Efficiency in river discharge measurement: combining Chiu’s method with particle image velocimetry techniques

Eficiência na medição de vazão em rios: combinando o método de Chiu com técnicas de velocimetria por imagem de partículas

ABSTRACT

This study evaluated the efficiency of combining Large-Scale Particle Image Velocimetry (LSPIV) with Chiu’s entropy method to estimate river discharge in the Brazilian semi-arid region. Measurements were conducted at six cross-sections of the Acaraú River basin, which covers 10% of the state of Ceará, using Acoustic Doppler Velocimetry (ADV) as the reference method. LSPIV was applied using smartphone-recorded videos, processed with the RIVeR, PIVlab, and PTVlab software. In the second approach, LSPIV was combined with the entropy method, considering the maximum surface velocity and calculating the entropy coefficient (ϕ) for each section. Central sections showed lower error values, while camera positioning directly influenced the results. The ϕ coefficient proved to be unstable, indicating the need for further calibration. Compared to ADV, Method II (LSPIV + Entropy) presented deviations ranging from 2.97% to -33.33%, with a mean absolute error of 20.7%. Method III (LSPIV + Entropy + Elliptical Distribution) showed deviations of up to -45.34%, with a mean absolute error of 23%. Despite the variability, the combination of LSPIV and entropy proved to be a promising, low-cost, and non-intrusive alternative for discharge monitoring in small to medium-sized rivers located in semi-arid regions.

Keywords:
Flow velocity; LSPIV; Entropy; Semi-arid; Watershed management

Resumo

Este estudo avaliou a eficiência da combinação entre a Velocimetria por Imagem de Partículas em Grande Escala (LSPIV) e o método de entropia de Chiu para estimar a vazão em rios na região semiárida brasileira. As medições foram realizadas em seis seções transversais da bacia do rio Acaraú, que abrange 10% do estado do Ceará, utilizando a Velocimetria Acústica por Efeito Doppler (ADV) como método de referência. A LSPIV foi aplicada por meio de vídeos gravados com smartphone, processados com os softwares RIVeR, PIVlab e PTVlab. Na segunda abordagem, a LSPIV foi combinada com o método de entropia, considerando a velocidade máxima de superfície e calculando-se o coeficiente de entropia (ϕ) para cada seção. As seções centrais apresentaram menores valores de erro, enquanto o posicionamento da câmera influenciou diretamente os resultados. O coeficiente ϕ mostrou-se instável, indicando a necessidade de calibrações adicionais. Comparado à ADV, o Método II (LSPIV + Entropia) apresentou desvios entre 2.97% e -33.33%, com erro absoluto médio de 20.7%. O Método III (LSPIV + Entropia + Distribuição Elíptica) apresentou desvios de até -45.34%, com erro absoluto médio de 23%. Apesar da variabilidade, a combinação entre LSPIV e entropia demonstrou ser uma alternativa promissora, de baixo custo e não intrusiva para o monitoramento de vazão em rios de pequeno a médio porte localizados em regiões semiáridas.

Palavras-chave:
Velocidade do fluxo; LSPIV; Entropia; Semiárido; Gestão de bacias hidrográficas

INTRODUCTION

Intermittent rivers represent more than 50% of the global river network, including small-order streams (Messager et al., 2021). These water bodies are generally located in arid or semi-arid regions, making them more vulnerable in terms of meeting multiple waters demands. In the Brazilian semi-arid region, this vulnerability is exacerbated by the gradual increase in water scarcity, driven by reduced water availability (Araújo & Bronstert, 2016; Figueiredo et al., 2016), growing water demand (Medeiros & Sivapalan, 2020), and climate change (Gondim et al., 2018). In this region, intermittent rivers play a crucial role in hydrological connectivity for reservoir recharge, which forms a complex water supply structure. These natural open channels are also important in regulating flow for supplying scattered populations through a dense network of reservoirs (Mamede et al., 2012).

In this context, monitoring the hydrological variables of rivers in regions where water availability is limited must be even more rigorous. Streamflow is one of the most important hydrological variables in a watershed (Uliana et al., 2015). According to Depetris (2021), monitoring streamflow in rivers is an important tool for interpreting the changes occurring in drainage basins and river transit losses. The modernization of river flow monitoring networks should occur, using techniques such as improving precipitation-runoff modeling (Anderson & McDonnell, 2005), remote sensing, Doppler acoustic instrumentation (Fulton & Ostrowski, 2008; Herschy, 2008; Merz, 2010; Morgenschweis, 2010; Gravelle, 2015; Welber et al., 2016; Moramarco et al., 2017), particle tracking velocimetry (PTV) (Tauro et al., 2014; Tauro & Salvatori, 2017) or large-scale particle image velocimetry (LSPIV) (Fujita et al., 1998; Lee et al., 2010; Huang et al., 2018; Hutley et al., 2023).

These alternatives are especially valuable for hydrometeorological networks with limited infrastructure, as they allow for the supplementation of flow data using low-cost cameras (Muste et al., 2011; Dal Sasso et al., 2018; Perks et al., 2020) or drones (Detert et al., 2017; Dal Sasso et al., 2018, 2021; Eltner et al., 2020). The accuracy of LSPIV in discharge estimation directly depends on the correct characterization of the river cross-section. Geometric irregularities, morphological changes during floods, and errors in water level measurement affect the conversion of surface velocity into mean velocity, resulting in uncertainties in the estimated discharge (Fujita et al., 1998; Muste et al., 2008).

In the study by Hutley et al. (2023), discharge measurements using surface velocimetry (Hydro-STIV) showed relative errors ranging from 5% to 15%, which were reduced to 5% to 10% with the use of machine learning. Similarly, Lee et al. (2010) obtained estimates with a maximum error of 5% using an experimental video system with LSPIV. These studies emphasized that the accuracy of the technique strongly depends on the quality of the captured images, being affected by factors such as inadequate lighting, reflections, shadows, low surface texture, morphological changes in the cross-section, as well as camera positioning and the distribution of control points. Image preprocessing is also essential to correct distortions and enhance tracer detection, even when advanced processing techniques are used.

Moreover, maximum entropy algorithms have emerged as an effective solution for improving river flow estimation (Farina et al., 2014; Vyas et al., 2021). The concept of entropy is widely applied to predict velocity distributions and other parameters in open channel flow (Chiu, 1989; Sterling & Knight, 2002). Studies such as those by Moramarco & Singh (2010) have explored the relationship between the entropy parameter and the geometric and hydraulic characteristics of channels. The Chiu method relates the hydrodynamic characteristics of rivers to an entropy coefficient, providing a physical-mathematical connection with image velocimetry. In this sense, combining the maximum entropy method with surface velocity measurement techniques using images should offer a promising approach to streamflow measurement, especially in regions like the Brazilian semi-arid, where there are still few studies on the topic.

The method developed by Chiu (1989) is an innovative approach for modeling velocity distribution in open channels, based on a probabilistic framework combined with the maximum entropy concept and cross-sectional geometry data. The velocity distribution is expressed as a logarithmic function of a curvilinear coordinate, which allows the adjustment of different velocity profile patterns (Chiu & Said, 1995; Fulton & Ostrowski, 2008; Moramarco et al., 2019; Vyas et al., 2021). The method also includes a parameter fitting technique that combines the theoretical model with experimental data, enabling efficient calibration even with limited data.

According to Chiu (1988), in wide rivers where the flow can be treated as one-dimensional, the maximum velocity typically occurs at the water surface (Araújo & Chaudhry, 1998; Chiu & Tung, 2002; Corato et al., 2011; Moramarco et al., 2017; Araújo; Simões; Porto, 2022). In narrow rivers, however, the maximum velocity tends to occur below the water surface. This difference results from the combined influence of boundary roughness, channel slope, and cross-sectional shape, factors that directly affect secondary currents and velocity distribution. Thus, the application of the Chiu method can enhance the quality of data acquisition processes, reducing data collection time and providing more accurate and reliable measurements. Although LSPIV requires good visibility and appropriate lighting conditions, it is considered a relatively simple and low-cost technique for measuring surface velocity in rivers (Muste et al., 2011). Therefore, the use of LSPIV as a discharge measurement method for river monitoring is highly promising.

Therefore, the aim was to evaluate the efficiency of combining Large-Scale Particle Image Velocimetry (LSPIV) with Chiu’s entropy method (Chiu, 1988), considering two different approaches, both compared to a reference method based on measurements using ADV devices.

MATERIALS AND METHODS

Study area

Located in the semi-arid region of Brazil, the Acaraú River Basin (ARB) extends over approximately 14,000 km², accounting for 10% of the territory of the state of Ceará. With 15 strategic reservoirs, the basin supplies the water demands of 28 municipalities (Ceará, 2009). The ARB's annual average flow rates ranged from 0.223 m³/s to 8,191 m³/s. With a coefficient of variation of 160.25% among the analyzed flow rates, the basin exhibits high spatial variability in discharge.

The climate of the ARB Hydrographic Region is predominantly warm throughout the year, with high temperatures and low thermal amplitudes, as well as an irregular rainfall regime and high solar radiation, which contributes to high evaporation rates. According to the Köppen climate classification in Brazil, as described in the study by Alvares et al. (2013) and in the study by Muniz et al. (2017) for the state of Ceará, the Acaraú Basin can be classified into two climate types: tropical with a dry winter (Aw), characterized by average temperatures above 18 °C, a dry season in winter, and concentrated rainfall in summer, commonly found in tropical coastal regions, and hot semi-arid (BSh). In the southern portion of the region, the hot semi-arid climate (BSh) predominates, with an annual average temperature above 18 °C, low humidity, and insufficient precipitation to support dense vegetation, with a predominance of this type.

The region is also characterized by annual precipitation ranging from 800 to 1,000 mm, distributed irregularly between January and May, and a high evaporation rate of 2,000 to 2,500 mm (Almeida et al., 2007). The area predominantly consists of crystalline bedrock, which hinders the recharge of underground water sources. In terms of soil types, luvisols, litholic neosols, and planosols are predominant (Companhia de Gestão dos Recursos Hídricos, 2016), generally shallow and with low permeability.

The study area covered six river sections (Figure 1) located within the Acaraú River Basin, whose main source is in the municipality of Monsenhor Tabosa, at approximately 800 meters of altitude, in northeastern Brazil. Of these, six sections were selected for discharge measurement analysis using the methods proposed in this study (Figure 1). Two discharge measurement campaigns were conducted: the first in September 2023 and the second in November 2024.

Figure 1
Map of the location of control sections in the Acaraú River basin, CE.

Methodological approach

In this study, three distinct methodological approaches were applied to estimate surface velocity and measure discharge in river cross-sections. The first approach used the FlowTracker I device, by SonTek (2007) — a portable Acoustic Doppler Velocimeter (ADV) capable of measuring three-dimensional water velocity with high accuracy (±1% or ±0.25 cm/s), particularly suitable for shallow sections and complex flow conditions. FlowTracker measurements were combined with the Float Method (FM), which served as the reference for both surface velocity and discharge. The second approach applied Large-Scale Particle Image Velocimetry (LSPIV) together with Chiu’s entropy method (Chiu, 1988), implemented using open-source software developed by Patalano et al. (2017), to estimate both surface velocity and discharge. Lastly, the third approach combined LSPIV with Chiu’s method and further refined discharge estimates by incorporating the elliptical hydrodynamic distribution model proposed by Corato et al. (2011).

Chiu’s method adapted

Chiu’s method is based on the distribution of flow velocities using a probabilistic approach grounded in the entropy concept (Chiu,1991). The probabilistic model developed by Chiu allows the mean velocity (ū) to be expressed as a function of the maximum velocity (u_max), as demonstrated by Moramarco and Singh (2001). According to Chiu and Said (1995), the relationship between the mean velocity and the maximum velocity can be represented by the following equation:

$\overline{u}=\varphi \cdot u_{max}$ (1)

where: ϕ is the slope of a line generated from a linear regression between ū and umax.

Chiu & Tung (2002) and Xia (1997) observed that M remains constant when the coefficient ϕ stabilizes based on the (ū, u_max) pairs. Additionally, they concluded that the parameter M is specific to each channel section and does not vary with discharge or water depth. This characteristic allows for accurate flow estimation using only a few velocity measurements (Farina et al., 2014).

To apply Chiu’s method (Chiu, 1989) in this study, it was necessary to estimate the maximum velocity (u_max) using the indirect approach proposed by Vyas et al. (2021). Additionally, the ϕ coefficient was later calibrated and validated.

The calibration of ϕ was performed using 50% of the highest observed discharge values, selected alternately throughout the historical data series. The optimal value of ϕ was determined by minimizing the root mean square error (RMSE) between the observed (Q) and simulated (Qs) discharge values, adjusted after optimization using the minimize function from the SciPy library. The BFGS (Broyden-Fletcher-Goldfarb-Shanno) algorithm was employed, as it is suitable for minimizing differentiable nonlinear functions and is based on Hessian matrix approximations throughout the iterations (Virtanen et al., 2020; Nocedal & Wright, 2006).

Validation was conducted using the remaining 50% of the observed data to assess the suitability of the calibrated ϕ(M) value. To evaluate its efficiency in both the calibration and validation stages, the Nash-Sutcliffe efficiency coefficient (NSE) was used, a widely applied metric for assessing the accuracy of the fit between Q and Qs over the historical data series.

To apply Chiu's method (Chiu, 1989) in this study, it was necessary to estimate the maximum velocity (u_max) using the indirect approach proposed by Vyas et al. (2021). Additionally, the values of ϕ and M were indirectly estimated at the deepest point of the cross-section. The adaptation of these maximum velocity values was feasible based on discharge measurement data collected by the Water Resources Management Company between 2017 and 2022 in two rivers located in the Acaraú Basin, in Northeastern Brazil. Thus, with the estimation of the ϕ coefficient, it was possible to apply the following approaches.

Surface velocity and discharge measurement - Method I

To measure surface velocity in the monitored cross-sections, the Float Method (FM) was adopted. This method involves determining the velocity of a floating object by timing how long it takes to travel a pre-measured section of the river (Santos et al., 2001).

Although the surface float moves at the same velocity as the water surface due to its low weight, it is highly susceptible to wind drag. In the absence of strong winds, the FM measures only the surface water velocity. For this reason, Chiu’s coefficient must be applied to estimate the average velocity across the cross-section. However, in this study, the FM was used to complement surface data obtained from the FlowTracker I device, as the employed model does not allow for direct surface flow measurements.

Surface velocity was determined by timing the float’s travel time over a predefined distance, as shown in Figure 2. Three measurements were taken, positioning the float as close as possible to the central axis of the cross-section. Among these measurements, the highest recorded velocity was considered for analysis.

Figure 2
Measurement of surface velocity using floats. (a) Spherical floats; (b) Movement of floating spheres over predetermined distances.

For each average velocity value measured at each station using the ADV device, a surface velocity was extrapolated based on the relationship between average velocity and surface velocity obtained through the Float Method (FM), using the highest average velocity recorded in the analyzed cross-section as a reference. In this way, it was possible to extrapolate the surface velocity data based on the average velocity data measured by the ADV.

The reference discharge measurement was carried out using the FlowTracker I device, an ADV-type instrument, which determined the total discharge in the river cross-sections. This method was adopted as the reference for the other two discharge measurement methods (Method II and Method III). The measurement procedure followed the mid-section method, where measurement points were distributed along vertical lines or subsections, with known depths and distances relative to the riverbank (Figure 3).

Figure 3
(a) Discharge measurement using an ADCP-based device (FlowTracker); (b) Report generated by the FlowTracker after field data collection. Source: Adapted from SonTek (2007).

The discharge (Q, in m³/s) of the river at each analyzed cross-section was determined by summing the product of the average velocity (u̅_i, in m/s) and the partial area of each measurement profile ($A_i$, in m²), as shown in Equation 1. The area (A_i, in m²) was obtained using the ADCP and served as a reference for the discharge calculation.

$Q={\displaystyle \sum }_{i=0}^n\left({\overline{u}}_i.A_i\right)$ (2)

Application of the Large-Scale Particle Image Velocimetry technique

The Large-Scale Particle Image Velocimetry (LSPIV) technique allows for the measurement of surface water velocity using image analysis. The system employs video cameras to capture images of the water surface, applying cross-correlation methods to estimate the displacement of natural tracers — such as turbulence patterns, ripples, leaves, and branches — between successive images, resulting in the surface velocity distribution. To scale the digital images to real-world measurements, field markers were installed, and the distances between them were recorded. Additionally, a fixed platform was set up to minimize image shaking caused by improper camera handling.

Data acquisition for velocity measurement followed these steps: pre-processing (camera setup and positioning), installation and acquisition of Ground Control Points (GCP), image processing, and post-processing.

To capture the images for LSPIV application, a camera mounted on a tripod was installed at the side of each river cross-section (Figure 4a). The camera was configured to record video for 2 minutes, with an azimuthal and inclination angle between the surface flow and the camera perspective of approximately 60º (Figure 4b).

Figure 4
(a) Mobile support for image capture; (b) Camera perspectives from the installation of the camera on the support with a lateral viewing angle of 60°.

For image processing in real-world coordinates, scaling techniques using Ground Control Points (GCP) were applied in the field. Due to the oblique view provided by the camera lens relative to the captured image, four GCPs were used, with the distances between them measured using a measuring tape or determined through the processing of coordinates collected on-site. This procedure allowed for the definition of straight-line segments between the GCPs, generating six distances between the ground points (P01 - P02, P02 - P03, P03 - P04, P04 - P01, P01 - P03, and P02 - P04).

It is important to highlight that all four GCPs must be positioned at the same water level and sequentially numbered in a counterclockwise direction (Patalano et al., 2017). The GCPs were marked using materials available on-site, such as branches, rocks, bottles, or any other structure that could be identified or highlighted in the images. These markers were distributed along the riverbanks (Figure 5).

Figure 5
Identification and measurement of the distances between Ground Control Points (GCP) located along the banks of the measurement sections.

Video recordings of surface flow in the cross-sections were captured using a smartphone camera (iPhone 11) with image resolution set to 1280 x 720 pixels (HD) at a frame rate of 30 frames per second. For each velocity measurement in each analyzed section, 15-second recordings were made. These recordings were then divided into grayscale frames, stabilized, and orthorectified using RIVeR software, version 2.6.

The image processing was performed using the PIVlab application (Thielicke & Stamhuis, 2014) and PTVlab (Brevis et al., 2011), linked to the graphical user interface (GUI) of RIVeR, which interconnected the functions of LSPIVlab and PTVlab in a systematic way. During the image preprocessing, the Contrast Limited Adaptive Histogram Equalization (CLAHE) technique was selected to enhance the contrast of the images within an analysis window (the area where the particle displacement estimate from one image to another will be calculated using cross-correlation), set to 64 pixels. The Fast Fourier Transform (FFT) processing algorithm was chosen to improve the performance of the cross-correlation. Afterward, post-processing of the images was carried out by adjusting limits between standard deviation filters, magnitude limit filters for the vectors, interpolating missing data, analyzing the limit velocities, among other configurations to fine-tune the velocity vectors in the image.

After processing and post-processing, the images in PIVlab, the velocimetry data were orthorectified based on the measurements between the GCPs (ground control points). The data were then adjusted with geometric information from the sections, filtering the velocity vectors that intersected only the line corresponding to the analyzed cross-section. This process resulted in the extraction of the surface velocity vectors of interest for each section. The processing and co-processing of these images were used to estimate the surface velocity and discharge for Methods II and III.

Surface velocity and discharge measurement – Method II

Based on image processing, it was possible to obtain the distribution of surface velocities for each analyzed cross-section. The surface velocity estimates for each section were subsequently used to estimate the discharge using Method II. For each surface velocity vector extracted through LSPIV processing, a correction coefficient (ϕ) was applied to estimate the average velocity ($\overline{u}$) and, consequently, the discharge for each section. This coefficient originates from the flow model proposed by Chiu (1989), in which velocity distribution is influenced by the maximum velocity.

Quantifying the aspect ratio (B/D), defined as the ratio between the section width and its depth, is essential for simplifying the complex flow phenomena in rivers. Ferro (2003) demonstrated that the lateral influence of the channel decreases as one moves away from the central axis toward the banks, and the effects of the sidewalls become insignificant for B/D values above 5.5, where velocity distribution follows a logarithmic law. Rajaratnam & Muralidhar (1969) found B/D values greater than seven, further confirming that velocity distribution in wide channels follows a logarithmic profile. Similarly, Tracy & Lester (1961) observed that depth becomes insignificant for B/D > 5. Thus, when the B/D ratio is lower, obstacles influence the velocity distribution, but as this ratio increases, the influence of the sidewalls diminishes, resulting in an exponential velocity distribution.

Chiu (1988) also classified rivers as either wide or narrow based on the aspect ratio (B/D), using it to characterize the velocity distribution and the position of the maximum velocity. According to Chiu (1988), in narrow rivers (typically B/D < 10), maximum velocity tends to occur below the water surface. This effect is more pronounced when the roughness (Manning’s coefficient) is low, and the channel slope is steep.

Furthermore, Chiu (1988) observed that as the B/D ratio decreases, the position of the maximum velocity progressively shifts toward deeper regions of the flow. On the other hand, in wide rivers (typically B/D > 10), the maximum velocity occurs at the water surface. In wide channels and natural rivers with high B/D ratios, maximum velocity tends to remain at the surface because secondary flow effects are attenuated. These findings have important implications for hydraulic modeling and for understanding velocity behavior in different types of rivers.

In this study, the geometric characteristics of the analyzed cross-sections indicate a tendency for the maximum velocity to occur at the free surface. Thus, in Method II, the discharge estimate is calculated under the assumption that, according to Chiu (1988), the maximum velocity occurs at the free surface of the section.

According to Chiu et al. (2005), the application of the velocity distribution equation can be associated with sufficiently stable coefficients, enabling flow estimation under various conditions based on velocity data. This relationship was applied to both regular cross-sections (artificial channels) and irregular sections (natural rivers and streams), covering a wide range of discharges under both steady and unsteady flow conditions. The results consistently demonstrated excellent correlation between the ($\overline{u}$, u_max) pairs collected in river cross-sections, enabling reliable discharge estimation (Chiu & Tung, 2002).

Therefore, Chiu’s coefficient (ϕ) is estimated using Equation 1, based on the ratio of the ($\overline{u}$, u_max ​) pairs, and is subsequently applied in Equation 3 – Continuity Equation and Equation 4 to estimate the discharge in each section using the surface velocity captured through LSPIV.

$Q=\varphi .u_{max}.A$ (3)

$Q={{\displaystyle \sum }}_{i=1}^n\left(\varphi .u_{LSPIV}.A_i\right)$ (4)

where: $\varphi$ is Chiu’s coefficient;$u_{LSPIV}$ is the surface velocity (m/s) recorded by LSPIV in the section; A is the measured cross-sectional area (m²); $u_{max}$ is measured maximum velocity data (m/s).

Surface velocity and discharge measurement - Method III

The surface velocity distribution in the entropy approach can be based on elliptical or parabolic distribution scenarios, developed by Corato et al. (2011). The first, the elliptical surface velocity distribution, is considered, according to Moramarco et al. (2013), more suitable for rivers classified as “wide.” The elliptical equation used to describe the velocity profile in a river cross-section is a common empirical form, frequently applied in hydraulic models. The elliptical equation used in this study can be expressed as:

$u_{max}\left(xv\right)=u_{maxs}\sqrt{1-{\left(\frac{xv}{xh}\right)}^2}$ (5)

where: x_h represents the distance from the left or right riverbank along the vertical, with reference to the y-axis ($xv$ = 0), along which the maximum surface velocity — $u_{maxs}$​ is recorded.

Statistical data analysis

In this study, the performance of the surface velocity and discharge estimation methods was evaluated by comparing them with reference measurements, using the percentage deviation (PD) and mean absolute error (MAE), calculated for each of the six measurement sections. Additionally, the average relative error between the surface velocities along the riverbanks was estimated by comparing the velocity distributions obtained using Methods II and III with the reference method (Method I).

To analyze the behavior of the measured data and the effectiveness of the proposed approaches, two discharge measurement campaigns were conducted using the methodologies presented here, one in 2023 and another in 2024.

RESULTS AND DISCUSSION

Geometry and flow characteristics of the sections by the reference method

Table 1 shows the results of the section geometry and discharge data for each evaluated section, measured between 2023 and 2024 in the Acaraú and Groaíras rivers, both part of the Acaraú River Basin. The sections exhibited varying widths, ranging from 4.6 to 24 meters. The maximum depth of the watercourses showed less variation between sections, with a minimum of 0.4 meters and a maximum of 0.9 meters.

Table 1
Geometry of the measurement sections, average velocity, and discharge by ADV.

Seção	B (m)	D (m)	B/D (-)	So	umax (m/s)	A (m2)	ū (m/s)	Q (m3/s)
S1	4.60	0.54	8.52	0.016	0.94	2.03	0.50	1.05
S2	5.60	0.50	11.20	0.003	0.68	2.04	0.44	1.06
S3	4.95	0.41	12.07	0.001	0.21	1.16	0.16	0.19
S4	20.86	0.89	23.44	0.001	0.67	8.69	0.42	3.68
S5	12.7	1.02	12.45	0.003	0.74	7.43	0.45	3.35
S6	23.92	0.63	37.96	0.007	0.45	6.17	0.40	2.47

• Legend: B – Section width; D – Maximum section depth; B/D – Aspect ratio; A – Section area; ū – Average velocity; Q – Discharge.

It is worth noting that, in all analyzed sections, the width-to-depth ratio (aspect ratio: B/D) was significantly greater than five, thereby reducing the influence of the lateral walls on the water velocity distribution, which, in this case, likely follows an exponential model (Tracy & Lester, 1961).

The highest discharges were observed at the section 04, while the lowest discharges occurred at the section 03. This result was expected, considering that section 04, located in the Acaraú River, receives greater inflows due to the drainage contribution of its watershed, along with the influence of the Paulo Sarasate Reservoir, which helps maintain higher levels of perennial flow. In contrast, section 03, located in the Groaíras River, is supplied solely by the Edson Queiroz Reservoir, which provides lower-magnitude discharges.

Figure 6 presents the distribution of water velocity along the profile of the measurement sections, obtained using the ADV method. It was observed that, in 80% of the sections analyzed in this study, the highest velocities occurred near the water surface, except for section S1, where the maximum velocity was recorded below the surface. This behavior is consistent with the classification proposed by Chiu (1988), which relates the aspect ratio (B/D) to the position of the maximum velocity in open channels.

Figure 6
Velocity distribution in various cross-sections of the Acaraú and Groaíras rivers, collected between 2023 and 2024.

Additionally, it was found that the lowest velocities occur near the riverbed, while the highest velocities are recorded in the surface layers of the section, indicating a velocity distribution that follows a logarithmic profile in the vertical direction. This behavior supports the theoretical assumption proposed by Prandtl & von Kármán (1925), as well as Faruque et al. (2014), who demonstrated the relationship between maximum surface velocity and local depth under different roughness conditions in regular channels. However, Araújo et al. (2017) point out that, in natural channels, these velocity distributions often exhibit non-uniform patterns, making such behavior less common.

Estimation of Chiu’s coefficient

Chiu's coefficients (ϕ) were determined based on field data measured using the paired sample per section approach, considering the estimation of maximum velocity (u_max), as described in the previous subsection. The obtained values, including the observed discharge (Q), the simulated discharge (Qs), RMSE, the NSE, and ϕ, are presented in Figure 7.

Figure 7
Coefficients of determination (R²), RMSE calibration curve, and Nash-Sutcliffe Efficiency (NSE) for the calibrated and validated ranges of ϕ(M) in the cross-sections.

The entropy relationship has been tested and evaluated in several rivers, demonstrating that, for the same cross-section, the entropy parameter remains constant even under different river flow conditions (Xia, 1997; Corato et al., 2014). Following the same logic, Farina et al. (2014) concluded that this parameter does not vary with water level or time, as it is an intrinsic characteristic of the cross-section, such as geomorphology and bed slope.

However, Farina et al. (2014) point out that the entropy parameter can undergo significant variations due to changes in velocity distribution and hydraulic conditions, especially when substantial alterations occur in the river's cross-section. These geomorphological variations may be related to both natural processes and human interventions, such as the construction of dams (Carvalho Neta, 2007). Furthermore, the hydrodynamics of natural watercourses are significantly influenced by erosion and deposition processes, which modify the bed roughness and channel shape, particularly in non-regulated reaches (Moramarco & Singh, 2010).

The stability of the entropy coefficient offers significant advantages for flow monitoring, allowing measurements without direct contact with the water, thus enhancing the safety of hydrological technicians. Studies, such as those by Moramarco et al. (2017), show that the application of this coefficient in estimating average velocities across the depth results in average errors of less than 12% for vertical velocities and 6% for the average flow velocity. Other works further reinforce the potential of this approach by utilizing different devices for measuring surface velocity, such as radar (Alimenti et al., 2020), ADCPs (Bahmanpouri et al., 2022a), and image-based methods (Bahmanpouri et al., 2022b), demonstrating promising results for accurate estimation of flow rates and average velocities under different river conditions.

The entropy coefficients (ϕ) found in this study ranged from 0.53 to 0.64 (see Figure 8), which corroborates the findings of Greco & Moramarco (2016). Based on field data from different river stations, their study demonstrated that the entropy parameter can vary with changes in the aspect ratio but remains within the range 0.5 < ϕ(M) < 0.8. For Chiu et al. (2000), the typical values range from 0.66 to 0.8 in natural channels.

Figura 8
Distribution of maximum velocities obtained from three approaches: directly measured data (ADCP) – Method I (reference), the LSPIV method – Method II, and the LSPIV + Elliptical Distribution method – Method III, in each river cross-section.

Sections 1 (ϕ = 0.58), 2 (ϕ = 0.62), 4 (ϕ = 0.64),5 (ϕ = 0.57) and 6 (ϕ = 0.61) fell within the range 0.57 < ϕ < 0.79 observed in rivers located in the states of Bahia and Sergipe, in the Northeast region of Brazil (Araújo et al., 2017). However, section S3 (ϕ = 0.53) exhibited values comparable to those observed in extremely wide rivers, ranging between 0.41 < ϕ < 0.61, such as the Amazon River in Brazil, as reported by Bahmanpouri et al. (2022a).

Chiu et al. (2000) demonstrated that lower values of the ratio coefficient ϕ indicate less uniformity in the velocity distribution, making the section more susceptible to velocity variations. When ϕ is small, the difference between the mean and maximum velocity increases, reflecting a more heterogeneous velocity distribution along the cross-section, which can significantly influence the accuracy of discharge estimation using the entropy method.

As shown in Figure 7, the number of samples between the pairs (um, u_max) ranged from 35 to 112 in the analyzed sections. After calibrating and validating ϕ, it was observed that the coefficient of determination (R²) varied between 0.80 and 0.95 when comparing the observed discharge data (Q) with the simulated data (Qs). These results indicate an excellent fit of the ϕ coefficient in the discharge estimation for the sections using the Chiu’s method adapted.

Surface velocity distribution

As shown in Figure 8, the maximum velocity vectors generated by method II - LSPIV were overestimated in sections S2, S4, and S6 compared to the observed values, demonstrating heterogeneous behavior in relation to the surface velocity vectors. Additionally, peaks of maximum velocity and an anomalous distribution of vectors near the banks were observed. In sections S3 and S5 the recorded maximum velocity was underestimated compared to the measured data. On the other hand, in the S1 section, the recorded intensities almost correspond to the maximum velocity values observed.

Regarding Method III (LSPIV + Elliptical Distribution), the velocity distribution data demonstrated improved accuracy, particularly in capturing the velocity profile between the section margins. This highlights the contribution of the elliptical distribution model in minimizing errors and inconsistencies in data acquisition, especially near the banks. However, since the maximum velocity in Method III is also derived from the LSPIV technique, the same estimation errors observed in LSPIV are carried over into this method.

Thus, the accuracy of velocity measurements using the method II and III depends on several factors related to camera positioning, calibration, and lighting conditions. However, despite existing limitations, image velocimetry techniques have proven capable of providing satisfactory flow measurements, with error margins of up to 15% for flow measurement in rivers, as observed in the works of Trieu et al., (2021).

Relative mean error

According to Figure 9, it was observed that, for both evaluated methods, the average errors in surface velocity estimation along the river cross-section showed considerable variations across different measurement points. In general, measurements using both the LSPIV approach and the LSPIV + Elliptical Distribution method exhibited lower average error values in the central region (41-60% of the distance) of the section.

Figure 9
Mean relative error of maximum velocities generated by LSPIV (a) and LSPIV + Elliptic Distribution (b) in relation to observed data by distance range from the left bank river for each section.

These results were also reported by Ioli et al. (2020), who conducted velocity and discharge flow measurements in the monitoring section of the Limmat River in Switzerland using drones and three different image-based velocimetry approaches (BASESURV, Fudaa-LSPIV, and RIVeR). The errors, compared to reference data (ADCP), indicated that the largest sources of error occurred along the riverbanks rather than at the center of the monitoring section.

According to Le Coz et al. (2021), one of the main sources of uncertainty in surface flow measurement using images is associated with orthorectification, which depends on accurate camera calibration and can be influenced by improper positioning, especially at very vertical or low grazing angles. The use of Ground Control Points (GCPs) is essential for reducing these errors, being more effective when well distributed around the area of interest. The article highlights that a minimum of six GCPs (3.5%) already significantly improves the accuracy of orthorectification, while 19 GCPs can reduce discharge uncertainty to less than 1.3%. Moreover, it emphasizes that GCP quality is more important than quantity, meaning that a few well-measured points can be more effective than many imprecise ones.

A low-altitude oblique view (grazing angle view), as applied in the present study, may have introduced perspective distortions, making orthorectification more challenging and amplifying uncertainties in surface flow measurement. These distortions may have led to underestimation or overestimation of velocity vectors, compromising the accuracy of the analysis.

According to Le Coz et al. (2021), this type of camera positioning creates a strong ambiguity between the camera height and its tilt angle, making calibration more sensitive to errors and hindering the precise identification of parameters. Small variations in tilt angle can result in significant changes in image scale, directly impacting the conversion of image coordinates to real-world coordinates and increasing errors in velocity and discharge estimation. The author highlights that the best perspective for discharge measurement is an oblique view from an elevated point, such as a mast, building, or elevated structure, as this positioning reduces correlations between camera parameters and improves calibration stability, leading to lower uncertainties in orthorectification.

Perks et al. (2020) also highlight that the geometry of the riverbed, characterized by irregularities such as rocks, sinuous banks, and depth variations, can influence flow dynamics, generating secondary flows and vortices (Torres et al., 2024), which may distort surface velocity measurements. The study emphasizes that accurate camera calibration, proper placement of GCPs, and selecting optimal capture conditions are essential to reducing uncertainties and improving measurement reliability.

Additionally, Perks et al. (2020) stress the importance of establishing a standardized operational procedure for image-based velocimetry approaches, ensuring consistency in data acquisition, image preprocessing, and velocity estimation. Such standardization would facilitate inter-comparisons between different methodologies, enhance accuracy, and promote the broader adoption of these techniques in hydrological monitoring.

Performance of the water discharge estimation methods

Table 2 presents the results of discharge estimates using the two evaluated methods for the different measurement sections and compares them with the reference method.

Table 2
Performance of discharge estimation between 2 different approaches compared to the reference method.

Sections	Date of measurement	Method I	Method II	Method III	(II) * 1	(II) * 2
		(m3/s)	(m3/s)	(m3/s)	PD (%)	PD (%)
S1	1/11/2024	1.01	1.04	0.88	2.97	-12.87
S2	1/11/2024	1.06	1.29	1.14	21.70	7.55
S3	29/09/2023	0.18	0.12	0.11	-33.33	-38.89
S4	28/09/2023	3.68	5.02	4.00	36.41	8.70
S5	28/09/2023	3.35	2.95	2.47	-11.94	-26.27
S6	27/09/2023	2.47	2.03	1.35	-17.81	-45.34
			MAE (%)		20.7	23.27

• ^*
  ¹Efficiency between the reference method and Method II;
• ^*
  ²Efficiency between the reference method and Method III. PD = Percentual Deviation (%); MAE = Mean Absolut Error (%).

In Table 2, it can be observed that Method II exhibited percentage deviations in discharge ranging from 2.97% to -33.33%, with an average absolute error of approximately 20.7%. On the other hand, Method III showed variations in discharge between 7.55% and -45.34%, with an average absolute error around 23.27%.

The extrapolation of surface velocities considering an elliptical-type distribution (Method III) resulted in an increase in discharge estimation errors compared to Method II. This effect may be related to low discharges associated with the section geometry, which can contribute to the formation of vortices and recirculation zones along the section margins. The lower water depth near the riverbanks may lead to estimation and measurement errors in these zones (Ansari et al., 2023), with high variation and underestimated velocity flow (Dal Sasso et al., 2021).

In the analyzed sections, the margins frequently exhibit recirculation zones (with zero or near-zero surface velocity) and vortex formation, making it challenging to model the velocity distribution. In this context, the discharge extrapolation using the elliptical distribution model may amplify errors in the measurement process, making it less reliable when applied to natural channels with low discharges (0.2 to 3.8 m³/s).

Overall, Method II exhibited a lower mean absolute error and more moderate percentage deviations, suggesting that the use of a previously calibrated entropic coefficient (ϕ), combined with the maximum velocity data recorded via LSPIV, may represent a promising approach for non-contact discharge monitoring (Corato et al., 2014; Bahmanpouri et al., 2024; Moramarco et al., 2017). This approach demonstrated more consistent and accurate performance compared to Method III.

Conversely, Method III showed greater variation in percentage deviations, including cases of significant underestimation of discharge, as observed in Section S6. These results indicate that the LSPIV + Elliptical Distribution combination may be more susceptible to errors under low discharge conditions, being strongly impacted by the absence of natural tracers, lack of artificial seeding, presence of shadows, and improper camera positioning, especially when installed at oblique angles and low heights.

According to Le Coz et al. (2021), the uncertainty in discharge measurement using LSPIV is directly related to the accuracy of ground control points (GCPs). Reducing these uncertainties depends not only on the number of GCPs but also on their spatial distribution in the field, as both are essential factors in minimizing errors in camera calibration and image orthorectification applied to velocimetry.

In the present study, four GCPs were used for orthorectification. According to Le Coz et al. (2021), this number can generate errors on the order of 45%, which can be significantly reduced to less than 30% with the use of at least six GCPs. This factor also influences the variability of discharge data in both Method II and Method III.

Another factor that may have influenced the results in Table 2 is the presence of few or no natural tracers in the analyzed sections - such as leaves, branches, or wave patterns - combined with the absence of artificial seeding during flow data collection by LSPIV, which may have contributed to the underestimation or overestimation of velocity vectors and, consequently, to inaccuracies in discharge estimation.

According to Dal Sasso et al. (2018), the absence of seeding can significantly compromise the quality of surface velocity data obtained through image-based velocimetry techniques. The same author highlights that in rivers with a low number of natural tracers, it is possible to mitigate these effects by increasing the number of analyzed frames, thereby compensating for the scarcity of visible particles.

Additionally, the study by Patalano et al. (2017) recognizes that the efficiency of these techniques depends on the presence of visible tracers on the water surface, whether natural or artificial. However, the study discusses strategies to mitigate the dependency on tracers and improve the quality of LSPIV data, which can reduce the need for tracers but do not eliminate the necessity of visible patterns on the water surface. An alternative already used by RIVeR is the application of filters such as CLAHE (Contrast Limited Adaptive Histogram Equalization) to enhance image contrast and highlight natural patterns on the water surface, as well as the use of Fourier-based image correlation, which can improve the detection of small patterns on the surface.

Yu, Kim and Kim (2015) emphasizes that LSPIV faces limitations under low discharge conditions, especially when there are few tracer particles, insufficient lighting, or minimal movement, resulting in weak correlations and high velocity estimation errors. As an alternative, STIV (Spatio-Temporal Image Velocimetry) has proven to be more efficient in capturing small flow variations, even with subtle particle displacements. Additionally, this approach visualizes displacement patterns over time, reducing the dependency on tracer particles and offering better performance under adverse conditions, such as low lighting and image noise.

The practical application of the LSPIV (Large-Scale Particle Image Velocimetry) technique in the field presented significant challenges related to the adjustment of intrinsic parameters (focal length, distortion coefficient, among others) and extrinsic parameters (position and orientation) of the cameras. Specifically, the proposed monitoring sections lacked preexisting physical structures, such as bridges or suitable elevations for camera installation.

To overcome these limitations, it is recommended to use poles with sufficient height to cover the entire flow field of view. This approach should consider both flood conditions, which increase river width, and drought periods, which reduce these widths, ensuring effective monitoring in different hydrological scenarios.

To enhance the application of the LSPIV technique, it is essential to standardize and validate image processing algorithms that, during preprocessing, evaluate the feasibility of surface flow analysis in the absence of tracers. If the presence of tracers is necessary, these algorithms should determine the adequate density to ensure more accurate measurements.

To reduce uncertainties related to image rectification, it is crucial to calibrate the intrinsic and extrinsic parameters during the preprocessing stages. Integrating this functionality directly into the Graphical User Interface (GUI) allows for a more intuitive and efficient calibration process. For this purpose, tools based on the OpenCV library or other open-source software can be used. OpenCV, for example, provides robust functionalities for camera calibration, enabling distortion correction and the acquisition of precise parameters.

The study by Le Coz et al. (2021) addresses the uncertainty in image orthorectification for video-based discharge measurements and investigates the influence of the number and accuracy of Ground Control Points (GCPs) in the camera calibration process. According to the tests conducted in the study, the uncertainty in discharge estimation decreases significantly as the number of GCPs increases, but there is a saturation point where adding more points does not significantly improve accuracy. The main findings suggest that a minimum of 6 well-distributed GCPs can already substantially reduce orthorectification uncertainty under oblique perspective conditions.

CONCLUSIONS

The velocity distribution along the analyzed profiles showed that in 80% of the sections, the highest velocity occurred near the surface, in accordance with the classification proposed by Chiu (1988).

The analysis of the entropy coefficient (ϕ) revealed values ranging from 0.53 to 0.64, aligning with previous studies, reinforcing the stability of this parameter for different flow conditions. The calibration and validation of the coefficient resulted in determination coefficients (R²) between 0.80 and 0.95, confirming the efficiency of the method in discharge estimation.

The application of the LSPIV technique in the field also posed operational challenges, particularly related to camera calibration, the positioning of ground control points (GCPs), and the incidence of shadows or inadequate lighting, factors that influenced the accuracy of velocity and discharge estimates. The results showed that using at least six GCPs can significantly reduce the uncertainties associated with image orthorectification and improve measurement reliability.

Despite the challenges, the results demonstrated the great potential of the LSPIV + Chiu combination for non-contact discharge monitoring, offering a viable alternative for small- and medium-sized rivers in the Brazilian semi-arid region.

ACKNOWLEDGEMENTS

For the support provided during the flow measurements, as well as for providing the historical flow databases for the study area, the authors acknowledge the Company for Water Resources Management of the State of Ceará (COGERH), and the Federal Institute of Science, Education, and Technology of Ceará (IFCE) for granting paid leave to the first author, enabling the completion of his doctoral studies.

• Data Avaiability:
  Research data is not available

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Edited by

• Editor-in-Chief:
  Adilson Pinheiro

• Associated Editor:
  Rodolfo Luiz Bezerra Nobrega

Data availability

Research data is not available

Publication Dates

• Publication in this collection
  07 July 2025
• Date of issue
  2025

History

• Received
  22 Nov 2024
• Reviewed
  24 Mar 2025
• Accepted
  28 Apr 2025