https://orcid.org/0000-0001-9659-0452
ABSTRACT
River flow measurements are of great significance for flood prevention. Traditional immersion measurement methods not only consume manpower and material resources but are also greatly affected by the environment. Non-immersion image flow measurement methods have attracted widespread attention owing to their simplicity and safety. This paper proposes a flow measurement algorithm based on the scalar transport equation. Combining the scalar transport equation with fluid mechanics is more consistent with hydrodynamic characteristics and provides a better physical basis than other methods. Simultaneously, to calculate the two-dimensional velocity vector, it is assumed that the 3 × 3-pixel area at the speed measurement point exhibits exactly the same motion trend between two adjacent frames. Because only a few key speed measurement points are calculated, the calculation time of this method is significantly reduced compared with other methods. To verify the feasibility of the algorithm, two rivers at hydrological stations were selected to estimate their flow. The experimental results show that the total flow error of the proposed method is only 3.29 and 2.79% compared with the total flow measured by the current meter, which is a significant improvement over other methods.
HIGHLIGHTS
In fluid mechanics, based on the scalar transport equation, endow the model with better physical interpretability.
Assumption of small-scale motion consistency.
Describes the Eulerian motion of the flow field rather than Lagrangian motion.
Calculations are made more efficient by only a few specific speed measurement points.
INTRODUCTION
River flow measurements are crucial for predicting and managing floods. Accurate flood warning information can be provided by monitoring river flow changes in real-time, helping to take timely response measures and reduce losses caused by floods (Jiang et al. 2022). It is also crucial to protect and restore river ecosystems. By understanding flow changes, we can assess the impact of rivers on ecosystems, develop appropriate protection measures, and maintain river biodiversity and ecological balance. It is difficult to perform normal measurements during the flood season by using traditional immersion flow measurement methods. With the development of computer vision, nonimmersion image flow measurement methods have become more mature and efficient. Currently, image velocimetry methods are divided into two categories: particle and variational optical flow-based image velocimetry.
Particle image velocimetry (PIV) technique (Adrian 1991), as a classical method in the field of image velocimetry, is used to calculate the flow rate by dispersing tiny particles as tracers into the fluid and capturing the motion trajectories of these particles using a high-speed camera, which is then used to calculate the flow rate by image analysis techniques. However, the effectiveness of the PIV technique is highly dependent on the uniform distribution and proper size of the tracers, which is often challenging in practice. To address this limitation, Fujita et al. (1998) innovatively proposed large scale particle image velocimetry (LSPIV), which does not require the artificial addition of tracers but rather utilizes the natural texture of the river surface or floating objects (e.g., leaves and bubbles) as the markers of velocimetry, which avoids the potential impact of tracers on the environment and broadens the scope of velocimetry techniques. Nevertheless, LSPIV still faces the problems of high computational complexity and difficulty in real-time data processing.
To overcome this difficulty, Fujita et al. (2007) further developed the spatio-temporal image velocimetry (STIV) method, which calculates the flow velocity by setting up velocimetry lines parallel to the main flow direction in the river and utilizing the texture angle formed by the velocimetry lines in the consecutive frames of the image. STIV is well known for its simplicity and efficiency but is limited to the acquisition of one-dimensional flow velocity information, which makes it difficult to comprehensively reflect the 2D characteristics of the flow field and is susceptible to image noise interference. In order to improve the accuracy and robustness of velocimetry, researchers have explored a variety of image preprocessing and post-processing techniques, such as Hu et al. (2023), who proposed a sample-enhanced hybrid frequency domain dataset and improved MobileNet-STIV algorithm based on the sample-enhanced hybrid frequency domain dataset to optimize the flow velocity estimation through deep learning techniques.
Meanwhile, the velocimetry methods based on variable spectral flow have been developing continuously. The classical model of Horn & Schunck (1981) is based on the assumption of luminance constancy, which is mathematically concise and clear but lacks a physical basis, which restricts its application in real complex flow fields. In order to enhance the physical meaning of the model, researchers have attempted to replace the brightness constancy assumption with the mass conservation equation (Béréziat et al. 2000) or the continuity equation (Corpetti et al. 2000; Corpetti et al. 2002; Corpetti et al. 2003; Corpetti et al. 2006) and to incorporate higher-order regularization techniques, such as second-order scattering-rotation regularization, in order to better capture the scattering and rotational characteristics of the fluid. However, while these improvements improve the physical accuracy of the model, they also increase the computational complexity and demand for computational resources.
In order to balance the model accuracy and computational efficiency, Liu & Shen (2008) proposed the projected equations of motion for fluid visualization, which provides physical constraints by introducing the Navier–Stokes equations, but the resulting nonlinear equations are complicated to solve, and an approximate solution method is required for practical applications. Cassisa et al. (2011) and Zille et al. (2014) studies, on the other hand, focus on the combination of the large vortex simulation with the subgrid transport equation model to improve the ability to describe complex flow fields by considering the small-scale velocity components of turbulence, but the turbulent diffusion coefficients in the model still need to be selected empirically, which limits the generalization of the model.
To overcome these limitations, Cai et al. (2018) constructed a stochastic optical flow constraint equation based on the theoretical framework of Mémin (2014) by decomposing the fluid motion into a large-scale component and a small-scale component (positional uncertainty) and combining it with a stochastic expression of the Reynolds transport theorem, in which the parameters can be obtained by estimation without relying on empirical values. In addition, the grouped flow measurement method (FD-DIS-G) combining inter-frame differencing and fast dense optical flow proposed by Wang et al. (2022) effectively improves the accuracy and efficiency of the velocity measurement by calculating the motion saliency map and optimizing the dense optical flow displacement. Zach et al. (2007), in order to solve the problem of the discontinuity of the flow field in the variationally divergent optic flow, proposed a method based on the full-variation (TV) regularization and the data term in the robust L1 paradigm, which is more robust to illumination errors and outliers.
In recent years, with the rapid development of computational technology, the integration of variational optical flow algorithms with other advanced techniques has become a new research hotspot, such as orthogonal decomposition (Stapf & Garbe 2014), wavelet expansion with higher-order regularization (Dérian et al. 2013), optimal control scheme (Papadakis & Mémin 2008), and Bayesian stochastic filtering (Beyou et al. 2013), and the introduction of these methods has provided a richer and more flexible solution for flow velocity measurement, which further promotes non-immersive image flow measurement technology development and application.
The classical variational optical flow is not derived from physical derivation, without a certain physical basis, and is not robust to light changes. Other methods transform the data term and regular term of classical variational photocurrent, due to the need to seek the optimal solution of photocurrent for all pixel points of the whole image by iterating continuously, which leads to low computational efficiency and cannot be well calculated in real-time, the existing algorithms need to be improved. Based on the scalar transport equation, Batchelor (1967) can better conform to the hydrodynamic characteristics of the fluid, with certain physical interpretations, and only for a few characteristics of the velocimetry points, the computational efficiency is more efficient compared to other methods. Therefore, this study proposes a novel optical flow computation method based on the scalar transport equation, aiming to address the deficiencies of existing algorithms in terms of physical basis and computational efficiency. By incorporating the scalar transport equation, which has a clear physical interpretation, this method can better adapt to optical flow estimation under complex conditions and significantly enhance computational efficiency. It provides an innovative solution for real-time and efficient optical flow computation, paving the way for new possibilities in applications related to computer vision, fluid dynamics, and other relevant fields.
METHODS
Algorithmic principle
The classical variational optical flow method is based on the brightness constant assumption, in which the brightness of the same target remains constant between adjacent frames. The target's movement is small: the target's displacement between adjacent frames is very small; that is, the speed change rate is basically zero.
, 
represents the horizontal and vertical coordinates of the pixel and moves to 
at time 
. From the constant brightness assumption, we know that according to the brightness constant assumption.
formula (3) can be rewritten as:
is the partial derivative of the grayscale of the pixel in the image along the direction of 
direction. 
is the velocity vector of optical flow along the x and y axes. At this time, there are two unknown variables but only one constraint equation. This is called the aperture problem in optical flow. A second assumption was introduced to solve this problem: the velocity change rate of the target is zero. That is:
denotes the weight coefficient at which the equation is solved to obtain the minimum solution.
is the Hamiltonian operator, and 
represents the direction vectors along the 
axes. 
is the velocity vector and 
is the Laplace operator. Where 
and 
denote the Reynolds and Schmidt numbers, respectively.
to estimate cloud motion, where Z is the observation depth. In a two-dimensional scalar image, the observation depth Z is zero. Therefore, in 
, Equation (9) can be rewritten as:
it is related to the first-order difference, as shown in the following equations:
. In order to solve this problem, the condition is set that the 3 × 3-pixel area around the calculation point and it have the same motion, as shown in Figure 2. Therefore, 9-pixel points within the 3 × 3-pixel window in the range of the calculation point are selected to be associated, and the equation is established, as shown in the following equation:
Schematic diagram of the motion consistency assumption.
, 
, 
, and after matrix operation, we can get:
Projective transformations and displacement transformations
. 
Ground calibration and image coordinate conversion.
, establish a one-to-one correspondence between 
and 
, and utilize the principle of projection transformation to find the transformation matrix 
, and the pixel point coordinates 
correspond to the points 
in the actual coordinates, whose relationship is:
, i.e.:
The initial coordinates 
of the velocimetry points on the section line can be accurately mapped to their corresponding positions 
in the real environment by applying a projection transformation matrix p. When the optical flow information is updated, the pixel coordinates of these velocimetry points on the image change accordingly to 
. Subsequently, these updated pixel coordinates are again transformed back to the coordinates in the actual environment using the projection transformation matrix 
. After this conversion process is complete, the actual displacement of each velocimeter point between two consecutive frames can be easily calculated 
. This displacement value is obtained by comparing the difference in the actual coordinates of the velocimeter point in the two frames before and after. Next, in order to find the surface flow velocity of each velocimetry point as well as the plumb line average flow velocity, it is only necessary to divide this displacement value by the time interval between the two neighboring frames.
Calculation of average river flow rate and total flow
, 
, n is the number of velocity points on the cross-section, the cross-sectional area corresponding to the velocity points 
to i is 
, and the average surface velocity between velocity points 
to i is 
:
is the water depth corresponding to the velocity point i, 
are the shore depths, 
is the distance between points i and 
, 
is the distance from the velocity point n to the shore, and m is the shore coefficient. The relationship between the plumbline mean flow velocity and the surface flow velocity is:
is the vertical average velocity and k is the surface velocity coefficient. According to the velocity-area method (River Flow Test Specification 2015), the flow rate of the water-passing section 
can be obtained as follows:
Schematic diagram of river cross-section.
of the entire river can be obtained from the river flow and area, as follows:
RESULTS
Experimental platform
The processor of the experimental equipment was Intel(R) Core(TM) i7-6700K, and the operating system was Windows 10 (X64), which runs Python 3. 8 and opencv4. 2.
Evaluation indicators
Working diagram of the current meter.
In order to comprehensively evaluate the accuracy of different methods in measuring the plumb mean flow rate, the total flow rate, and the mean flow rate in this cross-section. Root mean square error (RMSE) and relative error were selected as evaluation metrics (Alakbar & Burgan 2024).
, which is recorded as dataset A. The other is the result of the vertical mean flow rate at the corresponding velocity point measured by the standard instrument LJ-20 
, which is recorded as dataset B. By calculating the RMSE between datasets A and B, the accuracy and reliability of the measurement of vertical mean flow rate by different methods can be quantified. By calculating the RMSE between dataset A and dataset B, it is possible to quantify the degree of deviation of different methods from the standardized values of the mean vertical flow velocity measurements and to evaluate their accuracy. The RMSE is a commonly used metric for assessing the accuracy of predictive models. The calculation of RMSE involves squaring the difference between each pair of measurements and the true value of the current meter, then averaging these squared values, and finally taking the square root. The formula for RMSE is shown in the following equation:
For measurements of mean flow rate and total cross-section flow, the relative error is used as a key indicator to assess the performance of different methods. The relative error is obtained by calculating the difference between the measured value and the true value (or standard value) and dividing by the true value (or standard value), which visualizes the extent to which the measured value deviates from the true value. By comparing the relative errors between the average flow rate and the total cross-section flow rate measured by different methods and the standard values, we can gain insight into the reliability and accuracy of these methods.
Experimental design and analysis
Rivers of rules
In order to verify the feasibility of the algorithm, the artificial nullah section of Dali Train City Hydrological Station in Yunnan Province was selected for velocity measurement. According to the years of actual measurement experience of this hydrological station, the bank coefficient is 0.80 and the surface flow velocity coefficient is 0.82. The camera model is Hikvision DS-2CD1225-I3/I5 with a focal length of 6 mm. The camera is 20.3 m from point A, 32.4 m from point B, 16.2 m from point C, and 6.3 m from point D The experimental video was shot for 10 s, cut into 325 frames, and captured the image resolution of the video at 1,920 × 1,080 pixels.
Current meter measurement results
| Starting point distance b/m | Vertical average velocity/(m·s−1) | Partial average flow velocity /(m·s−1) | Partial area/(m2) | Partial flow/(m3·s−1) |
|---|---|---|---|---|
| 0 (shore) | 0 | – | – | – |
| 0–2 | – | 0.29 | 1.45 | 0.42 |
| 2 | 0.36 | – | – | – |
| 2–3 | – | 0.40 | 0.85 | 0.34 |
| 3 | 0.43 | – | – | – |
| 3–4 | – | 0.39 | 0.92 | 0.36 |
| 4 | 0.35 | – | – | – |
| 4–5 | – | 0.40 | 0.99 | 0.40 |
| 5 | 0.45 | – | – | – |
| 5–6 | – | 0.60 | 0.98 | 0.59 |
| 6 | 0.76 | – | – | – |
| 6–7 | – | 0.72 | 0.95 | 0.68 |
| 7 | 0.68 | – | – | – |
| 7–8 | – | 0.64 | 0.80 | 0.51 |
| 8 | 0.61 | – | – | – |
| 8–9 | – | 0.54 | 0.77 | 0.42 |
| 9 | 0.48 | – | – | – |
| 9–10 | – | 0.43 | 0.86 | 0.37 |
| 10 | 0.38 | – | – | – |
| 10–11 | – | 0.44 | 0.71 | 0.31 |
| 11 | 0.49 | – | – | – |
| 11–11.9 | – | 0.39 | 0.42 | 0.16 |
| 11.9 (shore) | 0 | – | – | – |
| Starting point distance b/m | Vertical average velocity/(m·s−1) | Partial average flow velocity /(m·s−1) | Partial area/(m2) | Partial flow/(m3·s−1) |
|---|---|---|---|---|
| 0 (shore) | 0 | – | – | – |
| 0–2 | – | 0.29 | 1.45 | 0.42 |
| 2 | 0.36 | – | – | – |
| 2–3 | – | 0.40 | 0.85 | 0.34 |
| 3 | 0.43 | – | – | – |
| 3–4 | – | 0.39 | 0.92 | 0.36 |
| 4 | 0.35 | – | – | – |
| 4–5 | – | 0.40 | 0.99 | 0.40 |
| 5 | 0.45 | – | – | – |
| 5–6 | – | 0.60 | 0.98 | 0.59 |
| 6 | 0.76 | – | – | – |
| 6–7 | – | 0.72 | 0.95 | 0.68 |
| 7 | 0.68 | – | – | – |
| 7–8 | – | 0.64 | 0.80 | 0.51 |
| 8 | 0.61 | – | – | – |
| 8–9 | – | 0.54 | 0.77 | 0.42 |
| 9 | 0.48 | – | – | – |
| 9–10 | – | 0.43 | 0.86 | 0.37 |
| 10 | 0.38 | – | – | – |
| 10–11 | – | 0.44 | 0.71 | 0.31 |
| 11 | 0.49 | – | – | – |
| 11–11.9 | – | 0.39 | 0.42 | 0.16 |
| 11.9 (shore) | 0 | – | – | – |
Schematic diagram of the surface calibration points of the open channel.
In order to verify the superiority of this method over other methods, the LSPIV method (Fujita et al. 1998), STIV method (Fujita et al. 2007), and FD-DIS-G method (Wang et al. 2022) were selected to compare with the new-method proposed in this paper. The implementation steps of several other algorithms are shown as follows:
Specific steps of the LSPIV method: the captured video image is affine transformed to a bird's-eye view. Grayscaling and Contrast Limited Adaptive Histogram Equalization (CLAHE) image enhancement are performed to enhance the texture of the image. Subsequently, flow field intercorrelation calculation is performed using the LSPIV algorithm. Finally, the pixel displacements of the velocimetry points are extracted and transformed into actual displacements and actual surface flow velocities by means of the projection transformation matrix.
Specific steps of the STIV method: the captured video image is subjected to CLAHE image enhancement, the corresponding velocity lines are set at the velocimetry points, and then the spatio-temporal image is generated. Finally, the generated spatio-temporal image is subjected to the gradient tensor method for texture angle calculation. It is converted to actual surface flow velocity by equations.
Specific steps of the FD-DIS-G method: the images of the two frames before and after are processed for frame difference calculation. A threshold is set, and the regions with frame difference value greater than the threshold are processed by the method to get the motion saliency map. Determine the block region calculation centered on the velocimetry point. Perform the grid gray value difference squared and minimize the searched optical flow vectors. Finally, anomalous data are processed. The optical flow vectors are finally converted to actual surface flow velocities.
The flow measurement results of the LSPIV method, STIV method, FD-DIS-G method, and this method are shown in Table 2.
Comparison of perpendicular average flow velocity results measured by different methods
| Method comparison | Current meter | LSPIV | STIV | FD-DIS-G | New-method | |||||
|---|---|---|---|---|---|---|---|---|---|---|
| Measurement value (m/s) | Measurement value (m/s) | Relative error (%) | Measurement value (m/s) | Relative error (%) | Measurement value (m/s) | Relative error (%) | Measurement value (m/s) | Relative error (%) | ||
| Distance (m) | d = 2 m | 0.36 | 0.23 | 36.11 | 0.38 | 5.56 | 0.33 | 8.33 | 0.31 | 13.89 |
| d = 3 m | 0.43 | 0.01 | 97.67 | 0.46 | 6.98 | 0.41 | 4.65 | 0.31 | 27.91 | |
| d = 4 m | 0.35 | 0.45 | 28.57 | 0.03 | 91.43 | 0.45 | 28.57 | 0.44 | 25.71 | |
| d = 5 m | 0.45 | 0.58 | 28.89 | 0.45 | 0.00 | 0.52 | 15.56 | 0.55 | 10.22 | |
| d = 6 m | 0.76 | 0.38 | 50.00 | 0.13 | 82.89 | 0.46 | 39.47 | 0.53 | 30.26 | |
| d = 7 m | 0.68 | 0.41 | 39.71 | 0.79 | 16.18 | 0.51 | 25.00 | 0.59 | 13.23 | |
| d = 8 m | 0.61 | 0.57 | 6.56 | 0.72 | 18.03 | 0.54 | 11.48 | 0.71 | 16.39 | |
| d = 9 m | 0.48 | 0.62 | 29.17 | 0.72 | 50.00 | 0.75 | 56.25 | 0.71 | 47.92 | |
| d = 10 m | 0.38 | 0.37 | 2.63 | 0.61 | 60.53 | 0.68 | 78.95 | 0.83 | 118.42 | |
| d = 11 m | 0.49 | 0.62 | 26.53 | 0.40 | 18.37 | 0.64 | 30.61 | 0.25 | 48.98 | |
| Average speed(m/s) | 0.47 | 0.39 | 17.02 | 0.43 | 12.77 | 0.49 | 4.26 | 0.49 | 4.26 | |
| Average flow(m3/s) | 4.56 | 3.77 | 17.32 | 4.21 | 7.68 | 4.75 | 4.17 | 4.71 | 3.29 | |
| Method comparison | Current meter | LSPIV | STIV | FD-DIS-G | New-method | |||||
|---|---|---|---|---|---|---|---|---|---|---|
| Measurement value (m/s) | Measurement value (m/s) | Relative error (%) | Measurement value (m/s) | Relative error (%) | Measurement value (m/s) | Relative error (%) | Measurement value (m/s) | Relative error (%) | ||
| Distance (m) | d = 2 m | 0.36 | 0.23 | 36.11 | 0.38 | 5.56 | 0.33 | 8.33 | 0.31 | 13.89 |
| d = 3 m | 0.43 | 0.01 | 97.67 | 0.46 | 6.98 | 0.41 | 4.65 | 0.31 | 27.91 | |
| d = 4 m | 0.35 | 0.45 | 28.57 | 0.03 | 91.43 | 0.45 | 28.57 | 0.44 | 25.71 | |
| d = 5 m | 0.45 | 0.58 | 28.89 | 0.45 | 0.00 | 0.52 | 15.56 | 0.55 | 10.22 | |
| d = 6 m | 0.76 | 0.38 | 50.00 | 0.13 | 82.89 | 0.46 | 39.47 | 0.53 | 30.26 | |
| d = 7 m | 0.68 | 0.41 | 39.71 | 0.79 | 16.18 | 0.51 | 25.00 | 0.59 | 13.23 | |
| d = 8 m | 0.61 | 0.57 | 6.56 | 0.72 | 18.03 | 0.54 | 11.48 | 0.71 | 16.39 | |
| d = 9 m | 0.48 | 0.62 | 29.17 | 0.72 | 50.00 | 0.75 | 56.25 | 0.71 | 47.92 | |
| d = 10 m | 0.38 | 0.37 | 2.63 | 0.61 | 60.53 | 0.68 | 78.95 | 0.83 | 118.42 | |
| d = 11 m | 0.49 | 0.62 | 26.53 | 0.40 | 18.37 | 0.64 | 30.61 | 0.25 | 48.98 | |
| Average speed(m/s) | 0.47 | 0.39 | 17.02 | 0.43 | 12.77 | 0.49 | 4.26 | 0.49 | 4.26 | |
| Average flow(m3/s) | 4.56 | 3.77 | 17.32 | 4.21 | 7.68 | 4.75 | 4.17 | 4.71 | 3.29 | |
From Table 3, we know that PCCs between the other methods and the current meter measurements are less than 0.2, while the PCC between the new-method and the current meter measurements is 0.213, which shows a stronger correlation than the other methods, and it is the most relevant to the real flow velocity distribution law compared with the other methods. For the RMSE, although the RMSE of this method is 0.205, it is larger than that of the FD-DIS-G method. But compared to the other two methods, the RMSE is reduced.
Comparison of PCC and RMSE of different methods at the Dali station
| Method | Pearson coefficient | Root mean square error |
|---|---|---|
| LSPIV | 0.193 | 0.218 |
| STIV | 0.176 | 0.254 |
| FD-DIS-G | 0.028 | 0.180 |
| New-method | 0.213 | 0.205 |
| Method | Pearson coefficient | Root mean square error |
|---|---|---|
| LSPIV | 0.193 | 0.218 |
| STIV | 0.176 | 0.254 |
| FD-DIS-G | 0.028 | 0.180 |
| New-method | 0.213 | 0.205 |
Vertical average flow velocity and open channel depth.
As for the mean flow velocity and flow rate calculation in the flow field, it can be seen from Table 2 that the present method is more accurate relative to several other methods. The relative error is only 3.29% relative to the true value of the current measured by the flow meter. The accuracy is improved by 14.03 and 4.39% relative to the LSPIV and STIV methods, respectively. Although the plumb line means velocity measured by this method fluctuates more at the three velocimetry points near the endpoint, it has the advantage of better accuracy in terms of the overall flow field flow and velocity.
Comparison of relative error and running time.
Natural river
In order to verify the universality of the method in this paper, a natural river at Meng provincial station in Yunnan province was selected for speed measurement comparison. According to the years of experience of this hydrological station, the bank coefficient is 0.70, and the surface velocity coefficient is 0.90. The camera model is Hikvision DS-2CD1225-I3/I5 with a focal length of 6 mm. The camera is 39.4 m from point A, 35.8 m from point B, 80.3 m from point C, and 85.6 m from point D. The experimental video was shot for 20 seconds, cut into 500 frames, and captured the image resolution of the video at 1,920 × 1,080 pixels.
Current meter measurement results
| Starting point distance (b/m) | Vertical average velocity/(m·s−1) | Partial average flow velocity/(m·s−1) | Partial area/(m2) | Partial flow/(m3 ·s−1) |
|---|---|---|---|---|
| 18.1 (shore) | 0 | – | – | – |
| 21–24 | – | 0.32 | 7.13 | 2.28 |
| 24 | 0.45 | – | – | – |
| 24–28 | – | 0.70 | 7.72 | 5.40 |
| 28 | 0.96 | – | – | – |
| 28–32 | – | 1.12 | 7.84 | 8.78 |
| 32 | 1.28 | – | – | – |
| 32–36 | – | 1.34 | 8.04 | 10.80 |
| 36 | 1.40 | – | – | – |
| 36–40 | – | 1.39 | 7.60 | 10.60 |
| 40 | 1.38 | – | – | – |
| 40–44 | – | 1.38 | 7.36 | 10.20 |
| 44 | 1.38 | – | – | – |
| 44–48 | – | 1.36 | 7.24 | 9.85 |
| 48 | 1.35 | – | – | – |
| 48–52 | – | 1.38 | 7.36 | 10.20 |
| 52 | 1.40 | – | – | – |
| 52–56 | – | 1.30 | 7.76 | 10.10 |
| 56 | 1.21 | – | – | – |
| 56–60 | – | 0.92 | 7.68 | 7.07 |
| 60 | 0.63 | – | – | – |
| 60–63 | – | 0.44 | 6.63 | 2.92 |
| 64.5 (shore) | 0 | – | – | – |
| Starting point distance (b/m) | Vertical average velocity/(m·s−1) | Partial average flow velocity/(m·s−1) | Partial area/(m2) | Partial flow/(m3 ·s−1) |
|---|---|---|---|---|
| 18.1 (shore) | 0 | – | – | – |
| 21–24 | – | 0.32 | 7.13 | 2.28 |
| 24 | 0.45 | – | – | – |
| 24–28 | – | 0.70 | 7.72 | 5.40 |
| 28 | 0.96 | – | – | – |
| 28–32 | – | 1.12 | 7.84 | 8.78 |
| 32 | 1.28 | – | – | – |
| 32–36 | – | 1.34 | 8.04 | 10.80 |
| 36 | 1.40 | – | – | – |
| 36–40 | – | 1.39 | 7.60 | 10.60 |
| 40 | 1.38 | – | – | – |
| 40–44 | – | 1.38 | 7.36 | 10.20 |
| 44 | 1.38 | – | – | – |
| 44–48 | – | 1.36 | 7.24 | 9.85 |
| 48 | 1.35 | – | – | – |
| 48–52 | – | 1.38 | 7.36 | 10.20 |
| 52 | 1.40 | – | – | – |
| 52–56 | – | 1.30 | 7.76 | 10.10 |
| 56 | 1.21 | – | – | – |
| 56–60 | – | 0.92 | 7.68 | 7.07 |
| 60 | 0.63 | – | – | – |
| 60–63 | – | 0.44 | 6.63 | 2.92 |
| 64.5 (shore) | 0 | – | – | – |
Schematic diagram of ground calibration and speed measurement line layout at the Mengxing Station.
Schematic diagram of ground calibration and speed measurement line layout at the Mengxing Station.
When comparing the algorithms for the natural river, the measurements revealed that the flow accuracy measured by the LSPIV algorithm and the FD-DIS-G algorithm was not high in this river, so the MobileNet-STIV algorithm was re-selected for comparison. The MobileNet-STIV algorithm flow is shown below (Hu et al. 2023):
Specific steps of the MobileNet-STIV method: a hybrid dataset is generated from the synthetic image dataset and the video-synthesized image dataset. The dataset is generated using the Multi-Scale Retinex method for image enhancement.
Then the Gaussian high-pass filtering process is performed to improve the quality of the samples and train the network. Corresponding speed lines are set at the speed points, and subsequently, spatio-temporal images are generated. The generated spatio-temporal images are utilized for angle prediction using the trained prediction network. Finally, it is converted into actual surface flow velocity.
The comparison of the measurement results of this method and the measurement results of other methods is shown in Table 5.
Comparison of perpendicular average flow velocity results measured by different methods
| Method comparison | Current meter | STIV | MobileNet-STIV | New-method | ||||
|---|---|---|---|---|---|---|---|---|
| Measurement value (m/s) | Measurement value (m/s) | Relative error (%) | Measurement value (m/s) | Relative error (%) | Measurement value (m/s) | Relative error (%) | ||
| Distance (m) | d = 24 m | 0.45 | 0.21 | 54.41 | 0.42 | 6.02 | 0.85 | 88.89 |
| d = 28 m | 0.96 | 1.20 | 24.72 | 0.79 | 17.60 | 1.22 | 27.08 | |
| d = 32 m | 1.28 | 0.88 | 31.28 | 0.93 | 27.60 | 1.31 | 2.34 | |
| d = 36 m | 1.40 | 0.85 | 39.62 | 1.04 | 25.70 | 1.22 | 12.86 | |
| d = 40 m | 1.38 | 0.78 | 43.50 | 1.10 | 19.99 | 1.32 | 4.35 | |
| d = 44 m | 1.38 | 1.46 | 6.03 | 1.09 | 20.76 | 1.41 | 2.17 | |
| d = 48 m | 1.35 | 2.04 | 51.28 | 1.20 | 11.06 | 1.19 | 11.85 | |
| d = 52 m | 1.40 | 1.05 | 26.97 | 1.53 | 9.28 | 1.31 | 6.43 | |
| d = 56 m | 1.21 | 1.23 | 1.55 | 1.07 | 11.16 | 0.89 | 26.45 | |
| d= 60 m | 0.63 | 0.42 | 33.98 | 0.95 | 50.79 | 0.93 | 47.62 | |
| Average speed (m/s) | 1.07 | 0.94 | 12.60 | 0.95 | 11.13 | 1.10 | 2.80 | |
| Average flow (m3/s) | 88.20 | 77.02 | 12.67 | 78.32 | 11.21 | 90.66 | 2.79 | |
| Method comparison | Current meter | STIV | MobileNet-STIV | New-method | ||||
|---|---|---|---|---|---|---|---|---|
| Measurement value (m/s) | Measurement value (m/s) | Relative error (%) | Measurement value (m/s) | Relative error (%) | Measurement value (m/s) | Relative error (%) | ||
| Distance (m) | d = 24 m | 0.45 | 0.21 | 54.41 | 0.42 | 6.02 | 0.85 | 88.89 |
| d = 28 m | 0.96 | 1.20 | 24.72 | 0.79 | 17.60 | 1.22 | 27.08 | |
| d = 32 m | 1.28 | 0.88 | 31.28 | 0.93 | 27.60 | 1.31 | 2.34 | |
| d = 36 m | 1.40 | 0.85 | 39.62 | 1.04 | 25.70 | 1.22 | 12.86 | |
| d = 40 m | 1.38 | 0.78 | 43.50 | 1.10 | 19.99 | 1.32 | 4.35 | |
| d = 44 m | 1.38 | 1.46 | 6.03 | 1.09 | 20.76 | 1.41 | 2.17 | |
| d = 48 m | 1.35 | 2.04 | 51.28 | 1.20 | 11.06 | 1.19 | 11.85 | |
| d = 52 m | 1.40 | 1.05 | 26.97 | 1.53 | 9.28 | 1.31 | 6.43 | |
| d = 56 m | 1.21 | 1.23 | 1.55 | 1.07 | 11.16 | 0.89 | 26.45 | |
| d= 60 m | 0.63 | 0.42 | 33.98 | 0.95 | 50.79 | 0.93 | 47.62 | |
| Average speed (m/s) | 1.07 | 0.94 | 12.60 | 0.95 | 11.13 | 1.10 | 2.80 | |
| Average flow (m3/s) | 88.20 | 77.02 | 12.67 | 78.32 | 11.21 | 90.66 | 2.79 | |
Analyzing the three methods in Table 5, it can be seen that the method proposed in this paper has the smallest relative error of 2.80% among the three methods for the measured mean flow velocity. The relative error is 2.80%, which is 9.80 and 8.33% higher than that of the STIV and MobileNet-STIV methods, respectively. The flow rate value measured by the proposed method is the closest to that measured by the current meter among the three methods, with a relative error of only 2.79%, compared with the STIV and MobileNet-STIV methods. The accuracy was improved by 9.88 and 8.42%, respectively.
From Table 6, it can be learned that the flow velocity calculated by this method has a high correlation with the flow velocity measured by the current meter. The Pearson's coefficient is 0.767, while the correlation of the results calculated by the STIV method is only 0.648. Although the correlation of the results calculated by MobileNet-STIV is slightly higher than that of the present method in the RMSE analysis. The present method exhibits less errorability and has good stability.
Comparison of PCC and RMSE of different methods at the Mengxing Station
| Method | Pearson coefficient | RMSE |
|---|---|---|
| STIV | 0.648 | 0.398 |
| MobileNet-STIV | 0.778 | 0.246 |
| New-method | 0.767 | 0.222 |
| Method | Pearson coefficient | RMSE |
|---|---|---|
| STIV | 0.648 | 0.398 |
| MobileNet-STIV | 0.778 | 0.246 |
| New-method | 0.767 | 0.222 |
Vertical average flow velocity and open channel depth.
Comparison of relative error and running time.
CONCLUSIONS
In this paper, we propose a river velocimetry method based on the scalar transport equation and assume that the 3 × 3 region around the velocimetry point has exactly the same motion trend between two frames to solve the aperture problem in the optical flow problem. Unlike the LSPIV and STIV methods, the above two methods aim to characterize the Lagrangian trajectories of objects in a river. In contrast, this paper is concerned with the distribution of the flow field velocity in a particular region at different times. It is more in line with the law that the velocity of the flow field is constantly changing with time.
After obtaining the surface flow velocity, the total flow rate was obtained by the surface flow velocity coefficient, the shore coefficient, and the section information, and the average flow rate was obtained by dividing the total flow rate by the total section area. Finally, the error of the total flow rate and the total average flow rate are compared. Through the first set of experiments, it was found that the measured values of the mean river flow velocity and river flow were closest to those of the current meter, with relative errors of 4.26 and 3.29%, respectively. Compared with the LSPIV method, which is widely used in hydrological stations, the accuracy of this method is improved by 12.76 and 14.03%. Through the second set of experimental results, it was found that the relative errors of the measured mean river flow and river flow values were 2.80 and 2.79%, respectively, and the measurement accuracies were higher than those of the STIV and MobileNet-STIV algorithms. During the process of projection transformation. The selection of the calibration points, the position of the camera can lead to different results in the calculation of the projection transformation matrix, which will have an impact on the experimental results. However, there can be multiple sources of data errors. For methods, such as LSPIV and FD-DIS-G, there are also reasons for the error such as unclear surface texture of the water flow. For methods such as STIV, the selection of the velocimetry line can also directly lead to different errors. The superimposed effect between the errors leads to the final data results.
Since the measurement method proposed in this paper only calculates the flow velocity for a specific region, the operation speed and accuracy are greatly improved. The accuracy and efficiency of the measurement are improved while reducing the cost, which can meet the monitoring needs of the river at hydrological stations.
However, the method has some limitations. Since the algorithm itself relies on clear and unobstructed images. The cases where the light intensity is not high at night and rainy and cloudy days make the image blurry are not resolved. The next step in the work will be to consider how to solve the problem of effectively restoring the flow field information in the image and solving for the velocity field in harsh environments. We will also consider how to put the algorithm into practical applications. To establish an efficient river flow monitoring system and make some contributions to the field of hydrology.
ACKNOWLEDGEMENTS
The authors would like to express sincere gratitude to the Yunnan Provincial Bureau of Hydrological and Water Resources for providing the necessary data to complete this research work.
FUNDING
This research was supported by the National Natural Science Foundation of China (No. 62363017) and the ‘Yunnan Xingdian Talents Support Plan’ Project (No. KKXY202203006).
DATA AVAILABILITY STATEMENT
All relevant data are included in the paper or its Supplementary Information.
CONFLICT OF INTEREST
The authors declare there is no conflict.
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