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Discharge estimation from H-ADCP measurements in a tidal river subject to sidewall effects and a mobile bed

Authors


Abstract

[1] Horizontal acoustic Doppler current profilers (H-ADCPs) can be employed to estimate river discharge based on water level measurements and flow velocity array data across a river transect. A new method is presented that accounts for the dip in velocity near the water surface, which is caused by sidewall effects that decrease with the width to depth ratio of a channel. A boundary layer model is introduced to convert single-depth velocity data from the H-ADCP to specific discharge. The parameters of the model include the local roughness length and a dip correction factor, which accounts for the sidewall effects. A regression model is employed to translate specific discharge to total discharge. The method was tested in the River Mahakam, representing a large river of complex bathymetry, where part of the flow is intrinsically three-dimensional and discharge rates exceed 8000 m3 s−1. Results from five moving boat ADCP campaigns covering separate semidiurnal tidal cycles are presented, three of which are used for calibration purposes, whereas the remaining two served for validation of the method. The dip correction factor showed a significant correlation with distance to the wall and bears a strong relation to secondary currents. The sidewall effects appeared to remain relatively constant throughout the tidal cycles under study. Bed roughness length is estimated at periods of maximum velocity, showing more variation at subtidal than at intratidal time scales. Intratidal variations were particularly obvious during bidirectional flow conditions, which occurred only during conditions of low river discharge. The new method was shown to outperform the widely used index velocity method by systematically reducing the relative error in the discharge estimates.

1. Introduction

[2] Continuous series of river discharge are crucial in studies of water resources. Rainfall-runoff models generally depend on water discharge series both for calibration and for validation of model concepts [McMillan et al., 2010]. Hydrodynamical models often rely on discharge series as boundary conditions [Liu et al., 2007]. Conventional methods to estimate water discharge series include a number of uncertainties which are dependent on flow conditions [Di Baldassarre and Montanari, 2009]. In large rivers, the relation between stage and discharge is often ambiguous [Petersen-Overleir, 2006]. Hysteresis effects often inhibit extrapolation of a rating curve beyond the range of measurements used for its derivation [Dottori et al., 2009]. In tidal rivers, the rating curve concept fails to describe the relation between water level and discharge because water level is not solely a function of river flow [El-Jabi et al., 1992]. Tides induce flows at time scales ranging from hours to days, invalidating the steady flow assumption. Time lags associated with these rapidly varying flows produce complex flow patterns across river channels. Horizontal acoustic Doppler current profilers (H-ADCPs) can measure water level and flow velocity across the river section, which in combination with moving boat ADCP measurements provide a promising alternative to conventional methods.

[3] Over the past decade, several methods to infer river discharge from H-ADCP measurements have been reported. Nihei and Kimizu [2008] developed a dynamic interpolation and extrapolation method, assimilating H-ADCP data with numerical simulations. Le Coz et al. [2008] compared the index velocity method (IVM) [see Simpson and Bland, 2000] and the velocity profile method (VPM) with several far-field extrapolation techniques. The IVM is widely being used and consists of regressing section-averaged velocity with an index velocity from the H-ADCP. The VPM computes discharge over the cross section (total discharge) from theoretical vertical velocity profiles made dimensional with the H-ADCP velocity measurements and integrated over the cross section. Recently, Hoitink et al. [2009] combined the IVM and VPM approaches in a semideterministic, semistochastic method to convert H-ADCP measurements to water discharge. The deterministic part relied on the validity of the law of the wall, to calculate discharge per unit width (or specific discharge) from single-depth H-ADCP velocity data. The obtained specific discharge is then regressed against time-shifted total discharge, which constitutes the stochastic part of the method. The method takes into account the time lag between specific and total discharge, which is relevant especially in tidal areas or wide inland rivers.

[4] Crucial in the deterministic part of the approach by Hoitink et al. [2009] is the determination of the effective hydraulic roughness length (z0), parameterizing river bed roughness. Throughout a single semidiurnal tidal cycle, they found that z0 remains relatively constant during periods of ebb and flood. Constancy of z0 assures the validity of the law of the wall, which allows for the estimation of depth mean velocity from measured single-depth velocity. In an alluvial channel, bed roughness depends on the nature of bed material and its spatial variations and on the dynamics of bed forms for a given bed material [Yen, 2002]. Although the former is linked to sediment grain properties such as grain size, the latter depends on flow depth and velocity. Changes in bed roughness due to bed forms can be substantial in flow over a sand bed [van Rijn, 1984, 2007]. In tidal environments, z0 exhibits variations between ebb and flood which are more likely to be caused by flow conditions than by bed composition [Dinehart, 2002]. Values of z0 may also vary over a spring-neap cycle [Cheng et al., 1999] and possibly over the long-term runoff fluctuations. Although the properties of bed material at a given cross section may be constant in time, the dynamic interplay of bed forms with flow conditions renders constancy of z0 questionable. Here we present data from five separate semidiurnal tidal cycles, providing insight into the spatiotemporal development of z0.

[5] H-ADCPs are typically being deployed at a river bank, which implies that the highest-quality flow measurements are obtained near the river bank. In open channels, the lateral wall is known to influence the shear stress distribution which may impact the cross-section averaged bed shear stress both in in-bank flows [Vanoni and Brooks, 1957; Cheng and Chua, 2005] and in compound channels [Shiono and Knight, 1991; Papanicolaou et al., 2007]. Sidewall effects affect not only the lateral shear stress distribution but also the velocity field in their proximity [Tominaga and Nezu, 1991]. In rivers, the position of maximum velocity in the water column generally appears below the surface, as opposed to the situation in tidal channels, where logarithmic velocity profiles prevail [Lueck and Lu, 1997; Sime et al., 2007]. The dip in the velocity profile is generally attributed to the generation of weak secondary flows [Cardoso et al., 1989; Nezu et al., 1993]. Whether these are driven by turbulence anisotropy or by channel geometry, secondary flows affect streamwise velocity profiles by redistributing momentum. Although the velocity dip is most pronounced in open channels having a width to depth ratio less than 5, wide open channels can also show this effect near the riverbank [Nezu and Nakagawa, 1993; Sukhodolov et al., 1998]. Based on an analysis of the Reynolds equations, Yang [2005] showed that the energy from the main flow is transported toward the nearest boundary through a minimum relative distance, or normal distance to the boundary. Accordingly, the flow region near the riverbank can “feel” the presence of the sidewall, resulting in a velocity distribution with the maximum velocity below the surface. In the present contribution we adopt a boundary layer model based on results from Yang et al. [2004a, 2004b] to account for sidewall effects in upscaling H-ADCP data.

[6] Discharge measurements obtained with a boat-mounted ADCP are affected by several sources of error [Gonzalez-Castro and Muste, 2007]. Errors in estimates of boat velocity, used to convert velocity data in instrument coordinates to Earth coordinates, can significantly bias discharge estimates. Boat velocity is measured with respect to a fixed reference by acoustic bottom tracking (BT) or by a Differential Global Positioning System (DGPS). BT-estimated velocities are biased by sediment transport and high sediment concentration near the bottom [Rennie et al., 2002]. In turn, DGPS velocity estimates are affected by boat operation, DGPS precision, and signal multipath artifacts related to riverbank vegetation [Rennie and Rainville, 2006]. Combining both systems, Rennie and Millar [2004] obtained spatial distributions of fluvial bed load sediment transport by linking the bias in boat velocity estimated with DGPS and BT and the apparent bed load velocity. In addition to the reference velocity, discharge measurements can be biased by heading errors [Kolb, 1995]. Heading errors can cause a bias in boat track and thus also in velocity measurements. Trump and Marmorino [1997] compared two independent estimates of boat velocity with BT in combination with a gyrocompass and a DGPS system. Results showed that boat speed estimates agreed, while direction estimates were strongly correlated to the boat heading from the gyrocompass. Here we present a correction method using a multiantenna system, which can minimize the errors posed by gyrocompasses and DGPS-derived headings after proper calibration and determination of the alignment between the ADCP and the compass.

[7] The structure of this paper is as follows: Section 2 introduces the boundary layer model accounting for sidewall effects, briefly repeating the work of Yang et al. [2004b]. Section 3 introduces the study area, data collection, and data processing methods. Section 4 describes the river bed composition, its morphology, and data referencing techniques. Section 5 presents an analysis of the flow structure, focusing on the H-ADCP measurement range. In section 6, the discharge estimation methodology is described, and section 7 presents the validation of the method. Sections 8 and 9 present a discussion and the conclusions, respectively.

2. Boundary Layer Model

[8] In a steady, uniform, and fully developed turbulent channel flow, the momentum equation in the streamwise direction can be written as

equation image

where s is defined as the streamwise direction; n is spanwise direction; z is normal distance from the bed; u, v, and w are mean velocity components in the s, n, and z directions, respectively; and τsz ≈ −ρequation image and τsn ≈ −ρequation image, where u′, v′, and w′ are turbulent velocity fluctuations; ρ is fluid density; g is gravity acceleration; and S denotes energy slope.

[9] Near the bed, the first term on the left-hand side of equation (1) is much greater than the second term [Yang et al., 2004b]. Integration along the vertical direction yields

equation image

where u*,b is shear velocity at the bottom. Equation (2) can be rewritten after the global shear velocity u* is introduced:

equation image

where α1 = (gHSu*2)/u*2, H is water depth, and c1 = (u*,b2u*2)/u*2. Measured profiles of Reynolds shear stress in open channel from the centerline to the sidewall show that −equation image/u*2 approaches 1 as z/H comes close to 0 [Immamoto and Ishigaki, 1988], indicating that c1 can be neglected.

[10] The third term on the right-hand side of equation (3), reflecting the influence of secondary currents, can be approximated by the linear relation [Yang et al., 2004b]

equation image

where α2 > 0. Therefore, an approximate relation for the Reynolds shear stress profile in open channel flow can be obtained as

equation image

where α = α1 + α2 > 0. The cross exchange of momentum by secondary flows is empirically modeled by steepening the dimensionless Reynolds shear stress profiles because generally, secondary currents near the surface act in the downward direction and near the bed in the upward direction [Nezu and Nakagawa, 1993; Yang et al., 2004b].

[11] Mean velocity profiles can be obtained assuming that

equation image

in which the turbulent eddy viscosity νt can be expressed as

equation image

where κ ≈ 0.4. Substituting equations (6) and (7) into equation (5), we obtain

equation image

which, after integration, yields the following expression for the velocity profile affected by sidewall effects:

equation image

where z0 is the roughness length.

[12] The second term on the right-hand side of equation (9) decreases with depth, thus creating the velocity dip at the surface. The parameter α can be directly related to the relative height above the bottom of the maximum velocity according to

equation image

Results from experiments including a wide range of channel aspect ratios [Yang et al., 2004b] indicate that α increases toward the banks, where the velocity dip becomes most pronounced. The dip correction factor α can be approximated with the expression

equation image

where n is the spanwise coordinate or distance from the bank.

3. Study Area and Data Collection

[13] Measurements were carried out in a 420 m wide cross section in the River Mahakam, East Kalimantan, Indonesia (Figure 1). Salinity intrusion generally reaches to about 10 km seaward from the delta apex. Salinity intrusion can reach beyond the delta apex only during extremely low flows, such as the El Niño-related drought in 1997. The study area is therefore generally subject to freshwater conditions. Due to the mild slope of the river, the tidal wave can propagate up to 190 km from the river mouth, depending on the river discharge. A 600 kHz H-ADCP manufactured by RD Instruments (RDI) was mounted for 525 days on a solid wooden jetty in a straight reach of the river, between two bends. This location was selected because of its relatively narrow cross section, maximizing the fraction of the river width covered by the H-ADCP (about one third). In addition, riverbanks at this particular location seem to be virtually fixed because they are naturally protected by the outcrops of a tertiary system [van Bemmelen, 1949]. The H-ADCP was mounted at 1.5 m below the lowest recorded water level and about 5 m from the bottom. Pitch and roll of the instrument remained constant during the measuring period, amounting to 0.06° and 0.55°, respectively. Because mean water depth rapidly increases to about 20 m, main and sidelobe beam interference due to bottom reflections was not expected because at 100 m range the vertical displacements of the beams would be 0.1 and 4 m, respectively. The measurement protocol for the H-ADCP consisted of 10 min bursts at 1 Hz, every 30 min. An ensemble was an average over 600 pings, and the horizontal cell size was 1 m. The range to the first cell center was 1.96 m.

Figure 1.

Location map showing the position of the H-ADCP in the River Mahakam. Salinity intrusion generally reaches to about 10 km seaward from the delta apex.

[14] Conventional boat-mounted ADCP discharge measurements were periodically taken in front of the H-ADCP. The research boat was equipped with a 1.2 MHz RDI broadband ADCP measuring in mode 12, a multiantenna Global Positioning System compass operating in differential mode (DGPS), and a single-beam echo sounder. The ADCP measured a single ping ensemble at approximately 1 Hz with a depth cell size of 0.35 m. Each ping was composed of six subpings separated by 0.04 s. The range to the first cell center was 0.865 m. The boat speed ranged between 1 and 3 m s−1.

[15] Along-channel (s) and cross-channel (n) coordinates for each ADCP campaign were defined on the basis of bed morphology following Hoitink et al. [2009]. Easting and northing coordinates of the depth map were rotated systematically in steps of 0.5°. For each rotation step, the root-mean-square deviations from mean values in the potential s direction were averaged. Depth variation along the s coordinate was found to be minimal when it deviated 165° from the north. Therefore, the positive s coordinate is defined as 165° with respect to the north. The n coordinate points perpendicular to the s coordinate, counterclockwise with its origin at the riverbank where the H-ADCP was deployed (Figure 2). The z coordinate was defined pointing upward with its origin at mean water level. Mean water level was defined as the mean over the 525 days of observations. The variation around the mean water level, η, ranging roughly from −1 to 1 m, was caused by the combination of tidal and subtidal fluctuations.

Figure 2.

Definition sketch (top view), where u is velocity in the flood direction, coinciding with the s axis; v is across channel velocity toward the inner bend along the n axis; ϕ is the H-ADCP beam separation angle; θ is the angular difference between the n axis and the axis of the central acoustic beam of the H-ADCP, measured positive as indicated; and u1, u2, and u3 are the radial velocities along the three acoustic H-ADCP beams.

[16] The H-ADCP measures along three beams in a horizontal plane, with ϕ = 25° angles between the beams (Figure 2). In the current deployment, the pitch of the HADCP was nearly 0°, and the central beam axis was rotated by an angle θ = 1.8° relative to the n coordinate, in an anticlockwise direction. The along-beam velocities, denoted by u1, u2, and u3, are positive toward the transducers and relate to the equation image and equation image velocity components according to

equation image
equation image
equation image

where the hat symbol is used to indicate that the velocity components can be considered to be a volume average over an acoustic target cell. Two of the three equations (12)(14) suffice to calculate equation image and equation image. The redundant beam is included in the instrument for error estimation. Because the n axis falls between the centerlines of beams 1 and 3, we chose to calculate equation image and equation image from equations (12) and (14), limiting the maximum beam separation to about 60 m at n = 150 m. Figure 3 shows the absolute difference between velocity estimates obtained using equations (12) and (14) and corresponding estimates using equations (13) and (14), as a function of velocity magnitude, computed as the average of both estimations. The error velocity is relatively large when the velocity magnitude is small (<0.2), which may be related to a reduction of flow homogeneity during weak flows. A minor systematic error increases with velocity magnitude, amounting to about 5 × 10−3 m s−1 at flows of 1.2 m s−1. The normalized frequency distribution for error velocities in the range between 0 and 0.01 m s−1 shows that 90% of the error is concentrated in the range 0–0.005 m s−1, confirming nonuniformities between the beams to be negligible.

Figure 3.

(left) Absolute difference between velocity estimates obtained using equations (12) and (14) and corresponding estimates using equations (13) and (14), as a function of velocity magnitude, computed as the average of both estimates. (right) Normalized frequency distribution of error velocity values lower than 0.01 m s−1. Bin centers are spaced by 0.001 m s−1.

[17] To categorize flow conditions at each of the moving boat ADCP campaigns, we computed an index velocity (uI) as the space-time average of the H-ADCP velocity components in the s direction. The mean and the linear drift were removed from water level time series recorded by the H-ADCP. Time series of water level elevation and index velocity were subjected to a linear low-pass filter with cutoff frequency corresponding to 4 days, to yield the subtidal fluctuations. Subtidal fluctuations were subsequently filtered with a cutoff frequency corresponding to 56 days, to yield seasonal fluctuations (Figure 4).

Figure 4.

(top) Water level elevation and (bottom) index velocity from H-ADCP velocity profiles in gray lines. The phase difference between water level and index velocity is approximately 1.5 h. White and black lines show subtidal and seasonal fluctuations, respectively. Thin dotted lines indicate mean values. Vertical dashed lines indicate the date of each moving boat ADCP campaign.

[18] The mean index velocity amounted to 0.61 m s−1 in the downstream direction. High- (low-) flow conditions were defined as periods above (below) the mean index velocity. Spring (neap) conditions were defined as those periods when the difference between subtidal and seasonal fluctuations was positive (negative). Some ADCP campaigns fall close to the intersection between the subtidal and the seasonal fluctuation curves, which therefore represent mean tide conditions. Five 13 h ADCP campaigns were carried out spanning high- and low-flow conditions during spring and neap tides. A summary of the tidally averaged quantities during the moving boat ADCP campaigns is presented in Table 1. During low-flow conditions, tidally averaged discharge reaches well below 1000 m3 s−1, with instantaneous flow in downstream as well as upstream directions. During high-flow conditions, tidally averaged discharge attains values ranging between 4000 and 7000 m3 s−1, with instantaneous flow in the downstream direction only. We used three ADCP campaigns for calibration purposes and the remaining two for validation of the method.

Table 1. Summary of the Boat Mounted ADCP Campaigns With Tidally Averaged Quantities
NameDateFlowTideW (m)A (m2)η(m)U (m s−1)Q (m3s−1)
Cal130 Nov 2008highspring42086500.360.806760
Val117 Jan 2009highspring42086100.270.756420
Cal212 Mar 2009highspring42086200.290.564780
Val224 May 2009lowmean40081100.040.524410
Cal36 Aug 2009lowneap4108140−0.380.10740

4. Bed Composition and Data Referencing

[19] Transect data across the river with a single-beam echo sounder were projected on a curvilinear grid based on linear interpolation [Legleiter and Kyriakidis, 2007] to produce the bathymetric map of the river (Figure 5). The bathymetry downstream of the H-ADCP location depicts a relatively shallow reach of about 12 m depth, while at the measurement section and upstream, very deep areas of up to 35 m occur. These deep trenches are most likely caused by confinement of the flow by nonerodible banks. The depth of these trenches is 2–3 times the mean water depth. Bed samples were obtained with a Van Veen grabber at locations near the H-ADCP. Samples from ten transects consisting of five bed samples each were sieved into 11 size classes to obtain a grain size distribution. Figure 5 also shows a map of the median grain size D50, based on interpolation of the samples. The spatial distribution of D50 indicates that the river bed is mainly composed of fine to medium sands (D50 = 200–300 μm). Riverbanks comprise fine sands and large amounts of silt and clay. It is interesting to note the presence of some patches of coarser sand in the middle of the section, at irregular parts of the bathymetry.

Figure 5.

Bathymetry of the River Mahakam. Easting and northing coordinates correspond to UTM (zone 50M) with respect to the position of the H-ADCP (denoted with the asterisk). The inset is a map of median grain size D50 in μm in the surroundings of the H-ADCP location.

[20] To construct a local depth map, range estimates from acoustic bottom tracking were corrected for pitch and roll of the instrument and referenced to the mean water level. The depth estimates were projected on a rectangular grid with a mesh size of about 2 m, which is slightly larger than a typical footprint of the ADCP beams, covering 1.5 m2. Considering each beam of the ADCP as an independent depth estimator, we computed the root-mean-square deviation (RMSD) between depth estimates from the four beams. Figure 6 shows the spatial distribution of RMSD for a particular ADCP campaign, where values above 2 m were discarded. It shows an increase in regions where the slope is higher (toward the bank and in the deeper section), highlighting the inaccuracy of the ADCP depth estimates due to errors primarily from pitch and roll angles. RMSD values in the region in front of the H-ADCP remain within 1 m, averaging to about 0.2 m. The coefficient of variation of the RMSD distribution, defined as the ratio of RMSD to the mean, indicates an overall error below 0.05, showing that the inaccuracy in bed topography estimates in the region in front of the H-ADCP with the bottom tracking system of the ADCP is acceptable.

Figure 6.

Spatial distribution of RMSD between depth estimates from the four beams of the ADCP, each averaged over a rectangular grid of 2 m spacing. The dashed line shows the 150 m range of the H-ADCP. The arrow points in the downstream flow direction.

[21] Conversion from single-point velocity to depth mean velocity requires an accurate, time-dependent description of bed topography along the measurement range of the H-ADCP. Therefore, we computed the bathymetric map from each ADCP campaign over the area that covers the measurement range of the H-ADCP. Figure 7 shows a series of depth maps produced with each of the ADCP campaigns, in which depth levels were confined to the range between 15 and 25 m. The local morphology shows a relatively flat bottom, gradually deepening toward the northeast of the cross section. A relatively large transverse bottom slope up to 6% can be found at about 100–150 m from the H-ADCP. A bed feature located in the center of the H-ADCP range appears to evolve in time, as suggested by the contour lines on the maps. The bathymetric data in Earth coordinates were transformed to local s-n coordinates and normalized with the width, computed as the length between the intersections of the n coordinate with the shorelines from a topographic map. Before normalization, we computed width, mean depth, and area of each transect. Mean and standard deviation of the width to depth ratios from all moving boat ADCP campaigns amounted to 22 and 2, respectively.

Figure 7.

Evolution of bed morphology in the area in front of the H-ADCP, produced from local bathymetric maps obtained during each of the moving boat ADCP campaigns. The dashed line shows the 150 m range of the H-ADCP. A bed feature at about half of the range of H-ADCP evolves in response to flow conditions.

[22] To compute flow velocity with respect to a fixed reference frame, boat speed must be subtracted first. Boat speed was computed for each ensemble with the BT and the DGPS compass system. BT-derived boat speed estimates were biased by sediment transport during high-flow conditions because the moving bed creates an apparent velocity in the same direction as the flow [Rennie et al., 2002]. Therefore, flow speed and discharge were biased low when using the BT system during high-flow conditions. Denoting the boat velocity vector as b, composed of a streamwise component bs and a normal component bn, we computed the difference between the boat speed vector from acoustic BT and from the DGPS compass, resulting in the bias vector:

equation image

Figure 8 shows width-averaged values of bbias,s as a function of width-averaged uI. The error bars indicate the standard deviation computed from the boat speed estimates from individual width cells. Focusing on the streamwise direction, the velocity bias correlates with flow speed during each of the five moving boat ADCP campaigns, which can be attributed to sediment transport. We transformed the flow velocity data in instrument coordinates to Earth coordinates with the DGPS system during high flows. During low flows (uI < 0.5), the BT estimate of the boat speed was used, for its smaller scatter.

Figure 8.

Scatterplot of bbias,s versus uI, showing the ship velocity bias as a function of flow speed in the streamwise direction.

5. Flow Structure

[23] The horizontal flow velocity vector u is composed of components u and v, defined in the s and n directions, respectively. Positive values of u coincide with downstream flow. Vertical profiles were transformed to relative height above the bottom according to

equation image

where η is water level variation. We normalized all transects within each ADCP campaign with the maximum width within that campaign, to yield a normalized spanwise n coordinate, β. We followed the same normalization procedure for the horizontal velocity profiles obtained with the H-ADCP. This way, all velocity measurements were consistently referenced in time and space, projected onto a uniform grid in (σ, β) coordinates. The grid spacing is typically 0.5 and 5 m in the vertical and spanwise directions, respectively. ADCP velocity measurements have contributions of mean flow, turbulence, and error components. To isolate the mean flow component from repeated transect measurements, we assumed the mass flux through (σ, β) grid cells to be constant in the streamwise direction within the measurement range. Therefore, the product of u and H + η is independent of s. The resulting time series were filtered with a cutoff frequency corresponding to 1.5 h, and the filtered values were divided by H + η and averaged in the s direction over the range that was covered during the measurements. Hereinafter, u denotes the mean flow component in the s direction.

5.1. Three-Dimensional Velocity Pattern

[24] Velocity profiles obtained from moving boat ADCP measurements were averaged over depth according to

equation image

[25] Figure 9 shows the spatiotemporal distribution of U and V for each of the five moving boat ADCP campaigns. The velocity patterns during the different campaigns feature similar spatial characteristics. A well-defined velocity core is centered at about β = 0.7. The magnitude and extension of the velocity core can be related to flow conditions. In Cal3, the velocity core becomes slightly shifted toward the center of the channel because the bathymetry downstream of the cross section has its thalweg in the middle of the river. Toward the opposite bank, U rapidly decreases to a zone of null velocity at about β = 0.8 and reverses for β > 0.8. Apparently, a recirculation cell in the horizontal plane is present, which may be the result of the sudden widening of the cross section (see Figure 4). The intensity of the recirculating flow is positively correlated with flow strength, suggesting that during high flows more momentum is withdrawn from the main flow in the form of a horizontal eddy. During low flows, however, the horizontal eddy persists even during flood tide, when the direction of depth mean flow reverses.

Figure 9.

Depth averaged (left) streamwise velocity U and (right) spanwise velocity V, as a function of normalized width and time, for each of the moving boat ADCP campaigns.

[26] Values of V contain contributions from the mean flow. A region where V = 0 is found in all campaigns at about β = 0.65 and in Cal1 at β = 0.7, which suggests that the along-channel direction obtained from the bathymetry coincides approximately with the mean flow over the deep trench.

[27] The three-dimensional velocity structure can be further understood from the tidally averaged flow field. For this purpose a velocity component u′ is defined which is aligned with the depth mean flow vector, according to

equation image

where the angular brackets denote averaging over a tidal cycle. Similarly, a zero-mean tidally averaged spanwise component reads as

equation image

Figure 10 shows patterns of 〈u′〉 and 〈v′〉 for each of the ADCP campaigns. The velocity core, about 280 m from the H-ADCP (β = 0.7), is located between middepth and the surface. Lines of equal velocity tend to compress much more over this portion of the section than in the region in front of the H-ADCP (β < 0.5 m). Close to the H-ADCP (β < 0.2 m), near-surface velocities tend to decrease, becoming small in comparison with the depth-averaged flow. This behavior can be related to the normal flows created by secondary circulation cells in the proximity of the bank [Nezu et al., 1993]. The tidally averaged flow field shows persisting secondary circulation cells across the section. The larger cell occupies half of the cross section and is likely triggered by curvature of the flow. The deeper part shows a more complex secondary circulation distribution, which may be linked to the three-dimensional flow pattern associated with the large bottom slopes in the deep trench.

Figure 10.

Spatial structure of (left) 〈u′〉 and (right) 〈v′〉 over the cross section during each of the moving boat ADCP campaigns. The vertical axes indicate depth in meters.

[28] The velocity field above the trench is intrinsically three-dimensional. Deterministic modeling of the flow would require a three-dimensional approach as the flow cannot be assumed to be uniform in the along-channel direction. The flow structure shows that the shape of the eddy varies systematically with flow strength in the velocity core, which suggests that the discharge through the eddying section can be predicted stochastically. The flow across the transect under study features two distinct zones. A section between β = 0 and 0.6 features a gradual increase of the flow strength with β and a strong secondary circulation that peaks in strength at β = 0.22. The second zone is between β = 0.6 and 1 and accommodates a complex three-dimensional eddy-type motion, which enhances the flow passing the trench and reverses the flow near the bank opposite to the H-ADCP. The measuring range of the H-ADCP is within the region β < 0.6, where the boundary layer model described in section 2 can be applied to convert H-ADCP data to specific discharge q = U(H + η).

5.2. Vertical Profiles of Streamwise Velocity in the H-ADCP Range

[29] Figure 11 presents vertical profiles of streamwise velocity equation image and spanwise velocity equation image, where the overline indicates width averaging over the range 0 < β < 0.6. Velocity profiles clearly deviate from the logarithmic distribution at about middepth and above. During the Cal3 campaign, velocity profiles show a pronounced peak near the surface during flood tide. Spanwise velocity fluctuations attain values up to 0.15 m s−1, with maximum spanwise velocities close to the bed and above middepth, suggesting curvature-induced secondary circulation.

Figure 11.

Width-averaged profiles of (left) equation image and (right) equation image as a function of normalized depth. Note that a different velocity scale was used in each ADCP campaign.

[30] Equation (9) can be rewritten in terms of σ, yielding

equation image

where u* is shear velocity and κ ≈ 0.4. The dip correction factor can be estimated as

equation image

where σmax is the relative height where the maximum velocity occurs. The degree to which the observed velocity profiles can be captured in the proposed boundary layer model was investigated in two steps. First, values of α were calculated by determining the relative depth of maximum velocity, σmax, for instantaneous velocity profiles. Since those velocity profiles are influenced by turbulence and noise, the relative depth where the mean flow velocity peaks is not readily obtained. To estimate σmax, we repeatedly fitted a logarithmic profile starting with the lowermost three ADCP cells, adding a velocity cell from bottom to top for each subsequent fit. The value of σmax is then established from the development of the goodness of fit, which decreases once cells above σmax are included in the fitting procedure. Figure 12 shows the goodness of fit of the data to the adopted velocity profile function, based on the mean absolute difference normalized with the mean velocity magnitude. The goodness of fit is consistently high in the range 0.1 < β < 0.55. In that range, it is typically below 2.5%, except for a small layer near the surface where it attains values up to 7.5%. This leads us to conclude that the model used to establish depth mean velocity is appropriate.

Figure 12.

Goodness of fit of the data to the adopted velocity profile function, based on the normalized, mean absolute difference.

[31] In the second step, u* and U were derived from the linear regression of u against (ln(σ) + 1 + α + αln (1 − σ))/κ. Figures 13 (top) and 13 (middle) present the tidally averaged estimates of u* and U, obtained from the regression analysis. The profiles of both 〈u*〉 and 〈U〉 prove to be highly consistent; that is, neighboring independent estimates are very similar. Figure 13 (bottom) shows the root-mean-square deviation (RMSD) between the estimated values of depth mean velocity from the regression, equation image, and direct estimates of U obtained from the observed velocity profiles, which were extrapolated to the bottom and bed. In the range of the H-ADCP (β < 0.6), the RMSD is below 0.06 m s−1 at all times, whereas in the trench zone the RMSD can reach up to 0.10 m s−1.

Figure 13.

Tidally averaged estimates of (top) U and (middle) u* computed from the regression based on equation (20). (bottom) Root-mean-square deviation between the direct estimates of U from the velocity profiles, and estimates from the regression analysis.

5.3. Roughness Length and Dip Correction Factor

[32] The estimates of equation image and u* can be used to estimate roughness length (z0), which proceeds from

equation image

[33] Figure 14 shows time series of the z0, geometrically averaged over the range 0 < β < 0.6, and α as a function of time since the start of the ebb in each of the moving boat ADCP campaigns. Herein, we apply high water slack (HWS) and low water slack (LWS) definitions to unidirectional flows with an appreciable semidiurnal tidal modulation by considering slack water to occur when U − 〈U〉 = 0. The start of the ebb is defined at HWS. During the period of maximum velocities, which lasts for 2 to 3 h, z0 and α estimates remain relatively constant in time during the ADCP campaigns. Substantial intratidal variations of z0 during Cal3 and Val2 can be primarily attributed to flow reversal, which renders velocity profiles unstable. Focusing on estimates of z0 during periods of maximum velocity shows that subtidal variations in roughness length are significantly larger than the intratidal variations. Roughness variations within a tidal cycle are only relevant during low-flow conditions, when bidirectional flows occur.

Figure 14.

Geometric mean (β < 0.6) of (top) z0, (middle) α, and (bottom) U as a function of time. Time series for each moving boat ADCP campaign commence at the start of the ebb period. Gray areas indicate the periods that are supposed to be void of slack water effects, during which z0 is relatively constant.

[34] Elaboration of equation (20) results in the following expression:

equation image

where

equation image
equation image
equation image

[35] Local values of z0 and α can be obtained from the bilinear regression with zero intercept through calculated values of g versus I + J and J. Figure 15 shows cross-river profiles of z0 and α over half of the channel width. Spatial variations in z0 are highest during bidirectional flows and increase systematically toward the bank. Cross-river profiles of α are consistent within transects featuring a minimum around β = 0.2 and an increment toward the center of the channel. In the vicinity of the riverbank (β < 0.2), width profiles of α according to equation (11) can be approximated with a Taylor expansion according to αA, where A = 1.3 and B = 1.3W/H. A linear fit yields A = 1.00, 1.24, and 1.38 and B = 5.23, 4.41, and 6.02 for each of the three calibration campaigns, respectively. These figures are consistent with A = 1.3 reported by Yang et al. [2004b]. However, the values of B are about 3 times smaller than the value obtained with the local aspect ratio of the river (W/H ∼ 22). This may relate to the fact that the velocity dip does not extend over the full river width.

Figure 15.

Cross-river profiles of z0 and α. Dotted lines remove spatial variations with wave lengths smaller than 50 m. The solid line is the best fit given by α = 1.2 – 6β for 0 < β < 0.16 and α = 0.24 for β > 0.16. The dotted line is given by equation (11).

[36] Figure 16 investigates the stage dependence of 〈z0〉, showing in Figure 16a a linear relation between 〈η〉 and the geometric mean over 0 < β < 0.6 of z0. Figures 16b and 16c explore whether the stage dependence of z0 varies over width, by splitting the region of interest in two. In the region 0.3 < β < 0.6 the relation shows a reduced linearity, whereas in the region 0 < β < 0.3 the stage-roughness relation agrees with one obtained for the region 0 < β < 0.6.

Figure 16.

Geometric mean of z0 as a function of tidally averaged water level for (a) 0 < β < 0.6, (b) 0 < β < 0.3, and (c) 0.3 < β < 0.6. Error bars indicate the standard deviation. Best fit lines represent log10(z0) = aη〉 – b with a = 2.5, 2.065, and 2.75 and b = 2.32, 2.3, and 2.3 for Figures 16a–16c, respectively.

6. Discharge Estimation Methodology

6.1. Deterministic Part

[37] Single-point H-ADCP velocity measurements uc can be translated to depth mean velocity U according to

equation image
equation image

where uc is the flow velocity array at normalized depth σc. Close to the riverbank, σc is typically in the range between 0.8 and 0.9, and α = 0.4 to 0.8. Within these ranges of variation, F takes values between 0.9 and 1.

[38] The following approach was followed to obtain continuous series of z0 and α over the range of the H-ADCP:

[39] 1. Values of z0 were predicted from the tidally averaged water level according to log10(z0) = 2.5 〈η〉 − 2.32 (see Figure 16a). Variation of z0 over width is ignored.

[40] 2. Values of α were predicted from a bilinear relation with β: α = 1.2 – 6β for 0 < β < 0.16 and α = 0.24 for β > 0.16. Variation of α in time is ignored.

6.2. Stochastic Part

[41] Using depth mean velocity estimates, specific discharge q = equation image(H + η) can be obtained from the H-ADCP velocity measurements. A stochastic model is adopted to relate q to the total discharge Q:

equation image

where W is the river width, f(β) is a constant amplification factor, and τ(β) is a time lag function to take inertial effects into account [Hoitink et al., 2009]. The time lag τ as a function of β was obtained by establishing the time difference between the occurrence of Q − 〈Q〉 = 0 and q − 〈q〉 = 0. The width-dependent amplification factor was obtained from a linear regression.

[42] Figure 17 shows width profiles of τ and f for the transition between ebb and flood (LWS) and vice versa (HWS) for each of the moving boat ADCP campaigns. Values of τ become remarkably large near the banks (β < 0.15). In the region β > 0.15 the agreement between values of τ and f from the different moving boat ADCP campaigns is high. In the region β > 0.7, τ exceeds half an hour. For β > 0.8, specific discharge does not follow total discharge anymore because of the flow reversals induced by the eddy. Profiles of f remain constant between the campaigns in the range 0.15 < β < 0.75. Discrepancies near the bank can be attributed to subtidal variations in flow strength, impacting the redistribution of specific discharge. Figure 17 also shows the relative root-mean-square error (rRMSE) in modeled values of Q, defined as the ratio of the RMSE in the modeled values of Q and the tidally averaged discharge 〈Q〉. Within 0.15 < β < 0.75, values of rRMSE remain below 0.1. Based on the rRMSE results in Figure 17 it can be concluded that the conversion of specific discharge to total discharge can best be based on H-ADCP measurements in the range β > 0.15. To calculate Q at any moment in time from estimates of q in that range, we use best fit lines of τ and f using the results from the three calibration campaigns, which read (1) τ = 8.9 − 4sin−0.9(1.25πβ) (see Figure 17, top) and (2) log(f) = −0.5 − 0.4log(β) − 0.33β (see Figure 17, middle).

Figure 17.

(top) Time difference between the occurrence of q − 〈q〉 = 0 and Q − 〈Q〉 = 0, in case of the transition from flood to ebb (HWS) and vice versa (LWS). (middle) Amplification factor f in equation (29). (bottom) Relative root-mean-square error in modeled values of Q.

7. Validation

[43] Nine months of velocity data obtained with the H-ADCP spanning a wide range of flow conditions were used to produce a continuous series of water discharge. Following the approach described in section 6, mutually independent estimates were obtained from each horizontal cell. Improved estimates of Q were obtained by averaging over 0.15 < β < 0.35. The validation was realized using data from two moving boat ADCP campaigns which were not used for the estimation of the model parameters z0, α, f, and τ. The first validation data set represents high-flow conditions during spring tide, and the second was during conditions of low river flow and a mean tidal range.

[44] Figure 18 shows the correspondence between discharge computed with the boat-mounted ADCP (Qtrans) and the present method (QHADCP). Peak discharges exceed 8000 m3 s−1. The RMSD between QHADCP and Qtrans amounted to 330 and 460 m3 s−1 for high- (Val1) and low-flow (Val2) conditions. The relative difference remains below 10% during the periods of validation and is highest during low discharges. Figure 18 also shows estimates of discharge obtained using the widely used index velocity method [e.g., Le Coz et al., 2008] and the relative difference with the boat-mounted ADCP campaign. Both during high flows and during low flows, the relative difference between modeled and measured discharges is larger using the IVM. During high flow, peak values of the relative difference reduce from 0.16 for the IVM to 0.11 for the present method. The comparison during Val1 seems to be better than during Val2. However, part of this relates to the difference in vertical scale, which was necessary because Val2 includes slack water, when relative errors become large. During slack water, the errors involved in the discharge estimates from shipborne velocity data, which are considered to be the “truth,” become of the same order of magnitude as the discharges derived from H-ADCP data.

Figure 18.

(top) Discharge as a function of time computed from the moving boat ADCP data (Qtrans), from H-ADCP data processed with the present method (QH-ADCP) and with the index velocity method QIVM for the two validation data sets. (bottom) Relative error in discharge measurements as a function of time for the present method and for the IVM.

8. Discussion

[45] Studies quantifying uncertainties in estimates of roughness length from velocity profiles have generally focused on rigid deployments [e.g., Wilcock, 1996]. Adopting an approach in which a shipborne ADCP is employed to estimate roughness length from velocity profiles is generally considered to be prone to errors [Sime et al., 2007]. Slight variations in the vertical placement of the profiles may cause significant variability in the results [Biron et al., 1998]. Poorly resolved velocity gradients, lack of knowledge of the extension of the bottom boundary layer, and measurement noise introduced by the moving vessel all may play a role in the determination of roughness length and shear velocity. The crucial argument we make to conclude that our estimates of roughness length and shear velocity can be accurate is that neighboring estimates of depth mean and shear velocity in Figure 13 are similar, while they have been obtained in a mutually independent way. We cannot think of a systematic error that would cause the neighboring measurements to be consistent but wrong. Our filtering in the time domain based on equations (23)(26) filters out the errors in vertical positioning, which can be expected to be different each time the boat passes by a location. Individual ADCP velocity ensembles are an average over six subpings of 0.04 s, which coincides with about 0.5 m when the boat speed is 2 m s−1. Therefore, the velocity profiles can be considered a spatial average over about 0.5 m in the boat direction, which reduces the effect of spatial variation of the reference height. The difference between smoothed lines of z0 during the different campaigns and the original estimates indicates the error, which is on the order of a decade.

[46] Converting H-ADCP velocity data to specific discharge crucially depends on the determination of the effective roughness length z0, parameterizing bottom roughness. The results presented in Figures 14 and 15 confirm conclusions by Hoitink et al. [2009], who claim that consistent estimates of z0 can be obtained from moving boat ADCP measurements when a large number of ADCP transects are available to filter out the contributions of noise and turbulence. It was shown that z0 was particularly dependent on the river stage (Figure 16), which may relate to changes in bed forms. During high flows, when the 18 m isobath migrates downstream, z0 is largest. Conversely, during low flows, when the 18 m isobath evolves upstream, z0 is lowest. Bed morphology in the River Mahakam is strongly influenced by the width confinement and is much more complex than in many other alluvial environments where morphodynamics are more predictable. The bed dynamics in the River Mahakam may be in response to the details of the three-dimensional flow patterns, which are stage dependent. It can be shown that the conversion factor F becomes more dependent on roughness when σc decreases, i.e., when the velocity dip becomes more pronounced. The effective influence of bed dynamics on the discharge estimates is thus dependent on the occurrence of sidewall effects.

[47] An essential assumption made in the analysis of sidewall effects is that the Reynolds equation in the bed region holds over the entire water depth. This assumption allowed Lueck and Lu [1997] and Cheng et al. [1999] to successfully use the logarithmic velocity profile over the entire water column, to compute bed shear stress and roughness in contrasting environments. In the same spirit, we applied the modified velocity profile with velocity dip to our observations to estimate hydraulic parameters, yielding consistent results that hold for measuring campaigns that took place months apart. Whereas the aim in the present contribution is to obtain accurate, continuous estimates of discharge, the iterative method to fit the velocity profile with a velocity dip to moving boat ADCP measurements can be readily used to estimate bed shear stress.

[48] Bedload sediment transport jeopardizes the use of acoustic bottom tracking for the transformation of flow velocity data from instrument coordinates to Earth coordinates. Flow velocity obtained accordingly is biased low, resulting in underestimation of discharge estimates. Figure 19 shows a comparison between the error in boat-mounted ADCP velocity measurements transformed to Earth coordinates using acoustic bottom tracking and using dual-antenna GPS measurements, considering H-ADCP velocity measurements as a reference of reality. It shows that the effect of the moving bed is progressively more noticeable for higher flow velocities, which may result from the nonlinear response of sediment transport to flow velocity. Sediment transport, in turn, can affect boundary layer processes, increasing z0. Hence, part of the stage dependency of z0 could be caused by higher sediment transport rates during high river stages. Better understanding of alluvial bed roughness will allow improvement of the accuracy of discharge estimates from H-ADCP measurements. In turn, regular calibration campaigns offer possibilities to perform such analyses, if the domain where bathymetry measurements are taken is made wider than merely the region where the H-ADCP is located. Surveys combining multibeam echo soundings and ADCP measurements would make it possible to link the roughness lengths inferred from ADCP velocity profiles to morphodynamic processes.

Figure 19.

Scatterplot of the relative difference R = ∣UtransUhadcp ∣/∣Uhadcp ∣ versus Uhadcp in the range 0.15 < β < 0.35, where Utrans and Uhadcp are depth mean velocity estimates from the transect boat and the H-ADCP, respectively, indicating the relative difference between Utrans and Ub as a function of flow strength for Utrans computed with (top) the BT and (bottom) the GPS.

9. Conclusions

[49] A new method to convert H-ADCP velocity measurements into continuous time series of water discharge is presented, which can be applied to large rivers with discharges exceeding 8000 m3 s−1. It extends an existing semideterministic, semistochastic approach developed for rivers of rapidly varied flows, adopting a boundary layer model that accounts for sidewall effects resulting in a dip in flow velocity near the surface. The method was applied to H-ADCP measurements taken at a site in the River Mahakam, where the flow is intrinsically three-dimensional. Data series covering five moving boat ADCP campaigns were used. Each campaign covered a semidiurnal tidal cycle, three of which were used for calibration, whereas the remaining two served for validation. The method includes four parameters: (1) a stage-dependent value of the bed roughness z0, which is geometrically averaged over the H-ADCP range; (2) a sidewall correction factor α that is assumed to be constant in time but varies over width according to a bilinear profile that fitted best to the calibration data; (3) a steady but width-dependent time lag τ between variation in specific and total discharge; and (4) a constant but width-dependent amplification factor f. With z0 and α, H-ADCP velocity measurements can be converted to specific discharge, using the boundary layer model. Specific discharge is being translated to total discharge using a regression model with parameters τ and f. The best estimate of total discharge is finally obtained by averaging over the width range where the linear correlation between specific and total discharge was highest. Further progress depends primarily on the predictability of z0, which depends on studies of alluvial bed morphology.

Acknowledgments

[50] This study is part of East Kalimantan Programme, supported by grant WT76-268 from WOTRO Science for Global Development, a subdivision of the Netherlands Organisation for Scientific Research (NWO). Fajar Setiawan and Unggul Handoko (Indonesian Institute of Sciences) are acknowledged for their contribution to the field campaigns. We thank Pieter Hazenberg and Johan Romelingh (Wageningen University) for the technical support. Two anonymous reviewers have helped improve the draft of this paper with constructive criticism.

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