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Divergence-FreeSpatialVelocityFlowFieldInterpolatorforImprovingMeasurementsfromADCP-EquippedSmallUnmannedUnderwaterVehicles

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Divergence-Free Spatial Velocity Flow Field Interpolator for Improving Measurementsfrom 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 simu-

lation that spatially and temporally varies with the M2 tide, the 2D DF-estimated velocity magnitude and ori-

entation 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, ver-

satile environmental surveying platforms that are capa-

ble 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 ca-

pability continue to increase, providing users with a new set

of tools for measuring the environment. There is a growing

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 scien-

tists can now use for model validation. UUVs are typically

equipped with a combination of positioning, depth, and

water quality sensors, and acoustic Doppler current pro-

filers (ADCPs) with bottom-tracking capabilities for nav-

igation 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.

Corresponding author address: Jamie MacMahan, Oceanography

Department, Naval Postgraduate School, 327c Spanagel Hall, 833

Dyer Rd., Monterey, CA 93943.

E-mail: jhmacmah@nps.edu

478 J O U R N A L O F A T M O S P H E R I C A N D O C E A N I C T E C H N O L O G Y VOLUME 29

DOI: 10.1175/JTECH-D-11-00084.1

� 2012 American Meteorological Society

2004a,b; Brown et al. 2011). Fong and Jones (2006), using

an ADCP mounted onto a UUV in the open ocean, sug-

gested averaging over 100 m (;2 min) to remove tempo-

ral 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., aver-

aging and filtering) do not give acceptable velocity esti-

mates, particularly in shallow-water (tidal) environments

that have large horizontal velocity gradients.

There are three limitations with acquiring velocity mea-

surements from ADCP-equipped UUVs in tidal environ-

ments. 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 peo-

ple or machinery are now required to deploy it. Additional

space can be obtained by decreasing the number of avail-

able environmental sensors, but this reduces the capa-

bilities and uniqueness of the UUV. The UUV can be

charged on site, but the charge time relative to the op-

eration 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 ve-

locities 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 re-

sults but led to spatially noisy tidal coefficients between

spatial bins. Therefore, 2D thin plate spline (TPS) inter-

polators 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 pos-

sible fit. In addition, they showed that tidal constituents

could be spatially smoothed by iteratively choosing a sub-

set of data locations to be node locations. The TPS ap-

proach 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 deploy-

ments. 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 depth-

averaged velocity vector, and h is local water depth. The

DF method assumes ›h/›t is negligible, which they dem-

onstrated 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

MARCH 2012 M A C M A H A N E T A L . 479

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 solu-

tion 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 ele-

vation as measured by the National Oceanic and Atmo-

spheric 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, dis-

cussed 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.

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 10-

beam 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 collec-

tion 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 de-

ployment 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 typ-

ically 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 con-

sumption 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 ap-

proximately 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 conser-

vative 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 av-

eraged to maintain a high spatial resolution (;10-m grid

spacing for 1 m s21 UUV vessel speed) while also re-

ducing 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

SE 5

ffiffiffiffiffiffiffiffis2

N,

s(2)

where s2 is the system and environmental variance, and

N is the number of independent observations, where

FIG. 1. UUV track lines (white lines) for the seven 1-h repeated

surveys in Bear Cut Inlet, Miami, FL, and UUV-derived bathym-

etry (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.

480 J O U R N A L O F A T M O S P H E R I C A N D O C E A N I C T E C H N O L O G Y VOLUME 29

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 mea-

surements 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 bathy-

metric variability, which varies across the channel. In the

end, averaging for particular scenarios can be too sim-

plistic, 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 sta-

tionary 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 dom-

inant. 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, vhAL=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 1-

and 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

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).

MARCH 2012 M A C M A H A N E T A L . 481

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 indepen-

dent 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 set-

ting. 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

E 5

"1

N�N

i51(Observations 2 Model)2

#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 be-

tween 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 ran-

dom observations by approximately 85% (as shown in

Fig. 3d, blue arrows).

b. Measured UUV MV-ADCP Data in Bear CutInlet, 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,

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 obser-

vations 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.

482 J O U R N A L O F A T M O S P H E R I C A N D O C E A N I C T E C H N O L O G Y VOLUME 29

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 rel-

atively 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

and the observed velocity magnitude, UA

5ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiu2

e 1 u2n

p,

and orientation, u 5 tan21(ue/un), from the easting and

northing velocity that occurred within 1 h (Table 1). Er-

rors (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 ra-

tio, inducing relatively more flow variability in the mea-

sured 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 environ-

mental noise is low at slack tide, creating inaccurate es-

timates 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 obser-

vations 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 re-

quirements, 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 veloci-

ties 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 es-

timate 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 Un-

dersea Warfare Center provided the second UUV for this

operation and useful insight for UUV operations. We ap-

preciate 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

TABLE 1. Hourly estimates of tidal stage, mean U, error U, and error u.

Hour 1 2 3 4 5 6 7

Stage Ebb 2 1 h Ebb Ebb 1 1 h Slack 2 1 h Slack Slack 1 1 h Flood 2 1 h

UA (m s21) 0.64 0.64 0.60 0.37 20.14 20.43 20.60

EU (m s21) 0.13 0.17 0.14 0.13 0.11 0.17 0.21

Eu (8) 15 41 15 37 86 21 25

MARCH 2012 M A C M A H A N E T A L . 483

three anonymous reviewers greatly improved the man-

uscript.

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