were identified as uncertainty factors that are not directly integrated into the velocity calculation equation. Based on the GUM standard, equations for calculating the uncertainty of surface velocity, depth-averaged velocity, and flow discharge measurements were developed. Furthermore, using the results from the standard uncertainty evaluation, assessments were performed for the velocity uncertainty of both surface velocity and depth-averaged velocity, as well as the flow rate measurement uncertainty of the SIV. We anticipate that the velocity and flow rate measurement uncertainty framework and the uncertainty analysis results for the SIV presented in this research will enhance the reliability of SIV-derived flow rate measurements, thereby contributing to more dependable flow rate determination.

1. Introduction

The flow discharge of a river serves as crucial foundational data in the design of hydraulic structures, water resource management, and river disaster prevention. In practice, river discharge is also used as input data for numerical models and for the purposes of calibration and verification. The common method for measuring discharge in rivers relies on the velocity-area method, which involves dividing the river’s cross-section into several measurement subsections, measuring the velocity at each subsection, and then multiplying it by the corresponding cross-sectional area to estimate the discharge.

When using the velocity-area method, if the water stage is measured appropriately, the cross-sectional area can be estimated with considerable accuracy (it is important to note that this paper intends to use the terms ’measurement’ and ’estimation’ distinctly). Measurement refers to directly obtaining data values using specialized equipment, while estimation refers to deriving new data through calculations based on these measured values. According to this distinction, water stage and velocity are measured, while discharge is estimated. However, when referring to both collectively, it will be termed discharge measurement.

On the other hand, velocity measurement is a very difficult task that requires significant time, expense, and manpower. Furthermore, physically, velocity in rivers varies greatly both temporally and spatially, making proper estimation or measurement very challenging. During normal flow conditions, river velocity is measured using traditional current meters that measure velocity at a single point (hereafter referred to as point current meters) or using an Acoustic Doppler Current Profiler (ADCP, RD Instruments, Poway, CA, USA). However, during flood periods when it is difficult to operate such equipment, floats are used. In all these measurements, it is necessary to clearly understand the limitations inherent in the measured data. Uncertainty can be used as a way to represent the reliability of such measured data. In particular, knowing the uncertainty of discharge, which is the ultimate goal of hydraulic quantity measurement in rivers, is very important for understanding the characteristics of the data.

For the traditional river discharge measurement method using point current meters, standards for the measurement method and uncertainty evaluation methods have been established [1,2]. However, as mentioned in many studies, measuring velocity and estimating discharge using such traditional equipment requires significant cost, time, and manpower, and more importantly, involves considerable risks. Therefore, efforts are underway to utilize non-contact methods such as the Surface Image Velocimeter (SIV) and microwave current meter, which allow for automation and are economically and temporally advantageous. However, appropriate uncertainty estimation methods for these new measurement techniques have not yet been established. In this study, we aim to present a method for calculating the measurement uncertainty of discharge that arises when measuring river velocity and estimating discharge using a surface image velocimeter, in accordance with the GUM standard [3].

2. GUM-Based Uncertainty Framework for SIV

2.1. Uncertainty Evaluation of Discharge Measurement Based on GUM

Various organizations and researchers involved in river discharge measurement have adopted the GUM (Guidelines to the Expression of Uncertainty in Measurement) as the method for calculating the uncertainty of discharge measurement [4,5]. The GUM standard is a guideline for calculating measurement uncertainty, published by seven international organizations—ISO (International Organization for Standardization); BIPM (Bureau International des Poids et Mesures); IEC (International Electrotechnical Commission); IFCC (International Federation of Clinical Chemistry); IUPAC (International Union of Pure and Applied Chemistry); IUPAP (International Union of Pure and Applied Physics); and OIML (International Organization of Legal Metrology)—by synthesizing past research achievements to standardize guidelines for measurement uncertainty evaluation and calculation. It provides a framework for estimating measured values and evaluating the uncertainty of measurement results through the relational expressions between the measured value and various influencing factors. This guideline is recognized as the most appropriate method for uncertainty evaluation, and therefore, most uncertainty evaluations follow it. According to the GUM standard, the procedure for evaluating uncertainty is as shown in Figure 1.

Recently, studies on measurement uncertainty evaluation based on the GUM standard have been conducted for traditional discharge measurement methods and ADCP [6].

On the other hand, there is very little research on the uncertainty of discharge measurement using the latest techniques, such as microwave current meters and surface image velocimeters. For microwave current-meters, it is virtually difficult to find suitable research, and for surface image velocimeters, research has been conducted in Japan [7] and Korea [8]. Watanabe et al. [7] estimated the uncertainty of measured data by assuming the distribution of measured values of the Space-Time Image Velocimeter (STIV). This is slightly different from the aforementioned GUM. In contrast, Lee’s [8] study is an application of GUM to LSPIV (Large Scale Particle Image Velocimetry) based on the cross-correlation method. This study faithfully followed GUM, but the target was LSPIV.

2.2. Surface Velocity Measurement Uncertainty

When measured values $y=f(x_1,x_2,x_3,\dots x_N)$ depends on $N$ input quantities $x_1,x_2,x_3,\dots x_N$, the measurement uncertainty for $y$ is calculated using the following equation:

$$u_c\left(y\right)={\left[{\sum }_{i=1}^N{\left({\displaystyle \frac{\partial f}{\partial x_i}}\right)}^2{u}_{x_i}^2+{\sum }_{k=1}^n{u}_{f,k}^2\right]}^{1/2}$$

(1)

where the value $u_{f,k}$ represents the measurement uncertainty for the $k$-th uncertainty factor, which, although not included in the measurement equation, indirectly affects the measured value.

First, the uncertainty factors in surface velocity measurement using a surface image velocimeter were categorized into those that can be included in the velocity calculation equation and those that cannot (

Table 1. Factors of uncertainty in surface velocity measurement.

Category	Uncertainty Factor
$∆D_{px}$	Displacement in pixel
	$D_{gs}$ (Ground Sampling Distance)	reference points survey
		reference point identification
$∆t$	Time interval between two images
Analysis parameter	Width of STI
	Height of STI (number of image frames)
Image quality	Standard deviation of contrast value in interrogation area

).

First, in a surface image velocimeter, velocity is calculated by dividing the image displacement in a spatio-temporal image by the time interval between still images. Therefore, the uncertainty factors that can be included in the calculation equation can be divided into the uncertainty in image displacement and the uncertainty in the time interval between still images. Uncertainty factors that are not included in the velocity calculation equation can be broadly categorized into uncertainty due to analysis parameters and uncertainty related to image quality.

$$V_s={\displaystyle \frac{∆D_{px}}{∆t}} D_{gs}$$

(2)

where $V_s$ is the surface velocity, $∆D_{px}$ is the image displacement (in pixels), $∆t$ is the time interval between two still images, and $D_{gs}$ is the ground sampling distance (physical distance in meters per pixel).

In the velocity calculation Equation (2), the input quantities, image displacement and time interval, are independent measured or input values. Therefore, the surface velocity uncertainty calculation equation can be expressed like this.

$$u_{Vs}^2={\left({\displaystyle \frac{\partial V_s}{\partial ∆D_{px}}}\right)}^2{u}_{∆D_{px}}^2+{\left({\displaystyle \frac{\partial V_s}{\partial ∆t}}\right)}^2{u}_{∆t}^2+{\sum }_{k=1}^n{u}_{Vs,k}^2$$

(3)

where $u_{Vs}$ is the measurement uncertainty of the surface velocity, the partial derivative term $\left({\displaystyle \frac{\partial f}{\partial x_i}}\right)$ is the sensitivity coefficient, $u_{∆D_{px}}$ is the measurement uncertainty of the displacement, $u_{∆t}$ is the measurement uncertainty of the time interval between still images, and $u_{Vs,k}$ is the measurement uncertainty of the surface velocity due to the $k$th uncertainty factor, such as analysis parameters and image quality, which are not included in the measurement equation but indirectly affect the velocity estimation.

2.3. Uncertainty in Depth-Averaged Velocity Estimation

The equation for calculating the measurement uncertainty, $u_V$, of the depth-averaged velocity, $V$, is as follows:

$$V=K·V_s$$

(4)

where $K$ is the velocity conversion factor, and $V_s$ is the measured surface velocity. when calculating the discharge. Typically, a value of 0.85 is used for this velocity conversion factor.

The measurement uncertainty of this surface velocity is calculated based on factors like image quality and ground sampling distance. It is then used, along with the depth-averaged conversion coefficient, by substituting it into the depth-averaged velocity measurement uncertainty equation to determine the depth-averaged velocity’s measurement uncertainty.

However, the vertical velocity distribution in a river varies and is complex depending on the river’s characteristics. Therefore, uncertainty arises from the conversion factor during the process of converting surface velocity to depth-averaged velocity. In this study, we aim to estimate the uncertainty of the velocity conversion factor using statistical methods based on existing research results on the velocity conversion factor. The basic form of the depth-averaged velocity uncertainty calculation is as follows:

$$u_V^2={\left({\displaystyle \frac{\partial V}{\partial V_s}}\right)}^2{u}_{Vs}^2+{\left({\displaystyle \frac{\partial V}{\partial K}}\right)}^2{u}_K^2$$

(5)

The depth-averaged velocity measurement uncertainty is then substituted into the discharge measurement uncertainty equation and used to ultimately determine the discharge measurement uncertainty.

And the summary of various research results on the depth-averaged conversion factor known to date is as shown in Table 2.

Table 2. Velocity conversion factors in former studies.

(Velocity Conversion Factor) K	Flow Characteristics	References
0.84~0.90	experiment result logarithmic profile	Rantz [9] Turnipseed and Sauer [10]
0.80~0.93	San Joaquin River	Cheng and Gartner [11]
0.80~0.90	logarithmic profile	Creutin et al. [12]
over 0.85	experiment channel	Roh et al. [13]
0.80	Rapid mountain stream	Kim et al. [14]
0.80 ± 15%	natural river (water depth < 2 m)	Hauet et al. [15]
0.90 ± 15%	natural river (large water depth) artificial concrete channel	Hauet et al. [15]

Table 2. Velocity conversion factors in former studies.

(Velocity Conversion Factor) K	Flow Characteristics	References
0.84~0.90	experiment result logarithmic profile	Rantz [9] Turnipseed and Sauer [10]
0.80~0.93	San Joaquin River	Cheng and Gartner [11]
0.80~0.90	logarithmic profile	Creutin et al. [12]
over 0.85	experiment channel	Roh et al. [13]
0.80	Rapid mountain stream	Kim et al. [14]
0.80 ± 15%	natural river (water depth < 2 m)	Hauet et al. [15]
0.90 ± 15%	natural river (large water depth) artificial concrete channel	Hauet et al. [15]

After reviewing the research results conducted under these various conditions, it was found that the velocity conversion factor ranges between 0.80 and 0.93, and its probability distribution is not precisely known. Therefore, this study assumes a uniform distribution. In this case, the expected value (mean) is 0.865, and the standard uncertainty is estimated to be 0.0735. Thus, the standard uncertainty of the depth-averaged velocity in Equation (5) can be expressed as follows:

$$u_V^2={\left(0.865\right)}^2{u}_{Vs}^2+{\left(0.0375\right)}^2 V_s^2$$

(6)

Now, a diagram illustrating the framework for the standard uncertainty evaluation of the depth-averaged velocity that we have outlined is shown in Figure 2.

2.4. Discharge Calculation Uncertainty

Muste et al. [5] presented the discharge calculation uncertainty that can occur when estimating discharge using the mid-section method with a current meter, in accordance with the GUM standard. In this study, to evaluate the discharge measurement uncertainty of a surface image velocimeter, we applied it to the discharge measurement uncertainty equation when measuring discharge using the mid-section method and summarized the uncertainty evaluation for river discharge measurement using a surface image velocimeter. At this time, the average velocity, width, and depth for each subsection are considered separate measured values, and it is assumed that there is no correlation between them. Therefore, the discharge calculation uncertainty of the surface image velocimeter can be calculated using the following equation:

$$u_c\left(Q\right)={\left[{\sum }_{i=1}^m{u}_{V,i}^2{\left({\displaystyle \frac{\partial q_i}{\partial V_i}}\right)}^2+{\sum }_{i=1}^m{u}_{b,i}^2{\left({\displaystyle \frac{\partial q_i}{\partial b_i}}\right)}^2+{\sum }_{i=1}^m{u}_{d,i}^2{\left({\displaystyle \frac{\partial q_i}{\partial d_i}}\right)}^2+u_{Qm}^2+u_{Vm}^2+u_{um}^2+u_{us}^2+u_{op}^2\right]}^{1/2}$$

(7)

where $u_c\left(Q\right)$ is the combined standard uncertainty of the discharge, $u_{V,i}$ is the standard uncertainty of the average velocity measurement, $u_{b,i}$ and $u_{d,i}$ are the standard uncertainties of the width and depth measurements, respectively, for the ith subsection. $u_{Qm}$ is the standard uncertainty of the discharge due to the discharge measurement model (mid-section method). $u_{Vm}$ is the standard uncertainty of the discharge due to the number of measurement subsections (number of verticals). $u_{um}$ is the standard uncertainty due to unmeasured flow at the banks of the cross-section. $u_{us}$ is the standard uncertainty of the discharge due to flow unsteadiness during the measurement. $u_{op}$ is the standard uncertainty of the discharge due to measurement conditions. The fourth to sixth terms on the right-hand side of Equation (7) represent the uncertainties due to the correlation between the average velocity, width, and depth measurements for each subsection, respectively.

2.5. Uncertainty Due to Image Analysis

In Equation (3) mentioned earlier (which detailed the uncertainty in surface velocity), when analyzing velocity using spatio-temporal images, the exact displacement at the pixel level cannot be known precisely. Therefore, the error range for image displacement can be considered to be ±0.1 pixels. Assuming a uniform distribution for this error, the standard uncertainty is 0.058 pixels. With this in mind, the uncertainty due to image displacement can be expressed by the following equation:

$$u_{{∆D}_{px}}={\displaystyle \frac{u_{{∆D}_{px}}}{∆D_{px}}}\times 100\%={\displaystyle \frac{0.058}{∆D_{px}}}\times 100\%$$

(8)

This implies that the relative uncertainty due to image displacement decreases as the image displacement increases.

On the other hand, the ground sampling distance ($D_{gs}$) is the physical size (in meters) of one pixel in the image. If the image coordinates and physical coordinates of known reference points are obtained through reference point surveying, the physical coordinates of all pixel points within the image can be determined. The uncertainty due to reference point surveying is small, typically less than 1% within the reference point area [16]. This is because, assuming the control point survey was diligently performed, errors due to the surveying equipment occur within ±10 mm at each point, and errors arising from the reference point identification process occur within ±0.5 pixels. It becomes smaller as the river size increases and as the distance from the camera decreases. Therefore, the uncertainty in the ground sampling distance calculation can be considered negligible. That is,

$$u_{D_{gs}}=0$$

(9)

Furthermore, the ground sampling distance increases with the size of the river and the distance from the camera. Consequently, the combined standard uncertainty due to potential image displacement can be expressed as follows:

$$u_{∆D}^2={\left({\displaystyle \frac{\partial \left(∆D\right)}{\partial \left(∆D_{px}\right)}}\right)}^2{u}_{∆D_{px}}^2+{\left({\displaystyle \frac{\partial \left(∆D\right)}{\partial \left(D_{gs}\right)}}\right)}^2{u}_{D_{gs}}^2={\left(D_{gs}\right)}^2{u}_{∆D_{px}}^2$$

(10)

Equation (10) indicates that as the ground sampling distance increases, the uncertainty regarding displacement also increases.

2.6. Uncertainty Due to Image Quality

Image quality is not directly included in the surface velocity measurement equation but is a factor that can introduce uncertainty when estimating surface velocity.

Lee [8] presented the velocity measurement uncertainty according to light brightness when using the cross-correlation method, as shown in Figure 3, by using the standard deviation of the grayscale values (0–256 for a grayscale image) within the correlation window. This study indicated that as the light brightness decreases, the standard deviation of the grayscale values decreases, leading to an increase in uncertainty. It was also reported that when the standard deviation of grayscale values is approximately 18 or less, uncertainty of 10% or more occurs.

Therefore, the surface velocity measurement uncertainty $u_{\sigma ,i}$ (m/s) due to the standard deviation of the grayscale values (luminance) can be determined for each measurement subsection based on the standard deviation of the grayscale values ${\sigma }_i$ (dimensionless) of the spatio-temporal image formed at that subsection, as follows:

$$u_{\sigma ,i}=\left\{\begin{array}{cc}9.046\times {10}^{-2} {\sigma }_i^{-0.746}{V}_i,& {\sigma }_i>18.0\\ 5.5551 {\sigma }_i^{-1.902}{V}_i,& {\sigma }_i\le 18.0\end{array}\right.$$

(11)

2.7. Uncertainty Due to the Number of Measurement Verticals

The uncertainty in discharge estimation due to the number of measurement verticals (number of measurement subsections), $u_{Qm}$, is calculated using the following equation.

$$u_{Qm}=0.01 u_{n}Q$$

(12)

where Q is the measured discharge (in m³/s) and $u_n$ is the discharge measurement uncertainty (%) according to the ISO number of verticals, which is given in the following

Table 3. Uncertainty according to the number of measurement lines [17].

No. of Measurement Lines	Uncertainty (%)
5	7.5
10	4.5
15	3.0
20	2.5
25	2.0
30	1.5
35	1.0
40	1.0
45	1.0

[17].

2.8. Uncertainty Based on Measurement Conditions

The measurement uncertainty based on different discharge measurement conditions, $u_{op}$, uses the following equation proposed by WMO [18].

$$u_{op}=0.027Q$$

(13)

where $Q$ is measured flow discharge (m³/s).

2.9. Final Equation for Discharge Measurement Uncertainty Evaluation

After going through the above process and removing unnecessary or difficult-to-calculate terms, the final equation for discharge measurement uncertainty is summarized as follows:

$$u_c\left(Q\right)={\left[{\sum }_{i=1}^m{u}_{V,i}^2 A_i^2+u_{Vm}^2+u_{op}^2\right]}^{1/2}$$

(14)

where $u_c\left(Q\right)$ is the combined standard uncertainty of the discharge, $u_{V,i}$ is the standard uncertainty of the average velocity measurement for the $i$th subsection, $u_{Vm}$ is the standard uncertainty of the discharge due to the number of measurement subsections (number of verticals), and $u_{op}$ is the standard uncertainty of the discharge due to measurement conditions. $A_i$ is the cross-sectional area of the $i$th measurement subsection.

The uncertainty in average velocity estimation is calculated for each measurement subsection using the following equation, which is an application of the previous Equation (6) to each subsection:

$$u_{V,i}^2={\left(0.865\right)}^2{u}_{Vs,i}^2+{\left(0.0375\right)}^2{\left(V_{s,i}\right)}^2$$

(15)

where $u_{Vs,i}$ is the surface velocity measurement uncertainty of measurement subsection $i$, and $V_{s,i}$ is the surface velocity of measurement subsection $i$.

The surface velocity measurement uncertainty $u_{Vs,i}$ for each measurement subsection $i$ in Equation (14) is calculated separately for each subsection using the following equation:

$$u_{Vs,i}^2={\left({\displaystyle \frac{0.058 D_{gs, i}}{∆t}}\right)}^2+u_{\sigma ,i}^2{\left(V_{s,i}\right)}^2$$

(16)

where $D_{gs, i}$ is the ground sampling distance (m/px) at measurement subsection $i$, and $u_{\sigma ,i}$ is the surface velocity measurement uncertainty due to the standard deviation of grayscale values at each measurement subsection, as given by Equation (10).

2.10. Procedure for Evaluating Discharge Measurement Uncertainty

The calculation procedure for discharge measurement uncertainty is as follows:

• Input the following for each measurement subsection i: ground sampling distance $D_{gs,i}$ (m/px), standard deviation of grayscale values of the spatio-temporal image ${\sigma }_i$ (dimensionless), measurement subsection area $A_i$ (m²), and measured surface velocity $V_{s,i}$ (m/s). Also, input the total measured discharge $Q$ (m³/s) of the entire cross-section, the time interval between image frames Δt (s), and the coefficient $K$ for converting surface velocity to average velocity (typically = 0.85).
• Calculate the uncertainty due to the standard deviation of grayscale values for each measurement subsection, $u_{\sigma ,i}$ (m/s) using Equation (11).
• Calculate the surface velocity measurement uncertainty for each measurement subsection, $u_{Vs,i}$ (m/s), using Equation (16).
• Calculate the average velocity measurement uncertainty for each measurement subsection, $u_{V,i}$ (m/s), using Equation (15).
• Calculate the discharge measurement uncertainty due to the number of verticals, $u_{Qm}$ (m/s), using Equation (12).
• Calculate the discharge measurement uncertainty due to measurement conditions, $u_{op}$ (m/s), using Equation (13).
• Substitute the average velocity measurement uncertainties $u_{V,i}$ (m/s), the measurement subsection areas $A_i$ (m²), the discharge measurement uncertainty due to the number of verticals $u_{Vm}$ (m/s) and the discharge measurement uncertainty due to measurement conditions $u_{op}$ (m/s) into Equation (13) to obtain the total discharge measurement uncertainty $u_c\left(Q\right)$ (m³/s).

3. Case Study: Discharge Uncertainty Assessment for SIV

3.1. Experiment Channel

Based on the standard uncertainty evaluation results for each factor, we intend to apply the GUM standard, as described in the previous section, to an actual river to calculate the discharge measurement uncertainty of a surface image velocimeter. The experiment for uncertainty evaluation was conducted at the Andong River Experiment Center of the Korea Institute of Civil Engineering and Building Technology (Figure 4). The Andong River Experiment Center is equipped with a channel similar in scale to actual small rivers, allowing for the simulation of various flow conditions. In Korea, according to the Small River Maintenance Act, a small river is defined as a river not subject to or governed by the River Act. It refers to an area where water flows or is expected to flow on a non-temporary basis, explicitly stated as having an average width of 2 m or more and a river length of 500 m or more. The Andong River Experiment Center is an artificial channel that is 560 m long and 11 m wide. This scale is deemed suitable for conducting discharge measurement uncertainty evaluation experiments, as it falls under the classification of small rivers according to Korean law.

Discharge measurement using the surface image velocimeter was performed in accordance with ISO 1088:2007 [17]. For discharge calculation, measurement verticals were spaced at 0.25 m intervals, and surface velocity was measured at a total of 19 points across the water surface (Figure 5). During surface velocity measurement, the spatial width of the STI was 64 pixels, and a temporal duration of 10 s (300 frames) was used.

The cross-section of the measurement reach is shown in Figure 6. The water depth required for discharge calculation was 0.517 m based on the deepest point of the channel, and the water surface width at this time was 5.0 m.

3.2. Calculation of Discharge Measurement Uncertainty

From the measured cross-sectional area, water depth, and surface velocity distribution, the cross-sectional area and discharge of each measurement subsection, as well as the total discharge of the cross-section, were calculated using the mid-section method. Given that the riverbed at this location was composed of gravel, the velocity conversion factor was set to 0.80. The total discharge was 3.43 m³/s. The calculated uncertainties for each measurement point using the measured data are shown in

Table 4. Combined standard uncertainty of discharge in measuring points.

No.	${\mathit{V}}_{\mathit{s},\mathit{i}}$ (m/s)	${\mathit{A}}_{\mathit{i}}$ (m2)	$\mathit{q}$ (m3/s)	${\mathit{u}}_{\mathit{V}}^2{\left(\frac{\partial \mathit{q}}{\partial \mathit{V}}\right)}^2$	${\mathit{u}}_{\mathit{b}}^2{\left(\frac{\partial \mathit{q}}{\partial \mathit{b}}\right)}^2$	${\mathit{u}}_{\mathit{d}}^2{\left(\frac{\partial \mathit{q}}{\partial \mathit{d}}\right)}^2$	${\mathit{u}}_{\mathit{c}}^2\left(\mathit{q}\right)$	${\mathit{u}}_{\mathit{c}}\left(\mathit{q}\right)$ (m3/s)	${\mathit{u}}_{\mathit{c}}\left(\mathit{q}\right)$ (%)
1	1.303	0.035	0.036	0.031	3.72 × 10−6	3.03 × 10−9	5.68 × 10−8	3.78 × 10−7	0.002
2	1.572	0.070	0.088	0.033	2.06 × 10−5	1.76 × 10−8	3.31 × 10−7	2.09 × 10−6	0.005
3	1.980	0.106	0.168	0.035	7.03 × 10−5	6.28 × 10−8	1.18 × 10−6	7.16 × 10−5	0.008
4	2.144	0.110	0.189	0.036	8.92 × 10−5	8.08 × 10−8	1.52 × 10−6	9.08 × 10−5	0.010
5	2.201	0.104	0.183	0.036	8.30 × 10−5	7.57 × 10−8	1.42 × 10−6	8.45 × 10−5	0.009
6	2.293	0.105	0.193	0.036	9.02 × 10−5	8.28 × 10−8	1.55 × 10−6	9.18 × 10−5	0.010
7	2.377	0.102	0.194	0.037	9.24 × 10−5	8.52 × 10−8	1.60 × 10−6	9.41 × 10−5	0.010
8	2.427	0.100	0.194	0.037	9.22 × 10−5	8.53 × 10−8	1.60 × 10−6	9.39 × 10−5	0.010
9	2.440	0.094	0.183	0.037	8.12 × 10−5	7.52 × 10−8	1.41 × 10−6	8.27 × 10−5	0.009
10	2.424	0.111	0.215	0.037	1.12 × 10−4	1.04 × 10−7	1.95 × 10−6	1.14 × 10−4	0.011
11	2.479	0.114	0.226	0.037	1.23 × 10−4	1.14 × 10−7	2.15 × 10−6	1.25 × 10−4	0.011
12	2.601	0.116	0.241	0.038	1.40 × 10−4	1.30 × 10−7	2.45 × 10−6	1.42 × 10−4	0.012
13	2.614	0.123	0.257	0.037	1.60 × 10−4	1.50 × 10−7	2.81 × 10−6	1.63 × 10−4	0.013
14	2.466	0.109	0.215	0.035	1.10 × 10−4	1.03 × 10−7	1.94 × 10−6	1.12 × 10−4	0.011
15	2.571	0.116	0.239	0.037	1.36 × 10−4	1.27 × 10−7	2.39 × 10−6	1.39 × 10−4	0.012
16	2.727	0.120	0.262	0.039	1.65 × 10−4	1.55 × 10−7	2.90 × 10−6	1.68 × 10−4	0.013
17	2.593	0.096	0.199	0.036	9.40 × 10−5	8.85 × 10−8	1.66 × 10−6	9.58 × 10−5	0.010
18	2.136	0.063	0.108	0.030	2.81 × 10−5	2.64 × 10−8	4.95 × 10−7	2.86 × 10−5	0.005
19	1.598	0.031	0.040	0.024	3.83 × 10−6	3.55 × 10−9	6.66 × 10−8	3.90 × 10−6	0.002
Total $Q$			3.430	-	-	-	-	-	-

.

Furthermore, the discharge measurement uncertainty for each measurement point and the overall data were used to evaluate the combined standard uncertainty of the discharge measurement. The summarized standard uncertainty for the total discharge is shown in

Table 5. Combined standard uncertainty of total discharge.

Uncertainty Factor	Uncertainty Evaluation (m3/s)	Uncertainty Factor	Uncertainty Evaluation (m3/s)
$Q$	3.430	$u_{us}^2$	0.0
${\displaystyle \sum _{i=1}^{19}}{u}_{V,i}^2{\left({\displaystyle \frac{\partial q_i}{\partial V_i}}\right)}^2$	1.69 × 10−3	$u_{op}^2$	8.57 × 10−3
${\displaystyle \sum _{i=1}^{19}}{u}_{b,i}^2{\left({\displaystyle \frac{\partial q_i}{\partial b_i}}\right)}^2$	1.57 × 10−6	$u_c^2\left(Q\right)$	0.018
${\displaystyle \sum _{i=1}^{19}}{u}_{d,i}^2{\left({\displaystyle \frac{\partial q_i}{\partial d_i}}\right)}^2$	2.95 × 10−5	$u_c\left(Q\right)$ (m3/s)	0.134
$u_{Qm}^2$	2.94 × 10−4	$u_Q$ (m3/s)	0.268
$u_{Vm}^2$	7.35 × 10−3	$u_Q$ (%)	7.810
$u_{um}^2$	0.0	$Q=3.430\pm 0.268$ m3/s (7.81%) (confidence interval 95%)

.

At this time, the contributions to measurement uncertainty were quantified and summarized as follows: (The literature provided after the summary indicates adherence to the criteria presented in this document).

(1)

Standard uncertainty of the mid-section method 0.5%: Muste et al. [19].

(2)

Uncertainty related to the number of transverse sections 2.5%: ISO 1088 [17].

(3)

Uncertainty regarding measurement conditions 2.7%: Muste et al. [5].

Using the velocity measurement results from the surface image velocimeter and the cross-sectional area and depth survey results, the discharge was calculated. Based on this discharge calculation, the combined standard uncertainty of the discharge measurement was then estimated. The evaluated discharge measurement uncertainty using the surface image velocimeter, based on the GUM standard, resulted in a combined standard uncertainty of 0.112 m³/s. Dividing this by the measured discharge yields the relative discharge measurement uncertainty, which was found to be 6.04%.

4. Conclusions

This study developed surface velocity and discharge measurement uncertainty evaluation techniques for surface image velocimeters based on the GUM standard. Furthermore, it presented a framework for evaluating the velocity and discharge measurement uncertainty of surface image velocimeters and conducted such an evaluation at the River Experiment Center of the Korea Institute of Civil Engineering and Building Technology. A budget uncertainty analysis was also performed to analyze the factors influencing the uncertainty.

The conclusions derived from this study and future research directions are as follows:

• For uncertainty evaluation based on the GUM standard, the uncertainty factors of a surface image velocimeter were categorized into those that can be included in the functional equation of the surface velocity calculation and those that cannot. Factors that can be included in the velocity calculation equation are image displacement measurement and the calculation of the time interval between still images. Uncertainty factors not included in the velocity calculation equation include the analysis method, parameters, and image quality.
• Formulas for calculating the measurement uncertainty of surface velocity, depth-averaged velocity, and discharge based on the GUM standard were developed. Using the categorized uncertainty factors and the relationships between velocity and discharge, a method for evaluating the velocity and discharge measurement uncertainty of surface image velocimeters was developed, and standard uncertainty evaluation for each uncertainty factor was performed. Additionally, the standard uncertainty evaluation results were used to evaluate the velocity uncertainty of surface velocity and depth-averaged velocity and the discharge measurement uncertainty of the surface image velocimeter.
• It is expected that the discharge measurement uncertainty evaluation method for surface image velocimeters presented in this study can provide the reliability of discharge measurement results obtained from surface image velocimeters, thereby enhancing the applicability of surface image velocimeters and significantly expanding the versatility of the equipment.