Estimating surface velocity coefficient from entropy theory for STIV discharge measurements

Abstract

River discharge is one of the most difficult hydrological quantities to measure both accurately and safely and several novel methods have been searched for in the past decades in addition to the conventional float method. The space-time image velocimetry (STIV), which measures streamwise surface flow velocity distributions from video images can be one of the candidates to meet the above requirements. However, as long as the measurement target is a surface flow, there raises a problem to be solved when converting the surface velocity distribution to discharge via the surface velocity coefficient α, because the standard value of 0.85 does not always provide an adequate discharge in practice. To improve the versatility of discharge measurement with STIV, the maximum entropy method (MEM) was utilized to derive a general equation for the surface velocity coefficient, taking into account the velocity dip phenomenon and wind effects. The applicability of the proposed method was demonstrated through application to two rivers, the Ohta River and the Nakamura River. The discharge estimated by the proposed method agreed well with the discharges measured by the acoustic Doppler current profiler (ADCP) which is considered to provide the most reliable data, and gave better results than those estimated by the conventional laws of vertical velocity distribution. A sensitivity check of the entropy parameter in flow estimation was also performed.

river flow measurement; STIV; maximum entropic method; surface velocity coefficient; ADCP; wind drift

Introduction

In response to the repeated large floods in recent years, the movement to improve the discharge measurement method is progressing internationally. As for the cases in Japan, in the July 2018 disaster on the Takahashi River in Okayama Prefecture, a tributary embankment was destroyed by flooding and a vast area was inundated, resulting in more than 200 casualties including even those who evacuated to the second floor of their homes. In the disaster of the Kuma River in Kumamoto Japan Prefecture in July 2020, heavy rainfall continued for several hours due to the stagnation of a linear precipitation zone in the river basin, causing a rapid rise of the water level and washing away many bridges. The maximum water depth in the inundated area reached more than 5 m. The river engineering problem with these disasters was that the peak flow rate could not always be measured, mainly due to the danger of on-site measurements and the destruction of the gauging station by the fluid dynamic force. In fact, among the fundamental data required for river plannings, discharge is the most difficult hydrological quantity to measure compared to the other values such as rainfall intensity or water level. In the past decades, various methods have been proposed to measure the reliable discharge data efficiently and safely1. Among the proposed measurement methods, there are methods using ultrasonic waves2,3, radio waves 4–6, and image analysis techniques 7–12 . Besides the above methods, the acoustic Doppler current profiler (ADCP) is considered to measure the discharge accurately because it can measure most of the cross-sectional velocity distribution, but it cannot be used when the flow velocity is very fast or when driftwood or other floating debris is flowing downstream. The surface velocity radar (SVR) is handy and convenient but safe measurement is not always possible as it requires a measurement from a bridge. In addition, since SVR measures only one point on a water surface it would take time to obtain the transverse velocity distribution efficiently. On the other hand, image-based technologies such as large-scale particle image velocimetry (LSPIV) 12–20 and space-time image velocimetry (STIV) 20–31 can measure velocity distributions simply by capturing surface video images by using either a river monitoring camera, a helicopter 32–34 or a drone 35–40. In general, STIV can measure the streamwise velocity component with higher accuracy than LSPIV, even for images taken from shallow inclination angles or with low resolution. In that sense, STIV can be regarded as the most suitable method for river discharge measurements1. However, even when a detailed surface flow velocity distribution is obtained by STIV, an accurate surface velocity coefficient (SVC) ${\alpha }_{surface}$ is required in order to convert it into a flow rate by the velocity-area method41–43. Hauet et al.43 analyzed a dataset of 3611 EDF (Electricité De France)-DTG’s discharge data measured by using a current meter and reported that the mean value of ${\alpha }_{surface}$ is 0.8 for natural rivers and 0.9 for artificial concrete channels, in which the mean discharge value was 8.45 m ${}^3$ /s and the maximum was 863 m ${}^3$ /s. Here, the velocity 3 cm below the free-surface was treated as the surface velocity. ADCP was not used due to the blanking distance of more than 15 cm from the water surface. According to Water measurement manual by United States Bureau of Reclamation (USBR)41 that introduced a rod float method as ‘Special Measurement Methods’, the conversion coefficient from the rod float velocity to the depth-averaged velocity ${\alpha }_{rod}$ varies from 0.66 for a depth of 30.5 cm (1 ft) to 0.80 for more than 6.1 m (20 ft) (Table [tbl1]). Here the rod float has to be immersed one-fourth or less of the flow depth. On the other hand, in Japan, ${\alpha }_{surface}$ of 0.85 is used for a surface float, ${\alpha }_{rod}$ of 0.88 for a 50 cm-long rod float, etc. (Table [tbl2]). In Japan, even a 4 m rod float has been used at depths of 5 m or greater. According to ISO 74844, ${\alpha }_{surface}$ is expressed as a function of roughness coefficients and ${\alpha }_{rod}$ is expressed as the relative submergence of the rod float, $d/H$ , where $d$ is draft and $H$ is water depth (Table [tbl3]). In order to compare the values of ${\alpha }_{rod}$ shown in Table [tbl2] and Table [tbl3], their variations are plotted against $d/H$ in Fig. 1 except in the case of surface float for the Japanese standard. It can be seen that both ranges are roughly overlapped. On the other hand, USBR showed much smaller ${\alpha }_{rod}$ values, which may be due to the difference of measurement and roughness conditions. For simplicity, in the following, $\alpha$ refers to both ${\alpha }_{surface}$ and ${\alpha }_{rod}$ .

Table 1. Relation between water depth and conversion coefficient ${\alpha }_{rod}$ using rod float by USBR41.

[tbl1]

Table 2. Relation among water depth $H$ , draft $d$ and $\alpha$ by Japanese method.

[tbl2]

Table 3. Relation among Manning’s coefficient $n$ , water depth $H$ and $\alpha$ by ISO.

[tbl3]

Fig. 1. Relative submergence of float $d/H$ vs $\alpha$ by ISO and Japan; $H$ : water depth, $d$ : draft.

Marjang and Merkley45 calculated surface velocity coefficients by three dimensional numerical simulations of rectangular and compound open channels for evaluating the accuracy of a float method. The result shows that $\alpha$ varies significantly with the distance from the wall, the water depth and the shape of the cross-section. The reason for their conclusions is that their simulation was for a channel with small aspect ratios between 0.5 to 4, where the influence of the secondary flow is dominant. The red lines in Fig. 2 delineate the relation between $\alpha$ and the Manning’s roughness coefficient indicated in Table [tbl3]. The greater the effect of roughness, the lower the value of $\alpha$ probably due to the development of the roughness boundary layer. However, it is known that especially in the center of the channel, where the aspect ratio of channel width divided by water depth is small, the maximum velocity is observed below the water surface due to secondary flow46. This is called the velocity dip phenomenon and the value of $\alpha$ can be greater than one at such a vertical section.

Fig. 2. Relation between $\alpha$ and Manning’s coefficient $n$ proposed by ISO; $\beta =\Delta /D_x$ , $\Delta$ : roughness height, $D_x$ : representative sediment diameter.

Based on the above discussion, there are still issues to be considered regarding the surface velocity coefficient. Since STIV can measure surface velocity distribution at high density, the accuracy of discharge measurement can be improved by providing appropriate lateral distribution of surface velocity coefficient. For that purpose, we utilized the vertical velocity distribution originally proposed by Chiu47 and further simplified by Moramarco et al.48,49 for practical usages. To estimate the discharge at an arbitrarily shaped cross section, we derived an equation for $\alpha$ using surface velocity values that can be varied laterally including the effects of velocity dip and wind. No such formulation has ever been presented in the past research. The accuracy of this method was examined by using the discharge measured by the ADCP as reference data and comparing it with the existing method using the conventional vertical velocity distribution equation.

Theory on vertical velocity distribution and surface velocity coefficient

Conventional velocity distributions and surface velocity coefficient

In order to convert the surface velocity distribution into river discharge, it is essential to estimate vertical velocity distributions at arbitrary cross sections. The research on the velocity distributions of open-channel flows has a long history46, but mean and turbulence structures of the flow especially in flood conditions is still a research target as of today, because large-scale coherent structures50 in flood flow condition are difficult to observe with conventional techniques. The most well-known formula for the vertical velocity distribution is the following logarithmic profile:

  

$$\frac{u\left(y\right)}{u_{*}}=\frac{1}{\kappa }ln{\frac{y}{y}}_0+\psi \left(\frac{y}{H}\right)\left(1\right)$$

where $u$ is the streamwise mean velocity at the height of $y$ from the riverbed, $u_{*}$ is the shear velocity, $\kappa$ is the von Karman constant, $y_0$ is bed roughness size that represents the distance from the bed where the velocity becomes zero, and $H$ is the local water depth. $\psi$ is the wake function describing the deviation from the logarithmic profile in the outer region close to the water surface and expressed as the following form with a wake parameter $\Pi$ 46,51.

  

$$\psi \left(\frac{y}{H}\right)=\frac{2\Pi }{\kappa }{sin}^2\left(\frac{\pi y}{2H}\right)\left(2\right)$$

where the bed roughness size $y_0$ is expressed as $\Delta /30$ with $\Delta$ being the roughness height which is related to the sediment size $D_x$ with a relation $\Delta =\beta D_x$ . Here, $D_x$ is the grain size for which $x$ % of the bed material is finer. The commonly used values of $D_x$ are $D_{50},D_{65},D_{84}$ or $D_{90}$ . For example, since the value of $\beta$ varies between 1 to 6 (Muste et al.52), $y_0=D_x/30-D_x/5$ . A typical value of $\beta$ is 3, which corresponds to $y_0=D_x/10$ . As a first approximation, the roughness height $\Delta$ (unit in meter) is related to the Manning’s roughness coefficient $n$ with the following relation53:

  

$$n=\frac{{\Delta }^{1/6}}{8.1g^{1/2}}=0.0394{\Delta }^{1/6}\left(3\right)$$

or given that the range of $\beta$ is 1 to 6 and the standard value is 3, we have the following equation:

  

$$n=(0.0394-0.0531)D_x^{1/6}\approx 0.0473D_x^{1/6}\left(4\right)$$

Neglecting the wake function, the surface velocity coefficient ${\alpha }_{log}$ can be expressed as the following relation for the case of $\beta =3$ 11:

  

$$\begin{array}{c}\hfill \left[c\right]{\alpha }_{log}=\frac{1}{1-y_0/H}-{[ln\left(\frac{H}{y_0}\right)]}^{-1}\\ \hfill =\frac{1}{1-\frac{D_x/10}{H}}-{[ln\left(\frac{H}{D_x/10}\right)]}^{-1}\left(5\right)\end{array}$$

In natural streams typical value of $H/y_0=H/\left(D_x/10\right)$ ranges ${10}^2$ to ${10}^4$ (Hauet et al.43).

An alternative profile can be expressed as the power law:

  

$$\frac{u\left(y\right)}{u_{surf}}={\left(\frac{y}{H}\right)}^{1/m}\left(6\right)$$

where $u_{surf}$ stands for the mean velocity at the surface and $m$ is an index. The surface velocity coefficient ${\alpha }_{power}$ can be expressed as:

  

$${\alpha }_{power}=\frac{m}{m+1}\left(7\right)$$

According to ISO74844, the values of $m$ and $\alpha$ are qualitatively specified according to the roughness conditions as shown in Table [tbl4]. It indicates that the value of $\alpha$ decreases with the increase of roughness. For delineating the effect of Manning’s roughness coefficient $n$ on the surface velocity coefficient $\alpha$ , Eqs. (3) to (5) are used to represent their relation as a function of $\beta (=\Delta /D_x)$ as shown in Fig. 2. It is interesting to note that the range of $n$ for different $\beta$ almost overlaps the range according to ISO’s standard. Therefore, we can utilize Fig. 2 for estimating the value of $\alpha$ as a first guess with the information of roughness. It should be noted that the log-law and the power law are applicable only to fully developed turbulent flows but not to the flow affected by secondary flows. Furthermore, in the case of the log-law, if the wake parameters are not known, a vertical velocity distribution that can be applied up to the water surface cannot be obtained.

Table 4. Value of $m$ and $\alpha$ under various bed conditions (ISO74844).

[tbl4]

Velocity profiles for expressing velocity dip phenomenon

Regarding the velocity dip phenomenon, it is considered to occur when the aspect ratio of a rectangular channel is less than five46. Velocity dips are caused by secondary flow effects due to the anisotropy of turbulence. In such a condition, the surface velocity coefficient can exceed unity. One of the extreme cases was measured at the deepest location in a channel section of the Mississippi River, in which the maximum velocity occurs at the one third of the depth below the water surface with the water depth of 33 m (Chiu and Said54). For describing such a velocity dip, the modification of wake function55 or the modification of log-wake law 56–58 have been proposed. However, the proposed profiles require the information of the dip depth for fitting the velocity profile, which cannot be used for the estimation of discharge from the surface velocity data.

Velocity profile by the maximum entropy method

In a completely different method from the above, Chiu 47,59 derived a velocity distribution equation based on the concept of probability theory, i.e., the maximum entropy method (MEM). This method has recently attracted attention by various researchers and has been successfully applied to actual river measurements60,61. Among others, the modification and simplification of the theory made by Moramarco et al.48,49 promoted the use of MEM significantly. Bechle and Wu62 additionally introduced the same idea of MEM to account for the wind effect while measuring the surface flow by LSPIV. One of the outstanding ideas of MEM is to hypothesize that the ratio of the maximum velocity $u_{max}$ at the section to the cross-sectional mean velocity $U$ does not vary with water level, i.e., $\varphi =U/u_{max}=$ const. This assumption has been verified by various researches61,63–68. Most of these researches estimate the river discharge by measuring only the maximum velocity at the section after confirming the above assumption.

To apply MEM to STIV measurements, we will focus on the surface velocity distribution, including wind effects, and review and modify the relevant equations that are easy to use in STIV analysis, which are not indicated in the past studies. According to Chiu’s theory47, $\varphi$ is related to the entropy parameter $M$ via the following equation:

  

$$\varphi \left(M\right)=\frac{U}{u_{max}}=\frac{e^M}{e^M-1}-\frac{1}{M}\left(8\right)$$

The velocity distribution derived by the maximum entropy theory is given as:

  

$$\frac{u\left(\xi \right)}{u_{max}}=\frac{1}{M}ln[1+(e^M-1)\frac{\xi -{\xi }_0}{{\xi }_{max}-{\xi }_0}]\left(9\right)$$

where $\xi$ is the dimensionless coordinate of $y$ measured from the bottom boundary and is expressed by the following relationship:

  

$$\xi =\frac{y}{H-h}exp(1-\frac{y}{H-h})\left(10\right)$$

Here, ${\xi }_0$ is the coordinate at the bottom boundary where velocity becomes zero, ${\xi }_{max}$ is the coordinate for the maximum velocity, $H$ is the water depth where the maximum velocity is observed, and $h$ is the depth from the water surface at the point where the cross-sectional maximum velocity occurs (Fig. 3). Using Eqs. (8) to (10), the following equation can be obtained:

  

$$\begin{array}{cc}\hfill \left[c\right]U& =u_{max}\varphi \left(M\right)=u_{surf}\hfill \\ \hfill & =\frac{M\varphi \left(M\right)}{ln[1+(e^M-1)\frac{H}{H-h}exp(1-\frac{H}{H-h})]}\left(11\right)\hfill \end{array}$$

This equation relates the cross-sectional average velocity $U$ to the velocity $u_{surf}$ recorded at the free surface on the vertical measurement line where the maximum velocity $u_{max}$ in the cross-section occurs. When $h=0$ , Eq. (11) can be simplified as follows:

  

$$U=u_{surf}\varphi \left(M\right)\left(12\right)$$

because $u_{surf}=u_{max}$ in this condition.

Fig. 3. Variables at a cross-section.

To facilitate the handling of this theory, Moramarco et al.48,49 assumed that Eqs. (9) and (10) holds on an arbitrary vertical line. With this simplification, the vertical velocity distribution at the distance $x_i$ from the riverbank can be expressed as follows:

  

$$\frac{u(x_i,y)}{u_{max,i}}=\frac{1}{M}F(M,h_i/H_i,y/H_i)\left(13\right)$$

where,

  

$$\begin{array}{cc}\hfill & F(M,h_i/H_i,y/H_i)=\hfill \\ \hfill & ln[1+(e^M-1)\frac{y/H_i}{1-\frac{h_i}{H_i}}exp(1-\frac{y/H_i}{1-\frac{h_i}{H_i}})].\left(14\right)\hfill \end{array}$$

The subscript $i$ indicates the value at the section $x_i$ as shown in Fig. 3. Note that the value of $M$ is calculated using the value of the entire cross section. The surface velocity at the $x_i$ section, $u_{surf,i}$ can be expressed as:

  

$$\frac{u_{surf,i}}{u_{max,i}}=\frac{1}{M}G(M,h_i/H_i)\left(15\right)$$

where

  

$$\begin{array}{cc}\hfill & G(M,h_i/H_i)=\hfill \\ \hfill & ln[1+(e^M-1)\frac{1}{1-\frac{h_i}{H_i}}exp(1-\frac{1}{1-\frac{h_i}{H_i}})].\left(16\right)\hfill \end{array}$$

It should be noted that when there is no velocity dip ( $h_i=0$ ), $G(M,0)=M$ and $u_{surf,i}=u_{max,i}$ . The depth-averaged velocity for each vertical section $U_i$ can be calculated from Eq. (13) as follows:

  

$$U_i=\frac{u_{max,i}}{M}{\int }_0^{1}F(M,h_i/H_i,y/H_i)d\left(y/H_i\right)\left(17\right)$$

The expression for the surface velocity coefficient ${\alpha }_i$ is obtained from Eqs. (15) and (17) as:

  

$${\alpha }_i=\frac{U_i}{u_{surf,i}}=\frac{{\int }_0^{1}F(M,h_i/H_i,y/H_i)d\left(y/H_i\right))}{G(M,h_i/H_i)}.\left(18\right)$$

When $h_i=0$ , Eq. (18) is simplified as:

  

$$\begin{array}{cc}\hfill & {\alpha }_i=\hfill \\ \hfill & \frac{1}{M}{\int }_0^{1}ln[1+(e^M-1)\frac{y}{H_i}exp(1-\frac{y}{H_i})]d\left(\frac{y}{H_i}\right).\left(19\right)\hfill \end{array}$$

On the other hand, to determine or draw internal velocity distribution using only the surface velocity, with or without the velocity dip phenomenon, the following equation obtained from Eqs. (13) and (15) are useful:

  

$$u(x_i,y)=u_{surf,i}\frac{F(M,h_i/H_i,y/H_i)}{G(M,h_i/H_i)}\left(20\right)$$

When $h_i=0$ , Eq. (20) reduces to the following form:

  

$$u(x_i,y)=\frac{u_{surf,i}}{M}ln[1+(e^M-1)\frac{y}{H_i}exp(1-\frac{y}{H_i})]\left(21\right)$$

Since the vertical velocity distribution given by Eq. (13) or Eq. (20) has two parameters, $M$ and $h_i$ , it has a high degree of freedom for expressing arbitrary distributions. The velocity gradient at the water surface is zero for $h_i=0$ , positive for $h_i<0$ and negative for $h_i>0$ . The negative dip depth is simply a parameter to represent the positive velocity gradient at the water surface for each method and has no physical importance. In fact, Chiu59 did not mention anything about negative dips. Fig. 4 shows the variation of $\alpha$ as a function of $M$ and relative dip depth. When the relative velocity dip depth $h/H$ is less than about 0.5, the value of $\alpha$ increases with $M$ .

Fig. 4. Variation of surface velocity coefficient $\alpha$ for different $M$ and dip depth $h$ .

Estimation of velocity dip

As mentioned previously, the velocity dip phenomenon mainly takes place due to the cross-sectional secondary flow generated near the side boundary. In reality, velocity dips can also occur in the central portion of the river due to large coherent turbulent structures. However, it is difficult to predict the dip depth in advance. On the other hand, the magnitude of the velocity dip in a rectangular channel has been studied in detail over the past several decades through detailed measurements of cross-sectional velocity profiles by laboratory experiments and field observations. Because of the variety of measurement data used in the evaluation, the dip depth estimation equation also yields somewhat different results. In this research, we examined the following formulas proposed by Yang et al.69, Bonakdari et al.70, Guo71, and Pu72. Yang’s formula is expressed as:

  

$$\frac{h_i}{H_i}=1-\frac{1}{1+1.3exp(-x_i/H_i)}\left(22\right)$$

where $x_i$ is the distance from the riverbank to the $i$ -th section indicated in Fig. 3. Bonakdari’s formula is given by:

  

$$\frac{h_i}{H_i}=1-\frac{42.4+{\left(\frac{x_i}{H_i}\right)}^{4.2}}{94.7+{\left(\frac{x_i}{H_i}\right)}^{4.2}}\left(23\right)$$

Guo’s formula is expressed as:

  

$$\frac{h_i}{H_i}=1-\frac{1}{1+exp[-{\left(\frac{2x_i}{\pi H_i}\right)}^{1.5}]}\left(24\right)$$

Pu’s formula is given by:

  

$$\frac{h_i}{H_i}=1-\frac{40.1+{\left(\frac{x_i}{H_i}\right)}^{4.4}}{80.5+{\left(\frac{x_i}{H_i}\right)}^{4.4}}\left(25\right)$$

The four formulae are compared in Fig. 5. When the relative distance from the river bank is small, $h_i/H_i$ takes a value of about 0.5 to 0.6, but they diminish asymptotically to zero for its value greater than about 5. In order to examine the performance of the above velocity dip formulae, the experimental data measured at the center of a laboratory flume73 is used for comparison. Fig. 6 shows the results for a flow with an aspect ratio of two. It can be seen that Yang’s and Guo’s formulas provide reasonable velocity profiles, whereas Bonakdari’s and Pu’s formulas overestimate the dip depth in this condition.

Fig. 5. Comparison of velocity dip formula; $h$ : dip depth, $x$ : distance from side wall, $H$ : water depth.

Fig. 6. Comparison of velocity dip formula with an experimental data (CW-1) at the center of a flume with a depth 10 cm and a width 5 cm (Nezu and Nakagawa73).

Approximate formula between $M$ and $\alpha$

For practical purposes, the approximate functional relationship between $M$ and $\alpha$ calculated by Eq. (18) is obtained by a least square method as follows, hereafter ${\alpha }_{MEM}$ is used instead of $\alpha$ to distinguish with the other definitions: For $h_i/H_i=0$ ,

  

$$\begin{array}{cc}\hfill {\alpha }_{MEM}& =0.7697M^{0.0946}\hfill \\ \hfill & for1<M<9(R^2=0.9976)\left(26\right)\hfill \end{array}$$

or

  

$$M=15.862{\left({\alpha }_{MEM}\right)}^{10.7549}(R^2=0.9962)\left(27\right)$$

As a reference, the value of $M$ for a conventional surface velocity coefficient of 0.85 with no velocity dip is about 2.86. In a practical situation, estimating velocity dip depth is of vital importance for applying STIV for discharge measurement. It is useful to note that Eq. (8) can be approximated by the following relation.

  

$$\begin{array}{cc}\hfill \varphi \left(M\right)& =0.5692M^{0.2139}=U/u_{max}\hfill \\ \hfill & for1.2<M<7.1(R^2=0.9984)\left(28\right)\hfill \end{array}$$

Introduction of wind effect

Regarding the wind effect on the surface flow measurement, Bechle and Wu62 proposed a velocity distribution formula based on the entropy theory as follows:

  

$$\begin{array}{cc}\hfill & u_w(x_i,y)=\hfill \\ \hfill & \frac{f{U}_{10}}{M_w}ln[1+(e^{M_w}-1)\{1-(1-\frac{y}{H_i})exp\frac{y}{H_i}\}]\left(29\right)\hfill \end{array}$$

Here, $u_w(x_i,0)=f{U}_{10}$ is the wind-induced drift current at the water surface, $U_{10}$ is the wind speed at a height of 10 m, $f$ is the wind drift coefficient that converts the wind speed to drift current and $M_w$ is the entropy parameter for a drift current profile. The value of $M_w$ was set to the default value of $-6$ using experimental data74. Therefore, the vertical velocity distribution including the wind-induced current $u_{total}$ at the section $i$ is expressed as:

  

$$u_{total}(x_i,y)=u_{surf,i}\frac{F(M,h_i/H_i,y/H_i)}{G(M,h_i/H_i)}+u_w(x_i,y)\left(30\right)$$

Here, $u_{surf,i}$ is the velocity that ignores the wind effect. Finally, the depth-averaged velocity at the section $i$ including the wind effect and velocity dip is given by:

  

$$\begin{array}{cc}\hfill & u_{total,i}=\hfill \\ \hfill & \frac{u_{surf,i}}{G(M,h_i/H_i)}{\int }_0^{1}F(M,h_i/H_i,y/H_i)d\left(y/H_i\right)\hfill \\ \hfill & +\frac{f{U}_{10}}{M_w}\hfill \\ \hfill & {\int }_0^{1}ln[1+(e^{M_w}-1)]\{1-(1-\frac{y}{H_i})exp\frac{y}{H_i}\}d\left({\frac{y}{H}}_i\right)\left(31\right)\hfill \end{array}$$

and the surface velocity coefficient at the section is expressed by:

  

$${\alpha }_i=\frac{u_{total,i}}{u_{surf,i}+f{U}_{10}}\left(32\right)$$

Eq. (32) is the general expression for the surface velocity coefficient including the effects by the velocity dip and wind-induced drift, although it includes the integral form. Using the surface velocity profile measured at STIV, velocity dip equation, wind speed data, and appropriate entropy parameters, it is expected that the discharge can be accurately estimated by the velocity area method.

In the literature, few studies have been conducted on wind effects in field measurements. Experimentally, the effects of wind on water surface have been investigated by using a wind tunnel in the 1970s74,75 for measuring wind drift currents. Among them, Wu75 suggested that the magnitude of drift currents is about 3.5% of the wind speed $U_{10}$ at long fetches. Bechle and Wu62 adopted a value of 3% in their model, i.e., $f=0.03$ in Eq. (29). Motonaga et al.76 and Onda and Hirakawa77 investigated the wind effect in the field condition. Motonaga et al.76 proposed a value of $f=0.074$ and Onda and Hirakawa77 suggested a value of $f=0.082$ according to their simultaneous field measurements by using an anemometer, ADCP, surface velocity radar, and STIV. However, it is still difficult to accurately assess wind effects in natural environments because wind speed and direction tend to change over time.

Fig. 7. Velocity distributions under the effect of wind drift current for $h/H=0$ .

In order to examine the wind effect on the internal velocity distribution, vertical velocity distributions subject to a drift current calculated by Eq. (30) with no velocity dip condition, i.e. $h_i/H_i=0$ are shown in Fig. 7. The plots are for typical flood surface velocities of 3 m/s. It is obvious that the drift current effect extends deeper with the increase of the wind speed. In the case of adverse wind, the maximum velocity tends to occur beneath the water surface. The effect of wind is relatively greater at slower flow velocities. Fig. 8 shows the variation of $\alpha$ calculated by Eq. (32) for the typical surface velocity of 3 m/s and 5 m/s. Since what we can observe by STIV is only the surface velocity accelerated or decelerated by the wind, the value of $\alpha$ can exceed unity when the adverse wind is relatively strong and the volumetric velocity is slow. To estimate discharge from the surface velocity, it is necessary to consider the vertical velocity distribution shown in Fig. 7. It is not recommended to simply estimate discharges by varying the surface velocity as if the wind effect extends to the bottom of the river. In reality, during storms, there are effects such as large undulations on the water surface, and such effects should be taken into account in future analyses. 　　

Fig. 8. Variation of surface velocity coefficient $\alpha$ by the wind effect.

Space-time image velocimetry (STIV)

Development of STIV

STIV is one of the local remote sensing techniques that can measure streamwise velocity distributions from video images of river surface flows21,22,78–83. It is called aerial STIV when the images are taken from the air by a helicopter or a drone33,34,36,37. It should be remembered that the term ‘STIV’ was first used in this Japanese paper in 2003. In STIV, the time evolution of image intensity along a search line set in the direction of the main flow is represented as a space-time image (STI). The advantage of STIV over the other methods is that the time-averaged velocity integrated along the search line can be obtained simply by measuring the orientation of the tilted texture appeared in the STI. Another advantage of STIV analysis is that it uses information from all consecutive frames, which is different from methods such as LSPIV, where the time interval between frames used in the analysis affects the results. The challenge for STIV has been how to determine this gradient automatically and accurately. If you use LSPIV to get the same information, you need to take the time to analyze hundreds of consecutive image pairs and then take averaging and filtering steps. In general, STIV is much faster than LSPIV. Conversely, STIV has the disadvantage of not being able to detect flow direction by itself, and thus cannot detect flow features such as circulating or curved flows. Therefore, STIV should be used specifically for flow discharge measurement.

As for nighttime flow measurements, a far-infrared camera was first used successfully by Fujita et al.84 and Fujita25. For airborne measurements, aerial STIV in hovering mode was developed by Fujita et al.36 by introducing the efficient image stabilization technique and the navigation mode was developed by Notoya et al.37. Further improvement was carried out by Fujita et al.34,85, in which masked 2D Fourier spectra of STI was used to filter out the effects by dispersive waves and other noises in order to pick up the predominant Fourier component related to the main flow. In the latest research85,86, a deep learning technique was used to establish a more robust STIV system. In another approach, the space time volume velocimetry (STVV) was developed by Tsuji et al.87 for measuring river surface velocity vectors efficiently. STIV (version with the gradient tensor technique) was first commercialized as a software KU-STIV88, and STIV (Fourier spectrum) and STIV (deep learning) were added to the latest software Hydro-STIV89. Recently, Han et al.90 developed 2D-STIV to mountain river flows and demonstrated superior performance to LSPIV.

Table 5. Specifications of the measurement sites.

[tbl5]

Fig. 9. Search lines for STIV measurement; (a) oblique image and (b) rectified image of the Shimada Bridge, at 14:30, Oct. 12, 2019, and (c) oblique and (d) rectified image of the Jinshin Bridge, at 4:00, June 19, 2020. The width of the river is 126 m at the Shimada Bridge and 35 m at the Jinshin Bridge. The pixel resolutions closest to and farthest from the riverbank are also shown.

STIV measurements at two locations

To demonstrate the performance of STIV, two field measurement cases are presented: the Nakamura River (Shimada Bridge), a small river approximately 35 m wide, and the Ota River (Jinshin Bridge), a medium-sized river approximately 126 m wide. These observations were made as part of an innovative river technology project led by the Water Management Bureau of the Ministry of Land, Infrastructure, Transport and Tourism Japan. The project started in 2019 aiming at exploring alternatives to the traditional float method for the purpose of labor saving and unmanned observation. In this project, simultaneous measurements by different techniques such as float method, radar, and imaging were conducted to compare the performance and the accuracy was examined with the data measured by ADCP (Teledyne RD Instr. River Pro). The specifications of the observation sites such as the drainage area, the bed slope, $D_{60}$ , the Manning’s roughness coefficient in the lower channel, the return period, the average annual maximum discharge and the design discharge are shown in Table [tbl5]. The measurements at the Jinshin Bridge were conducted by using a far infrared camera (FLIR, FC-632-O (19mm), 640 by 480 pix.) while the other was carried out using a conventional monitoring camera (Panasonic, WV-SUD638, 1920 by1080 pix.). The cameras were mounted on a long pole installed on the embankment with a height about 10 m above the water surface. At each site, the field of view of the camera is set to cover the measurement area by the ADCP in order to compare the result of the concurrent measurements. The ADCP measurements were carried out by slowly tugging the boat equipped with ADCP along the bridge back and forth once. This is a conventional method used in Japan. The bin thickness was 25 cm and the blanking distance from the water surface was 39 cm at the Jinshin Bridge and 54 cm at the Shimada Bridge. At each station an anemometer was installed near the bridge for wind speed measurements. This study will focus on cases where either wind or velocity dip effects can be examined.

In STIV measurements, the length of the search line was about 20 m at the Shimada Bridge and about 16 m at the Jinshin Bridge. With this setup, the camera depression angle covering the entire river width ranges from 5.2 to 21.6 degrees, satisfying the recommended minimum angle of 3 degrees83. STIV analysis was performed using 30 fps video footage of about 20 or 30 seconds. Wind speeds were converted to values at an altitude of 10 m.

Table 6. STIs and related data for two cases; Shimada Bridge, 14:30, Oct. 12, 2019 and Jinshin Bridge, 4:00, June 19, 2020.

[tbl6]

Fig. 10. Velocity distributions measured by various methods.

The oblique image with 40 search lines and its rectified image for each site are shown in Fig. 9. The measurements were performed at 14:30, Oct. 12, 2019 at the Shimada Bridge and 4:00, June 19, 2020 at the Jinshin Bridge. The small square marks seen in the left figure are GCPs used for the image rectification. At the Jinshin Bridge, the image is in black and white because it was taken late at night with a far-infrared camera. Although it was late at night, the far-infrared camera clearly captured the river surface textures. At the Shimada Bridge, the current was fast and there occurred a large surface swell, but the velocity distribution was analyzed well as will be shown below.

Table [tbl6] shows typical space-time images (STIs) for each site. At the Jinshin Bridge, the oblique pattern is nearly straight, indicating that the flow in each search line is nearly constant. In contrast, at the Shimada Bridge, the oblique pattern is mixed with a random pattern caused by water surface undulations and bubbles. Paying attention to the STI in No. 15, the white thick vertical pattern indicates that the position of the bubble above the standing wave has not moved in time, but the oblique pattern representing the flow crosses it, so the surface velocity can be determined from the slope of the pattern. The clear black vertical lines on No. 40 STI of the Shimada Bridge are due to the panels and markers placed on the riverbank. Table [tbl6] also shows the distance from the camera to the search line, pixel resolution on the search line, depression angle of the search line, and flow velocity. The No. 40 STI at the Jinshin Bridge has a coarse pixel resolution of 10.9 cm/pixel, but as the oblique pattern is clear, the velocity analysis can be conducted without any problems.

Fig. 11. Surface velocity distribution by LSPIV.

In the analysis, the spatial resolution of the measurement was first verified by changing the number of search lines in three different ways (10, 20, and 40 lines). STIV results for the case of 40 search lines are shown in Fig. 10. The effect of the number of search lines will be discussed in the next chapter. In the following discussions, results using 40 search lines are used unless otherwise stated. T.P. in the figure is the abbreviation of Tokyo Peil (Dutch word for datum), which is a measure of height from the average sea water level of the Tokyo Bay. The manual measurement, STIV measurement by a deep learning technique using Hydro-STIV86, LSPIV measurement by Fudaa-LSPIV 91, and the two ADCP data obtained by going back and forth between the left bank and the right bank are compared. The manual measurement method finds the pattern gradient visually.

In the case of the Shimada Bridge, the velocity distribution analyzed by STIV with the deep learning technique agrees well with the manually measured data as shown in Fig. 10(a). The STIV results also agree well with the two ADCP data measured 54 cm below the water surface. For comparison, surface velocity vectors obtained by LSPIV are shown in Fig. 11(a). In LSPIV, a window size of 60 by 60 pixels (3 m by 3 m) is used after some trial calculations using Fudaa-LSPIV. The lateral velocity distribution in Fig. 10(a) was obtained by taking the average of the velocity distributions of several measurement lines in the center. LSPIV captured well the central part of the profile including the maximum velocity but underestimated in the region far from the camera position. This difference may be due to the fact that the resolution is lower at locations far from the camera and that the measurement space is different between STIV and LSPIV.

In the case of the Jinshin Bridge, STIV with the deep learning technique also shows a comparable result to the manual measurement of STIs as shown in Fig. 10(b). Most of the STIV data is consistent with the ADCP data. The two localized reductions in velocity at approximately 50 m and 90 m from the left bank are due to the wake effects of the two piers. STIV has successfully captured these effects from surface flow measurements. In this case, LSPIV did not perform well in measuring the velocity distribution, yielding rather smaller velocities. Fig. 11(b) provides the velocity vectors obtained by LSPIV. This error is due to the small depression angle and low image resolution as seen in the background image, which causes the image of the distant camera points to be stretched by geometric correction, blurring the fine textures that are useful for pattern matching.

Table 7. Parameters on roughness $n$ and $\alpha$ defined by log-law, power-law and MEM at measurement sites; $y_0$ : roughness height, $H$ : mean water depth, $n$ : Manning $\prime$ s coefficient, $M$ : entropy parameter.

[tbl7]

Fig. 12. Comparison of vertical distributions in the center of the river; $X$ is the relative distance from the left edge of water surface.

Evaluation of vertical velocity distributions

Estimation of surface velocity coefficients

Table [tbl7] shows surface velocity coefficients calculated by various methods and related parameters. In the case of the logarithmic velocity distribution, the roughness height $y_0$ in Eq. (1) was calculated by $D_{60}/10$ as explained in section 2.1. Here $D_{60}$ is used as $D_x$ . Since the water depth is roughly 2.5 m at the Shimada Bridge and 3 m at the Jinshin Bridge, $y_0/H$ is 0.00268 in the former and 0.00367 in the latter. The Manning’s roughness coefficient calculated by Eq. (4) was 0.0251 to 0.0338 at the Shimada Bridge and 0.0272 to 0.0368 at the Jinshin Bridge. The river plan values of 0.033 and 0.038 presented in Table [tbl5] are roughly consistent with the upper end of the range. The surface velocity coefficients ${\alpha }_{log}$ calculated by Eq. (5) for log-law are 0.834 at the Shimada Bridge and 0.825 at the Jinshin Bridge, which are slightly smaller than the conventional value of 0.85. For the power law with $m=6$ , ${\alpha }_{power,m=6}$ is 0.857, which is comparable to the conventional value. For $m=7$ , ${\alpha }_{power,m=7}$ is 0.875. For MEM, the entropy parameter $M$ was calculated from the ADCP measurement data using Eq. (8) or Eq. (28) and was 2.98 for the Shimada Bridge and 1.85 for the Jinshin Bridge. ${\alpha }_{MEM}$ calculated by Eq. (27) was 0.853 at the Shimada Bridge and 0.816 at the Jinshin Bridge. Generally, surface velocity coefficients estimated from the entropy parameters are comparable to the default value of 0.85 at the Shimada Bridge and smaller than the other methods at the Jinshin Bridge.

Fig. 12 compares the vertical velocity distributions at the center of each river normalized by the surface velocity. Here, $X$ is the relative distance from the left edge of water surface. Compared in this figure are the vertical velocity distributions measured by ADCP and those estimated by MEM, the log-law, and the power law. In the case of the Shimada Bridge, there appears a small kink near the water surface in the MEM profile due to the effect of drift current by the wind with a speed of 4.89 m/s. Except for this point, MEM profile takes a similar profile by the log-law, while the power law takes relatively larger values closer to the bed. The ADCP plots takes similar profile to the other profiles. This degree of difference among the methods is not expected to make a significant difference in discharge estimation. At the Jinshin Bridge, the wind effect was negligible because of the small wind speed of 0.5 m/s. The log-law and MEM take similar profiles to ADCP data except the MEM overestimates the velocity near the water surface. Both the $m=6$ and $m=7$ power laws overestimate the flow velocity near the bottom, which is the reason for the relatively large surface velocity coefficient in this case.

Fig. 13. Comparison of vertical distributions near the bank of the Jinshin Bridge; X is the relative distance from the left edge of water surface.

Fig. 14. Variation of the Froude number and surface velocity coefficient $\alpha$ at Jinshin Bridge.

Evaluation of velocity dip

Because of the large aspect ratio of the river in this study, the effects of the dip are limited to the vicinity of the riverbank, which was observed by ADCP at several sections at the Jinshin Bridge. To examine the performance of the four dip equations, Eqs. (22) to (25), the well-observed vertical velocity distributions at $X=3$ m and $X=6$ m measured on July 14 are compared in Fig. 13. $X=3$ m corresponds to the location of $X/H_i=0.92$ and $X=6$ m corresponds to the location of $X/H_i=2.2$ , where $H$ is the local depth. In the ADCP data, there is a clear dip at $X/H_i=0.92$ , whereas at $X/H_i=2.2$ it almost disappears, so roughly $X/H_i\approx 2.5$ is considered to be the area of influence by the riverbank secondary flow at the Jinshin Bridge. At $X=3$ m, Guo’s equation reproduces the vertical velocity distribution well; Yang’s equation has a slightly smaller velocity but a similar profile. On the other hand, Pu’s equation and Bonakdari’s equation overestimate the entire velocity with deeper dip depth. At $X=6$ m, the effect of dip is negligible, but Guo and Yang’s equation reproduces the overall vertical velocity distribution well. Pu’s and Bonakdari’s equations failed to reproduce the profile. Fig. 14 shows the lateral distribution of surface velocity coefficient at the Jinshin Bridge using Yang’s equation, and as expected, the effect of the velocity dip is limited to a narrow region. The lateral distribution of the Froude number is also shown in Fig. 14, indicating that the flow is subcritical. The impact of the dip should also be influenced by the condition of the riverbank, i.e., whether or not there is vegetation near the bank. Therefore, further study is needed to evaluate the generation of secondary flows in the field conditions. It should be emphasized that it is impossible to reproduce such a distribution using conventional velocity distribution formulas. Therefore, when performing STIV analysis such as in canyons with small aspect ratios, care should be taken to account for dips or the discharge would be underestimated.

Fig. 15. Cross-sectional velocity distributions.

Cross sectional velocity distributions

Figs. 15(a) and (b) provide cross-sectional velocity distributions at the Shimada Bridge measured by ADCP and those reconstructed from the surface velocity distribution by using Eq. (30). The ADCP data is the average of left-to-right path and the right-to-left path. Note, however, that the ADCP is a direct measurement of internal flow, whereas the MEM distribution is an internal velocity distribution estimated using only the surface velocities on the search lines. The displacement of the main flow to the left bank is due to the pier located at $X=25$ m. The MEM results show a slight wind effect near the water surface, but the effect is not pronounced because the wind speed is not very high. The cross-sectional velocity distribution at the Jinshin Bridge is shown in Figs. 15(c) and (d). The distribution from the ADCP measures well the influence of the two piers; the MEM shows rather a smooth variation, but the velocity distribution, including the influence of the piers and the velocity gradient near the riverbed, are successfully estimated.

Discussions

Calculation of discharge

In the method proposed in this research, the surface velocity distribution by STIV and the surface velocity coefficient estimated by MEM can be used to calculate the flow rate by the velocity-area method, including the effects of wind and velocity dip. Velocities near the embankment outside the measurement range of STIV are calculated assuming the same Froude number as the measurement point closest to each embankment. Regarding the effect of the number of search lines, discharges calculated using 10, 20, and 40 search lines showed a relative error of less than 1% with each other as long as the coverage area is the same although not shown as a figure. Therefore, the number of search lines need not be very large as long as the lateral velocity distribution corresponding to the bathymetry can be covered. The discharge measured by ADCP also includes uncertainty that depends on how to fill in the data within blanking zones near the water and bottom. In the present analysis, the bottom zone was extrapolated by the wall law. The surface water zone was extrapolated by applying a three-point extrapolation. The three-point method is a linear approximation using the velocity values of the three ADCP layers closest to the water surface to obtain the velocity values at the water surface. Hereafter, ADCP data with the three-point extrapolation is treated as a standard reference discharge.

Fig. 16. Comparison of measured and estimated discharges.

Fig. 16 compares flow rates measured by the ADCP with those estimated by various methods. MEM with no wind was calculated using Yang’s equation as a dip formula. In the case of the Jinshin Bridge, MEM with Bonakdari’s equation. yields the closest data to ADCP, followed by log-law and MEM with Pu’s equation. In this case, as the wind speed was small, its effect is negligible as can be seen from $Q_{MEM\left(Yang\right)}$ and $Q_{nowind}$ . On the other hand, the conventional $\alpha =0.85$ and power-laws overestimate the discharge by about 3.3 to 6.2 percent. In Shimada Bridge case, MEM with Yang’s equation gave the closest data to ADCP, followed by MEM with Guo’s equation. and MEM with Pu’s equation. In this case, comparing $Q_{MEM\left(Yang\right)}$ and $Q_{nowind}$ , the error of 6.4% is improved to 1.6% by introducing the wind effect. Conventional methods such as log-law, power-law or $\alpha =0.85$ overestimate the discharge. It should be noted that the application of the log-law requires detailed data on the riverbed material, without which accurate velocity distribution cannot be estimated. With respect to the power law, an approximate velocity distribution can be reproduced by changing the value of $m$ , but it is difficult to know the exact value in advance. A comparison of the two cases shows that applying the MEM, which incorporates wind effect and velocity dip, to STIV provides more accurate and stable flow estimates than conventional methods.

Sensitivity analysis of $M$ on $\alpha$

Sensitivity of $M$ on surface velocity coefficient $\alpha$ can be seen in Fig. 4. With no velocity dip ( $h=0$ ), the relative errors of $\alpha$ when $M$ changes from 1 to 2 (relative error of 100%), 2 to 3 (50%), and 3 to 4 (33%) are 6.7%, 3.7%, and 2.8%, respectively. This indicates that changes in $\alpha$ are not sensitive to large changes in $M$ . Since $\alpha$ is directly related to flow discharge, the same is true for the relationship between discharge and $M$ . In other words, a slightly larger error in $M$ does not significantly affect the discharge or $\alpha$ .

Sensitivity analysis of wind effect

The wind effect on the surface velocity coefficient $\alpha$ has already been shown in Fig. 8. Numerically, when the surface velocity $u_{surf}$ ignoring the wind effect is 3 m/s, $\alpha$ decreases by $-4.3\%,-8.2\%,$ and $-15.0\%$ as the forward wind increases to 5 m/s, 10 m/s, and 20 m/s, respectively. Conversely, $\alpha$ increases by 4.7%, 10.0%, and 22.5% respectively as headwinds increase to 5 m/s, 10 m/s, and 20 m/s. For a surface velocity $u_{surf}$ of 5 m/s, $\alpha$ decreases by $-2.6\%,-5.1\%,$ and $-9.7\%$ as the forward wind increases to 5 m/s, 10 m/s, and 20 m/s, respectively, while as the headwind increases to 5 m/s, 10 m/s, and 20 m/s, $\alpha$ increases by 2.8%, 5.8%, and 12.3%, respectively. In other words, as surface velocity increases, the effect of wind velocity on $\alpha$ , and hence on flow rate, decreases. It should be noted that for the same observed surface velocity, the discharge increases with an adverse wind, because the wind drift increases with the wind as can be inferred from Eq. (32), and vice versa.

Conventionally, the wind drift coefficient $f$ is used to correct the surface velocity by using the following formula:

  

$$u_{surf,c}=u_{surf,o}-f{U}_{10}\left(33\right)$$

where, $u_{surf,c}$ is the corrected surface velocity and $u_{surf,o}$ is the observed surface velocity. When the wind drift coefficient of 0.03 (Bechle et al.62) is considered, it is found to produce errors that are almost equal to or slightly excessive compared to the MEM. On the other hand, $f=0.074$ (Motonaga et al.76) leads to a considerably excessive error compared to MEM. For example, in the case of the Jinshin Bridge, under a headwind of $-10$ m/s, the change in discharge due to wind is 8.8% by MEM and 11.1% at $f=0.03$ while it is 27.4% at $f=0.074$ , which seems to be too large. Note that the examples shown here are only estimates, as it is very difficult to observe flow in actual conditions where wind speeds exceed 10 m/s and water surface fluctuations are large.

Reevaluation of the entropy parameter

Another way to evaluate the entropy parameter is to compare with the formula proposed by Moramarco and Singh92:

  

$$\varphi \left(M\right)=\frac{e^M}{e^M-1}-\frac{1}{M}=\frac{\frac{1}{n}\frac{R^{1/6}}{\sqrt{g}}}{\frac{1}{\kappa }ln\left(\frac{y_{max}}{y_0}\right)}\left(34\right)$$

where $R$ is hydraulic radius, $g$ is the gravitational acceleration, $\kappa$ is the Karman constant, $y_{max}$ is the maximum depth. According to this formula, $\varphi \left(M\right)$ is 0.742 at the Shimada Bridge and 0.699 at the Jinshin River. Therefore, $M=3.45$ and $\alpha =0.865$ at the Shimada Bridge and $M=2.61$ and $\alpha =0.843$ at the Jinshin Bridge. The values of $\alpha$ are overestimated by about 1.4% at Shimada Bridge and 3.2% at Jinshin Bridge. Therefore, Eq. (34) can be used as a first guess of surface velocity coefficient when the data for bed friction such as Manning’s coefficient $n$ and $y_0$ are available.

To examine the consistency of the entropy parameter, several pairs of $U$ vs $u_{max}$ are collected to calculate $\varphi \left(M\right)$ as shown in Fig. 17. It can be noted that there is a linear relation between $U$ and $u_{max}$ for both bridge locations as has been reported by many researchers 63,65,92. From the plots, $\varphi \left(M\right)$ is estimated to be 0.720 at the Shimada Bridge and 0.643 at the Jinshin Bridge. The entropy parameter $M$ is 3.00 and $\alpha =0.854$ at the Shimada Bridge while $M=1.769$ and $\alpha =0.812$ at the Jinshin Bridge. These values are quite consistent with the data obtained from ADCP data. Therefore, at locations where ADCP observation data are available, highly accurate flow observation can be expected simply by analyzing surface flow velocity using STIV. In addition, at locations where the flow is stable and there are no significant changes in the velocity distribution, the discharge can be estimated simply by measuring the maximum surface velocity using STIV.

Fig. 17. $U$ vs $u_{max}$ for various discharges.

Conclusion

In this study, we paid attention to the surface velocity coefficient $\alpha$ from various points of view. Firstly, traditional tables of $\alpha$ are compared and summarized for a rough estimate from sediment hydraulics viewpoint. The variation of $\alpha$ is linked to the variation of equivalent roughness height in rough wall turbulent flows. Secondly, an equation for the cross-sectional distribution of the surface velocity coefficient was obtained by adding the effects of wind and velocity dip to the conventional entropy theory. This enabled accurate estimation of the discharge from the surface velocity distribution obtained by STIV. In addition, an approximate explicit expression between $\alpha$ and $M$ and a functional relationship between $\varphi \left(M\right)$ and $M$ are proposed for convenience. Thirdly, the surface velocity coefficients described above were applied to flow measurements in rivers of different scales. These were the Shimada Bridge (35 m wide) on the Nakamura River and the Jinshin Bridge (126 m wide) on the Ota River. To evaluate the measurement accuracy of the proposed method, simultaneous ADCP and STIV measurements were performed, showing that the proposed method can measure flow more accurately than other methods using vertical velocity profiles. Fourth, sensitivity checks showed that small variations in entropy parameters have little effect on discharge estimates, and that accounting for wind effects on flow measurements improves the accuracy of flow estimates. Finally, an applicability of the assumption of the $\varphi \left(M\right)=$ constant for different flow velocities was tested by collecting a series of discharge data. Note that the entropy method assumes a similarity between maximum and average velocities for a wide range of water levels, so care must be taken with complex cross-sectional profiles where this similarity breaks down such as compound channels.

Due to the increase of flood disasters in recent years, simple, safe and efficient methods for estimating discharge have come to be required. In that sense, the combination of image-based technique STIV and MEM proposed in this research is one of the best solutions to meet such requirements. The introduction of velocity dips is not important for large rivers, but is essential for measurements on rivers with small aspect ratios. Since many of the downstream sections of first-class rivers in Japan have compound geometries consisting of lower channel and floodplain, it is necessary to examine the assumption of the constant value of $M$ for different flow depths in the future study.

Acknowledgments

The field measurement data used in this research are obtained in a joint campaign lead by the Water and Disaster Management Bureau, MLIT, Japan. We are grateful to the people involved in the field campaign especially IDEA Consultants, Inc. who provided us the video footage of the Nakamura River.

Footnotes

Construction Engineering Research Institute

Chuden Engineering Consultants

Hydro Technology Institute Co., Ltd., Japan

Hydro Technology Institute Co., Ltd., Japan

Hydro Technology Institute Co., Ltd., Japan

Gifu University

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