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Divergence-Free Spatial Velocity Flow Field Interpolator for Improving Measurements from ADCP-Equipped Small Unmanned Underwater Vehicles JAMIE MACMAHAN Oceanography Department, Naval Postgraduate School, Monterey, California
ROSS VENNELL Department of Marine Science, University of Otago, Dunedin, New Zealand
RICK BEATSON Department of Mathematics and Statistics, University of Canterbury, Christchurch, New Zealand
JENNA BROWN Oceanography Department, Naval Postgraduate School, Monterey, California
AD RENIERS Applied Marine Physics, Rosenstiel School of Marine and Atmospheric Science, University of Miami, Miami, Florida (Manuscript received 26 April 2011, in final form 14 October 2011) ABSTRACT Applying a two-dimensional (2D) divergence-free (DF) interpolation to a one-person deployable unmanned underwater vehicle’s (UUV) noisy moving-vessel acoustic Doppler current profiler (MV-ADCP) measurements improves the results and increases the utility of the UUV in tidal environments. For a 3.5-h MV-ACDP simulation that spatially and temporally varies with the M2 tide, the 2D DF-estimated velocity magnitude and orientation improves by approximately 85%. Next the 2D DF method was applied to velocity data obtained from two UUVs that repeatedly performed seven 1-h survey tracks in Bear Cut Inlet, Miami, Florida. The DF method provides a more realistic and consistent representation of the ADCP measured flow field, improving magnitude and orientation estimates by approximately 25%. The improvement increases for lower flow velocities, when the ADCP measurements have low environmental signal-to-noise ratio. However, near slack tide when flow reversal occurs, the DF estimates are invalid because the flows are not steady state within the survey circuit.
1. Introduction Unmanned underwater vehicles (UUVs) are small, versatile environmental surveying platforms that are capable of being deployed and operated by one person, and are equipped with a sensor suite comparable to those mounted on larger-sized vessels. The size, weight, and cost of UUVs continue to decrease while vehicle functionality and capability continue to increase, providing users with a new set of tools for measuring the environment. There is a growing
Corresponding author address: Jamie MacMahan, Oceanography Department, Naval Postgraduate School, 327c Spanagel Hall, 833 Dyer Rd., Monterey, CA 93943. E-mail:
[email protected] DOI: 10.1175/JTECH-D-11-00084.1 Ó 2012 American Meteorological Society
need for collecting environmental data with UUVs in faster and more dynamic flows found in riverine and estuarine environments, with particular emphasis on the velocity flow field. With the increasing public availability of sophisticated numerical hydrodynamic models (e.g., Delft3D as of January 2011), UUVs are a tool that scientists can now use for model validation. UUVs are typically equipped with a combination of positioning, depth, and water quality sensors, and acoustic Doppler current profilers (ADCPs) with bottom-tracking capabilities for navigation below the water surface and water profiling (Shay and Cook 2003; Fong and Jones 2006; Hibler et al. 2008). ADCP measurements are inherently noisy and require time averaging to reduce the noise such that a statistically confident estimate of the mean is obtained (Muste et al.
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2004a,b; Brown et al. 2011). Fong and Jones (2006), using an ADCP mounted onto a UUV in the open ocean, suggested averaging over 100 m (;2 min) to remove temporal variability, which is often too coarse (e.g., tidal inlets). Brown et al. (2011) suggested 4 min of stationary averaging in riverine environments for describing the vertical profile, which would correspond to 240 m for 1 m s21 vessel speed. However, the ADCP data acquired from moving-vessel UUVs using standard statistical analysis tools (e.g., averaging and filtering) do not give acceptable velocity estimates, particularly in shallow-water (tidal) environments that have large horizontal velocity gradients. There are three limitations with acquiring velocity measurements from ADCP-equipped UUVs in tidal environments. First, the UUV performs best when traveling below the surface at .1 m s21, as this allows the UUV to navigate at depth and it avoids biofouling (e.g., seaweeds and grasses) of the small propeller, and is less prone to being hit by boaters. Though UUVs have the capability to perform quasi-station-keeping efforts when operating at the surface, this tends to be outside of their standard operation. Second, the operational duration is limited to a few hours by the available space for batteries within the compact UUV, and is too short to collect continuous measurements over a complete tidal cycle. Increasing space for additional batteries increases the weight of the UUV, which complicates logistics, as more than two people or machinery are now required to deploy it. Additional space can be obtained by decreasing the number of available environmental sensors, but this reduces the capabilities and uniqueness of the UUV. The UUV can be charged on site, but the charge time relative to the operation is approximately 2 to 1, making continuous high temporal resolution measurements problematic. The third limitation is the flow velocities measured by the ADCP are noisy, requiring time averaging (.2 min) (Fong and Jones 2006; Muste et al. 2004a,b; Brown et al. 2011), but since the UUV is moving, time averaging now consists of both time and space averaging, which can be problematic in environments with large horizontal velocity gradients or if the fluid motions are small in scale (e.g., narrow channels; Vennell 2006), requiring an increase in spatial resolution. More sophisticated statistical methods to improve mean flow estimates related to the noisy ADCP measurement, which are associated with the inherent UUV’s operational requirements for acquiring the spatial flow field in tidal environments, are discussed herein. ADCP-equipped UUVs are described as moving-vessel ADCP (MV-ADCP) measurements. There are a number of papers that focus on spatially detiding MV-ADCP velocities to reduce the inherent noise. In general, MV-ADCP velocities in tidal environments are acquired along a predefined track that is repeated every hour over a tidal
cycle. The total survey duration is dependent upon the relevant tidal constituents for the measurement site, which can range from 13 to 25 h when describing the M2, K1, and shallow-water overtides. Geyer and Signell (1990), among others, spatially binned the MV-ADCP and analyzed the tidal coefficients for each spatial bin separately, which were then collectively used to spatially describe a tidally forced flow field. Vennell and Beatson (2006, hereafter VB06) found that this method produced reasonable results but led to spatially noisy tidal coefficients between spatial bins. Therefore, 2D thin plate spline (TPS) interpolators that smoothly describe the tidal velocity over the spatial region are recommended (Candela et al. 1992, among others). VB06 demonstrated that when using TPSs (a class of radial basis functions), it is advantageous to place nodes at the data points and to constrain the weights by side conditions in order to ensure the smoothest possible fit. In addition, they showed that tidal constituents could be spatially smoothed by iteratively choosing a subset of data locations to be node locations. The TPS approach can be extended to create a spatial–temporal form of tidal analysis. However, since many UUVs cannot operate over a complete tidal cycle, the extension using the tidal analysis form of TPS to temporally smooth the data cannot be used for the presented UUV deployments. Hence, only spatial smoothing using TPS will be done following the techniques in Vennell and Beatson (2009, hereafter VB09), but without the sinusoids needed for the temporal tidal analysis. VB06 and VB09 developed a 2D divergence-free (DF) spatial interpolator that ensures mass is conserved and provides more realistic estimates of the depth-averaged velocities. The starting point is the continuity equation given by ›h/›t 1 $ U(h 1 h) 5 0,
(1)
where h is the sea surface elevation, U is the depthaveraged velocity vector, and h is local water depth. The DF method assumes ›h/›t is negligible, which they demonstrated is a good approximation for tidal flows with spatial scales less than a few kilometers. VB06 described the DF method, including tidal analysis (DF-tidal), and applied it to MV-ADCP measurements with success. VB09 subsequently demonstrated a modified DF method applied to 1-h MV-ADCP measurements for a tidal inlet. In addition, a number of synthetic and field applications along with other spatial interpolators are discussed in great detail in these two papers. The DF methods described in the two papers provide a framework and foundation for improving the ADCP-equipped UUV measurements. We expand upon the VB09 by first applying the DF method to a synthetic dataset that represents MV-ADCP data acquired in a channel with an M2 tide to demonstrate
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FIG. 1. UUV track lines (white lines) for the seven 1-h repeated surveys in Bear Cut Inlet, Miami, FL, and UUV-derived bathymetry (depth color lines associated with color bar, with depth given in m). The circle represents the location of the stationary Aquadopp and the triangle represents the location of the NOAA Virginia Key tidal station (station identification 8723214). Yellow box outlines bathymetry associated with the discussion in Fig. 2.
that a more accurate spatial map of the flow field can be acquired that is both temporally and spatially varying. Note that the MV-ADCP velocities are multiplied by the local water depth before the DF method is applied, and transformed back to water velocities afterward. Second, the DF method is applied to MV-ADCP data acquired by a YSI/Oceanserver EcoMapper Iver2 UUV deployed in Bear Cut Inlet, Miami, Florida. The significance of this manuscript is recognizing the limitations of the UUV’s operation that affect its ADCP data collection and the existence of a spatial interpolator that can provide a solution to improve the depth- and time-averaged velocities.
2. UUV field experiment The UUV experiment occurred in January 2011, in Bear Cut Inlet, Miami, Florida (Fig. 1). Bear Cut is a naturally occurring inlet between two barrier islands—Virginia Key and Key Biscayne—that connect the Atlantic Ocean with Biscayne Bay. Biscayne Bay has multiple openings to the Atlantic Ocean, reducing the shallow-water overtides typically found in many closed back bays. The water elevation as measured by the National Oceanic and Atmospheric Administration (NOAA) tidal gauge is dominated (84%) by the M2 tidal component (0.30 m) as related to the cumulative sum of 0.35 m for the daily tidal components (http://tidesandcurrents.noaa.gov). The K1 (0.03 m) and the shallow-water overtides (0.03 m) represent 9% and 7% of the tidal signal, respectively. The YSI/Oceanserver EcoMapper Iver2 UUV, discussed herein, is 1.6 m long with a diameter of 0.15 m, weighing 45 lb in air and can be deployed by one person.
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The UUV can operate at depths down to 60 m using four independent control planes and can travel at a speed of 0.5–2 m s21. For navigation, it uses GPS with Wide Area Augmentation System corrections when at the surface and bottom tracking when below the surface from a Sontek 10beam upward- and downward-facing Doppler velocimetry log (DVL) consisting of four velocity beams operating at 1.0 MHz and a vertical center beam operating at 0.5 MHz. Bottom tracking is functional up to 40 m below the UUV. The order of operation for the UUV navigation protocol is first GPS, followed by ADCP bottom tracking, then dead reckoning. In addition, the UUV is equipped with a dual-frequency side-scan sonar for bottom imaging and a full suite of water monitoring sensors with 10 GB of onboard data storage. The UUV runs on rechargeable lithium-ion batteries capable of up to 8 h of data collection at a speed of 1.3 m s21 in a zero flow environment. Two UUVs were deployed on the eastern side of Bear Cut and completed seven 1-h repeated tracks. The deployment started near ebb tide and finished near flood tide. There were a total of eight cross-channel transects (Fig. 1). At every even transect, the UUV traveled across the channel 1 m below the surface. At every odd transect, the UUV undulated to capture the vertical variation of water quality observations. ADCP measurements are typically discarded when the UUV performs the undulations, owing to the 308 pitch angle, but they are used in the analysis described herein. As this particular UUV never had been operated in these faster flows, alternating UUVs were deployed for each 1-h survey, allowing power consumption to be monitored. After each 1-h survey, data were downloaded and power usage was recorded. The UUV traveled at an operational speed of 1 m s21 in approximately 1 m s21 flow resulting in a 20% power draw for each 1-h survey. Note that when the power drops below 15%, the UUV executes a safety abort mission. Therefore, if only one UUV were available, its conservative mission time would be 4 h in the 1 m s21 current. The ADCP sampled at 1 Hz, and had a surface blanking distance of 0.25 m and bin size of 0.5 m with 30 depth bins. ADCP velocities were depth averaged and 10-s time averaged to maintain a high spatial resolution (;10-m grid spacing for 1 m s21 UUV vessel speed) while also reducing ADCP noise. An example of a 1-h 10-s-averaged velocity field is shown in Fig. 2 (left, blue vectors). The standard error (SE) is defined as sffiffiffiffiffiffiffiffi s2 , (2) SE 5 N where s2 is the system and environmental variance, and N is the number of independent observations, where
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FIG. 2. Velocity vectors for 1 h at ebb tide for depth-averaged and 10-s time-averaged ADCP observations (blue), 2-min moving-averaged ADCP observation (green), and 2D DF method (red) with measured velocity standard error circles (too small to be recognized).
N ; 10 for 10 s (Brown et al. 2011) is plotted as circles at the base of the vectors in Fig. 2. Note that the circles are too small to be seen in Fig. 2. Statistically, the ADCP measurements have minimal SE, but it is conceptually difficult to accept that the spatial flow variability plotted in Fig. 2 (left) is correct. Ignoring SE, Brown et al. (2011) found that the mean velocity did not asymptote until 3.5 min for a stationary observation. Applying a 2-min moving average, as recommended by Fong and Jones (2006), to the ADCP data and subsampling to 10 s further reduces SE (N 5 120), provides a better asymptotic estimate of the mean, decreases the spatial variability owing to the fact this represents a 120-m spatial average, and provides a more consistent result (Fig. 2, left). Increasing the averaging window increases the statistical confidence; it also smears out spatial variability that may be associated with bathymetric variability, which varies across the channel. In the end, averaging for particular scenarios can be too simplistic, requiring more physics within the interpretation, for which we recommend the 2D DF method. A self-contained, downward-looking 2-MHz Nortek Aquadopp ADCP mounted on a surface, nonmotorized mini-catamaran was deployed on the eastern side of the channel outside of the UUV tracks to provide a stationary estimate of the flow field (Fig. 1). This ADCP sampled at 1 Hz and had a surface blanking distance of 0.05 m and 0.5-m bins.
3. Results The in situ depth-averaged Aquadopp velocities were numerically rotated to a streamwise orientation. T-Tides tidal analysis (Pawlowicz et al. 2002) was applied to the
Aquadopp velocities resulting in an M2 tidal velocity amplitude of 0.7 m s21, which described 95% of the total velocity variance, indicating that the flow is tidally dominant. The remaining 5% represents instrument noise, platform-induced flow errors, other tidal constituents, and nontidal flows. Energy from other tidal components is included in the M2 constituent owing to the short 7-h record and differs from the long-term tidal analysis. To assume a flow to be approximately nondivergent the ratio of the first to second terms in Eq. (1) must be small—that is, vhA L=hUA , where v is the radian M2 tidal frequency; hA is the tidal elevation amplitude; L is the horizontal survey scale, which is O(1 km); h is the water depth; and UA is the velocity amplitude (VB06). For Bear Cut Inlet, the DF ratio is 0.06 and 0.01 for 1and 6-m water depths, respectively. Therefore, the flow features are well approximated by divergence-free, depth-averaged velocities.
a. Simulated UUV data for M2 tidal channel Ideally, MV-ADCP observations are obtained in a repetitive 1-h survey over a tidal cycle, so that harmonic analysis can be used to extract the tidal harmonics from the velocity signal, reducing the ADCP noise. However, can a nonrepetitive 3.5-h UUV mission be planned to maximize measuring the spatial flow variability over a large reach that is temporally varying with the tides? The answer as to whether the DF method can be applied to this scenario is simulated next. Streamwise only, depth-averaged velocities for a 0.7 m s21 M2 tide that is temporally changing (Fig. 3a), acquired every 10 s from a UUV traveling at 1 m s21 for
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FIG. 3. Synthetic tidal example. (a) The representative true ‘‘non-noisy’’ tidal velocity for the 3.5-h synthetic time frame, which mimics the operational duration of the UUV. (b) Planform of the spatial survey, where the 10-s measurement locations of noisy ADCP observations and nodes are described as small dots and large circles, respectively. (c) RMS difference as a function of number of centers/nodes. (d) Velocity output, where red vectors represent synthetic noisy velocities (offset in the streamwise direction by 250 m) and blue vectors represent DF velocities.
3.5 h in a continuous nonrepetitive track, were simulated for a unidirectional flow in a channel that is 500 m wide and 6 m deep (Fig. 3b), which is similar to the dimensions of Bear Cut. The track lines had a streamwise spacing of 600 m. The simulated velocities occurred around a flow maximum (Fig. 3a) and did not contain a flow reversal (discussed below). A random velocity noise of 3 times the standard error (s/ON 5 0.06 m s21 for N 5 10 independent observations for 10-s average) of measured ADCP noise was conservatively added to simulated velocities. Fifteen nodes/centers were chosen for the TPS smooth surface based on the root-mean-square (RMS) difference (Fig. 3c). As part of the method, a minimum node spacing is set at 5% of the size of the measurement area and thus will not pick up localized features smaller than this setting. Radial basis function (RBF) spline interpolation is a generalization of placing many weights on a thin plate to bend it to best fit the displacements given by the data values. Centers are the locations where the weights are placed on the plate. The greedy fit DF–RBF method places weights at a subset of the data locations—that is, centers are the middle of radially symmetric functions used to do the interpolation. The spatial locations of the centers are shown in Fig. 3b. A more detailed description of centers, in particular the number, is discussed in VB09. An example of the simulated flow with random noise is shown in Fig. 3d (red arrows). The computational time for this synthetic case was approximately 30 s using Matlab on a standard PC.
Error E is given by "
N
1 (Observations 2 Model)2 E5 N i51
å
#1/2 ,
(3)
where N is the total number of observational location (Fig. 3b). The error between the true simulated velocity magnitude (Eu) and direction (Eu) without noise and the random simulated velocity magnitude and direction with noise was 0.07 m s21 and 12.68, respectively, which is consistent with the imposed variability. The error between true simulated velocity magnitude and direction and the DF estimate associated with the random noisy velocity (Fig. 3d, blue arrows) was reduced to 0.01 m s21 and 1.48. The DF method thus improved simulated random observations by approximately 85% (as shown in Fig. 3d, blue arrows).
b. Measured UUV MV-ADCP Data in Bear Cut Inlet, Miami, FL For the Bear Cut Inlet deployment, a repetitive 1-h deployment was performed with two UUVs to describe temporal variability from ebb to flood tide. There is an improvement in the flow field using the DF method compared with the 10-s time-averaged flow field (Fig. 2, left), as there is a reduction in variability associated with the velocity amplitude and orientation. An example of the improvement is illustrated in Fig. 2 (right) at x 5 800,
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MACMAHAN ET AL. TABLE 1. Hourly estimates of tidal stage, mean U, error U, and error u.
Hour Stage
1 Ebb 2 1 h
2 Ebb
3 Ebb 1 1 h
4 Slack 2 1 h
5 Slack
6 Slack 1 1 h
7 Flood 2 1 h
UA (m s21) EU (m s21) Eu (8)
0.64 0.13 15
0.64 0.17 41
0.60 0.14 15
0.37 0.13 37
20.14 0.11 86
20.43 0.17 21
20.60 0.21 25
y 5 50, where the measured velocities are reduced, but farther upstream and downstream the velocities are larger, highlighting the inconsistency of the measurements, which are not supported by any bathymetric variability (Fig. 1, yellow box). In general, the bathymetry at this site is relatively straight and parallel, except as it shallows and horizontally diverges near the northeast boundary (Fig. 1). The DF-estimated velocities are more consistent in this region. Assuming that the DF estimates represent the true flow field, the error is estimated between the DF method pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi and the observed velocity magnitude, UA 5 u2e 1 u2n , and orientation, u 5 tan21 (ue /un ), from the easting and northing velocity that occurred within 1 h (Table 1). Errors (E) for UA and u are relatively consistent throughout the tide, except the slack tide orientation. The DF method improves UA by approximately 25%. The DF improvement increases with lower flow velocities because the ADCP measurements have a low signal-to-noise ratio, inducing relatively more flow variability in the measured velocities. The measured velocity orientation has a lot of variability between neighboring observations (Fig. 2, right). The DF estimate is more spatially consistent and thus provides a better representation of reality. The largest error (86 for u) occurs at slack tide. There are two reasons for the mismatch in orientation. First, the flows are not steady state—they are reversing within this 1-h survey—resulting in a flow convergence in the middle of the survey area, causing the orientation of the flows estimated by the DF method to be oriented toward the shoreline, and thus explaining the largest orientation error. Second, the measured velocity signal to environmental noise is low at slack tide, creating inaccurate estimates of flow orientation. The DF method improves noisy ADCP data, but there is a limit to its ability because it is using the measured ADCP data as a starting point. Thus, DF estimates from ADCP observations around flow reversals should be avoided. Therefore, a UUV can acquire ADCP data over a larger spatial reach without requiring subdivision, as long as there is not a flow reversal or reoccupation of a similar track to describe temporal variability, which may be of interest in describing eddy formation near tidal flow maxima. A priori knowledge of the tidal flows is required to maximize the UUV survey. It is best to obtain observations around the flow maxima to avoid flow reversals,
which tend to be the times that the more interesting flow features develop.
4. Summary The divergence-free spatial interpolator when applied to ADCP-equipped UUV data improves the utility of the UUV based on its limitations of power, moving-vessel requirements, and noisy velocity measurements, particularly in tidal environments. The UUV battery capacity does not allow for full tidal analysis, but the UUV can acquire useful data for a few hours in a tidal environment to spatially describe flow features around a particular tidal stage. Caution is required during flow reversals or low velocities with a low environmental signal-to-noise ratio, as the reliability of the results is diminished. The DF method provides the framework for improving the viability of the UUV by providing an accurate synoptic spatial estimate of the depth-averaged flow, which can be used to validate model output or discover regions of interesting flow behavior. Remember that the deployment of this UUV only requires one person, so exploration of flow features can be easily performed. As UUV usage moves farther upstream away from the coast into rivers away from the influence of tides, the DF method becomes ideal. Acknowledgments. JM and AR were supported by ONR (Grants N0001410WX21049 and N000141010379). The NSF (Grant OCE 0728324), ONR (Grant N0001410WX21049), and the National Defense Science and Engineering Graduate Fellowship supported JB. ONR DURIP (Grant N0001409WR20268) supported UUV. Mike Incze and Scott Sideleau from Naval Undersea Warfare Center provided the second UUV for this operation and useful insight for UUV operations. We appreciate the technical support from the YSI/Oceanserver team (Ben Clarke, Tony DiSalvo, and Daniel Osiecki). A special thanks to NPS Miami Winter OC4210 students: Bill Swick, David Paul Smith, Mark Hebert, Chris Tuggle, Stephanie Johnson, Chris Beuligmann, and Will Ashley and the UM students Zhixuan Feng, Atsushi Fujimura, and Patrick Rynne. We thank Virginia Key Beach Park. We appreciate additional funding from CNMOC and ONR Coastal GeoSciences. The constructive comments by the
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three anonymous reviewers greatly improved the manuscript. REFERENCES Brown, J., C. Tuggle, J. MacMahan, and A. Reniers, 2011: The use of autonomous vehicles for spatially measuring mean velocity profiles in rivers and estuaries. Intell. Serv. Rob., 4, 233–244, doi:10.1007/s11370-011-0095-6. Candela, J., R. C. Beardsley, and R. Limeburner, 1992: Separation of tidal and subtidal currents in ship-mounted acoustic Doppler current profiler observations. J. Geophys. Res., 97, 769–788, doi:10.1029/91JC02569. Fong, D. A., and N. L. Jones, 2006: Evaluation of AUV-based ADCP measurements. Limnol. Oceanogr.: Methods, 4, 58–67. Geyer, W. R., and R. P. Signell, 1990: Measurements of tidal flow around a headland with a shipboard acoustic Doppler current profiler. J. Geophys. Res., 95, 3189–3197. Hibler, L. F., A. R. Maxwell, L. M. Miller, N. P. Kohn, D. L. Woodruff, M. J. Montes, J. H. Bowles, and M. A. Moline, 2008: Improved fine-scale transport model performance using AUV and HIS feedback in a tidally dominated system. J. Geophys. Res., 113, C08036, doi:10.1029/2008JC004739.
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Muste, M., K. Yu, and M. Spasojevic, 2004a: Practical aspects of ADCP data use for quantification of mean river flow characteristics; Part I: Moving-vessel measurements. Flow Meas. Instrum., 15, 1–16. ——, ——, T. Pratt, and D. Abraham, 2004b: Practical aspects of ADCP data use for quantification of mean river flow characteristics; Part II: Fixed-vessel measurements. Flow Meas. Instrum., 15, 17–28. Pawlowicz, R., B. Beardsley, and S. Lentz, 2002: Classical tidal harmonic analysis including error estimates in MATLAB using T_TIDE. Comput. Geosci., 28, 929–937. Shay, L. K., and T. M. Cook, 2003: Submesoscale coastal ocean flows detected by very high frequency radar and autonomous underwater vehicles. J. Atmos. Oceanic Technol., 20, 1583–1599. Vennell, R., 2006: ADCP measurements of momentum balance and dynamic topography in a constricted tidal channel. J. Phys. Oceanogr., 36, 177–188. ——, and R. Beatson, 2006: Moving vessel acoustic Doppler current profiler measurement of tidal stream function using radial basis functions. J. Geophys. Res., 111, C09002, doi:10.1029/ 2005JC003321. ——, and ——, 2009: A divergence-free spatial interpolator for large sparse velocity data sets. J. Geophys. Res., 114, C10024, doi:10.1029/2008JC004973.
