DTIC ADA436031: Estimation of Directional Wave Spectra from an Autonomous Underwater Vehicle (AUV)

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Estimation of Directional Wave Spectra from an Autonomous Underwater Vehicle
(AUV)
Jeffery S. Riedel and Anthony J. Healey
Naval Postgraduate School
Center for AUV Research
Monterey, CA 93943-5000
831.656.1146 voice, 831.656.2238 fax
{jsriedel, healey}@me.nps.navy.mil
Abstract
Directional wave energy and wave spectra information, each of which has advantages and
play an important role in the physical processes disadvantages. These include deployable or in-situ
associated with the ocean environment. instruments such as directional wave buoys
Determining these directional wave characteristics [Grosskopf 1983], as well as arrays of current
is a fairly arduous task. An Autonomous meters and pressure sensors [Allender 1989].
Underwater Vehicle (AUV) provides a suitable Remote sensing techniques involving microwave
platform for making in-situ measurements in the radar systems [Tyler 1974], aircraft [McLeish
ocean environment. Small AUVs can typically cover 1980], or satellites [Monaldo 1984] are also
a 3-5 mile square region at a speed of 3-4 knots, commonly used. The issue associated with these
allowing for the survey of the water column over a measuring approaches is that they are too expensive
substantial range. Further, an AUV is much more for many users and are not suitable for rapid
versatile than the traditional bottom-mounted deployment without significant pre-planning.
ADCPs, suspended current meters or drifting buoys Currently many operational AUVs carry
in so much as the sensor (the AUV) is mobile and sensor suites, which allow the vehicles to record
can be programmed to collect data in any desired data that many be used to obtain directional spectra
area vice having to locate the sensors, recover estimates. This paper will outline the underlying
them, transit and re-deploy. principles used in identification of wave directions
In this paper we examine the potential of from standard wave following buoys. It will
an AUV to be used as a platform for making present the mathematical formulas used in
directional wave spectrum measurements, with determining the wave direction as a function of
particular reference to the Naval Postgraduate frequency. Extension of these algorithms to
School’s PHOENIX AUV. We show that by using subsurface velocity sensors will be made, where,
relative velocity measurements from a SONTEK through the use of the Doppler equation, a moving
Acoustic Doppler Velocimeter (ADV), ground- AUV can be used to determine wave directions.
referenced velocity measurements from a RDI Lastly, it will be shown how a control command
Navigator Doppler Velocity Log (DVL), vehicle can be obtained from the frequency dependent
motion data from a Systron-Donner Motion Pak wave direction estimates.
and employing a Maximum Entropy Method The information in this paper is not new,
(MEM) the wave directional spectra can be found. only the application to which this method is
The methodology used to determine the spectra, the applied. For more detailed descriptions of the
corrections that are required to account for the mathematical formulations presented in this paper,
Doppler shift due to the moving vehicle and the the reader is referred to [O'Reilly 19 96] dnaeht
results obtained from data collected during references therein.
AUVFEST ‘98 and experimental missions in
Monterey Bay will be discussed and analyzed. Wave Spectra and Directional Estimates
Introduction The elevation of the sea surface h (t) can
be described as the superposition of an infinite
Measurements of directional wave spectra have number of sinusoids of the form:
ianctceurreasttse that forecasting, range to from fundamental determination physics, of totactical h ((t))==(cid:229)(cid:229)¥¥ a cos( k x-- w t++f ) ==(cid:229)(cid:229)¥¥ h . (1)
n n n n n
information for military operations. There are n==1 n==1
Wave components with different frequencies are
many techniques for collecting spectral
usually assumed to be statistically independent
1
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Estimation of Directional Wave Spectra from an Autonomous
5b. GRANT NUMBER
Underwater Vehicle (AUV)
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Naval Postgraduate School,Center for AUV REPORT NUMBER
Research,Monterey,CA,93943-5000
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14. ABSTRACT
Directional wave energy and wave spectra play an important role in the physical processes associated with
the ocean environment. Determining these directional wave characteristics is a fairly arduous task. An
Autonomous Underwater Vehicle (AUV) provides a suitable platform for making in-situ measurements in
the ocean environment. Small AUVs can typically cover a 3-5 mile square region at a speed of 3-4 knots,
allowing for the survey of the water column over a substantial range. Further, an AUV is much more
versatile than the traditional bottom-mounted ADCPs, suspended current meters or drifting buoys in so
much as the sensor (the AUV) is mobile and can be programmed to collect data in any desired area vice
having to locate the sensors, recover them, transit and re-deploy. In this paper we examine the potential of
an AUV to be used as a platform for making directional wave spectrum measurements, with particular
reference to the Naval Postgraduate School’s PHOENIX AUV. We show that by using relative velocity
measurements from a SONTEK Acoustic Doppler Velocimeter (ADV), groundreferenced velocity
measurements from a RDI Navigator Doppler Velocity Log (DVL), vehicle motion data from a
Systron-Donner Motion Pak and employing a Maximum Entropy Method (MEM) the wave directional
spectra can be found. The methodology used to determine the spectra, the corrections that are required to
account for the Doppler shift due to the moving vehicle and the results obtained from data collected during
AUVFEST ‘98 and experimental missions in Monterey Bay will be discussed and analyzed.
15. SUBJECT TERMS
16. SECURITY CLASSIFICATION OF: 17. LIMITATION OF 18. NUMBER 19a. NAME OF
ABSTRACT OF PAGES RESPONSIBLE PERSON

a. REPORT b. ABSTRACT c. THIS PAGE 10
unclassified unclassified unclassified
because they are generated by random wind forces Datawell (cid:210) Directional Waverider measures
at different locations. From the central limit 3 -component accelerations of the buoy, which are
theorem it follows that the probability distribution integrated to yield the horizontal and vertical
of the surface elevation, h (t), is approximately displacements of the buoy. The hull and mooring,
Gaussian, consistent with many observations, of this buoy, were designed to give the buoy good
[Soong 1993]. wave following characteristics, thereby allowing
The procedure presently employed by the buoy displacements to approximate the
many of the operational data buoys in based on displacements of an actual water particle at the sea
Fourier analysis. In Fourier analysis it is surface.
convenient to work with complex exponentials For a wave train propagating in the
rather than sine and cosine functions, there fore positive x-direction, the fluid particle motion is
using the relation nevyibg
exp( i(w t++f )) ++exp( -- i(w t++f )) kg cosh( k( z++H))
cos( w t++f ) == (2) u ==a cos( kx-- w t) . (8)
2 w cosh( kH)
the expression for the surface elevation can be However, for the more general case of a wave train
nettirwsa propagating at some angle relative to the x-axis, it
h ( t) ==(cid:229)(cid:229) A exp( iw t) , (3) nac eb nwohs taht eht wolf dleif si nevigyb
w
w u( x, y, t) ==aw cos q exp( kZ) cos( k( xcos q ++ynis q ) -- w t)
where
v( x, y, t) ==aw nis q exp( kZ) cos( k( xcos q ++ynis q ) -- w t)
1
Aw == 2 aw exp( if w ) (4) w( x, y, t) ==aw exp( kZ) (nis k( xcos q ++ynis q ) -- w t)
and the summation is over both positive and (9)
negative frequencies. Let the average position of the wave buoy be given
The energy spectrum E(w ), is defined as yb x=y=z=0. For small amplitude waves, the
expected buoy displacements are small compared to
2
A the surface wavelength, therefore the buoy motion
w
E(w ) == (5) can be approximated by the fluid velocity at
D w
x=y=z=0.
where indicates an average over many data
For a full spectrum of waves, the buoy
records and D w is the spacing of the frequency displacements can be expressed using complex
bands. The spectrum is closely related to the noitatonsa
energy of the waves, and represents the distribution X( t) ==(cid:229)(cid:229) (cid:229)(cid:229) -- iA cos q exp( iwt)
w ,q
of wave energy as a function of frequency. w q
To describe the spatial and temporal Y( t) ==(cid:229)(cid:229) (cid:229)(cid:229) -- iA nis q exp( iwt) (10)
w ,q
characteristics of the sea surface linear w q
superposition of wave components is used. In Z( t) ==(cid:229)(cid:229) (cid:229)(cid:229) -- A exp( iwt)
w ,q
exponential terms this can be represented as w q
h ( x, y, t) ==(cid:229)(cid:229) (cid:229)(cid:229) A exp( i( k( xcos q ++ynis q ) -- wt)) erehw eht -i is due to the 90 (cid:176) phase difference
w ,q between the vertical and horizontal displacements.
w q
(6) The expressions in (10) can be written using
where x, y are the horizontal spatial coordinates, Fourier transforms as
and w and k obey the dispersion relation. The X( t) ==(cid:229)(cid:229) X(w ) exp( iwt)
frequency directional wave spectrum is defined as w
2 Y( t) ==(cid:229)(cid:229) Y(w ) exp( iwt) , (11)
Aw ,q w
E(w ) == D w D q (7) Z( t) ==(cid:229)(cid:229) Z(w ) exp( iwt)
w
and describes the distribution of energy as a
where the Fourier transforms are given by
function of frequency and direction.
X(w ) ==(cid:229)(cid:229) -- iA cos q
w ,q
q
Directional Estimates from Wave Buoys Y(w ) ==(cid:229)(cid:229) -- iA nis q . (12)
w ,q
q
The most commonly used instrument for
Z(w ) ==(cid:229)(cid:229) A
measuring waves in deep water is the "heave, pitch w ,q
q
and roll buoy" that measures the surface height and
To derive the relationships between the
slope in two orthogonal directions. The newer
measured time series and the unknown
2
ycneuqerf -directional wave spectrum the cross 1980], form the basis for most of the buoy analysis
spectrum must be considered. In general, the cross techniques currently used.
spectrum between two time series X(t) and Y(t) htiw
Fourier transforms X(w ) and Y(w ) is defined as Extension to Subsurface Sensors
X(w )Y* (w )
C (w ) == As discussed in the previous section, the
XY D w motion of a wave buoy is directly related to the
(13) fluid velocity, therefore, the cross-spectra of a tri-
where * indicates the complex conjugate, [Soong directional current meter yields the same low-
1993]. Substitution of (12) into (13) yields resolution directional wave information obtained
C (w ) ==(cid:229)(cid:229) cos q nis q E(w ,q ) from buoy measurements. Substituting the
XY
q normalized spectra of the vertical ( Z) velocity
(14) component w, and the horizontal ( X, Y) velocity
where it is assumed that the wave components components u and v into (18), the lowest four
propagating in different directions are statistically Fourier moments of the directional distribution of
independent. The cross spectrum C can be wave energy can be obtained and are given by
XY
expressed in continuous form as
mI ( C (w ))
C (w ) ==2(cid:242)(cid:242)p cos q nis q E(w ,q ) dq . a1 (w ) == [[C (w ) [[C (wwu) ++C (w ) ]]]]1/2 , (19)
XY ww uu vv
0 (mI C (w ))
Cross-spectra of the other time series pairs can be (15) b1 (w ) ==[[C (w ) [[C (wwv) ++C (w ) ]]]]1/2 , (20)
ww uu vv
obtained in a similar manner. The full set of C (w ) -- C (w )
relations for buoy cross-spectra can be found in a2 (w ) ==Cuu(w ) ++Cvv(w ) , (21)
[Dean 1984]. uu vv
It is convenient to define a normalized b (w ) == 2 eR ( Cuv(w )) , (22)
directional distribution of energy at a frequency w 2 C (w ) ++C (w )
uu vv
sa where C(ww ) is the spectral matrix of the velocity
S(q ;w ) == E(w ,q ) components u, v, w. Since the wave direction, q , is
E(w ) referenced to the navigation frame (N-E-D), vehicle
borne sensor measurements must be transformed
(16)
prior to use. It is interesting to note that the
htiw tinulargetni
estimates of these directional moments are
2p
(cid:242)(cid:242) E(w ,q ) dq insensitive to errors, so long as the errors are the
2(cid:242)(cid:242)p S(q ;w )dq == 0 == E(w ) ==1 . same on all measurement axes of the sensors, which
E(w ) E(w ) is typical with oceanographic sensors installed on
0
AUVs.
(17)
The objective of the data analysis is to
With this definition, (15) and the other referenced
spectral relations can be combined and expressed in infer the directional distribution S(q ), from the four
smret fo S( q ; w ) . Dropping the frequency measured moments a1, b1, a2, and b2 . ehTtsom
widely used techniques are described below.
dependence these relations can be expresses as
((((C (mI++CCX))ZC) ))1 /2 == 2(cid:242)(cid:242)p cos q S(q ) dq ”” a1 The Cosine Power Distribution
XX YY ZZ 0 [ Longuet-Higgins 1963] introduced a
((((CXX(mI++CCYYYZ))C) ZZ))1 /2 == 2(cid:242)(cid:242)0p nis q S(q ) dq ”” b1 simple cosine-power distribution, S(q ) == Acos 2 s(cid:231)(cid:230)(cid:231)(cid:230) q -- q mean (cid:247)(cid:246)(cid:247)(cid:246) (23)
CXX -- CYY == 2(cid:242)(cid:242)p cos 2 q S(q ) dq ”” a ŁŁ 2 łł
CXX ++CYY 0 2 htiw q mean the mean propagation direction, s a
2 Re( C ) 2p parameter that controls the width of the distribution
C ++CXY == (cid:242)(cid:242) nis 2 q S(q ) dq ”” b2 and A, a normalization coefficient. The parameters
XX YY 0 q and s are determined by fitting (23) to the
(18) mean
relations given in Equations (19)-(22).
These four relations between the cross-spectra of
The main drawback to this simple method is that
the buoy measurements and the directional
(23), with only two free parameters can describe
distribution of wave energy, derived by [Long
3
only unimodal distributions, and thus fails in Correction For A Moving Platform (The
snoitautis htiw a ladomib aes etats ,.g.e( -itlum Doppler Equation)
directional seas during a wind veering event or
swell arriving from two different sources). The equations for the wave directionality
estimations presented in the previous section are
The Maximum Entropy Method dilav rof anon -moving sensor. However, when
information is obtained from an AUV, corrections
[ Lygre and Krogstad 1986] introduced the must be made to account for the vehicle motion.
maximum entropy or MEM estimate of S(q ). ehT evaw ,ycneuqerf hcihw eht elcihevsretnuocne
Unlike Equation (23), this approach can describe while moving through the wave field, has been
both unimodal and bimodal distributions and shifted. This shift can be determined by using the
exactly fits the relations given in (19) -(22). This well-known Doppler equation,
directional spectrum is given by w 2
S(q ) == 1 (( 1 -- f 1 c1* -- f 2 c2* )) (24) w e »» w -- g Ucos b , (30)
2 p 1 -- f e-- iq -- f e-- 2 iq 2 where the spectrum that the vehicle encounters is a
1 2
htiw function of the vehicle's forward speed U dnasti
heading angle relative to the propagation direction
c ”” a ++ib
1 1 1 fo eht aes ,sevaw b . Using the techniques outlined
c2 ”” a2 ++ib2 ni eht suoiverp noitces lliw evig eht evaw
c -- c c* (25) directional distribution as a function of vehicle
f ”” 1 2 1
1 1 -- c 2 encounter frequency. If these estimations are used
1 in conjunction with the Doppler equation in a
f 2 ”” c2 -- c1f 1 recursive manner, the estimation of S(q ) can be
Still, the directional distribution is poorly found as a function of true frequency.
constrained by only four moments and the estimates The method by which this is performed is
require careful interpretation, [Krogstad 1991]. outlined below;
1. Determine the three components of the fluid
Mean Direction and Directional Spread velocity in vehicle body fixed coordinates.
2. Transform the fluid velocities into the global
An alternative approach that avoids the navigation frame.
pitfalls of S(q ) estimation, is to describe the 3. Compute the auto - and cross -spectra of the
directionality of waves by a few simple parameters. fluid velocity components.
For narrow S(q ) , a mean propagation direction q 4. Determine the Fourier moments using (19)-
m (22).
and a root-mean-square measure of the directional
5. Convert the Fourier moments into Krogstad
spreading energy s can be defined in terms of
q notation and use the MEM to determine the
ehttsrif -order and second -order Fourier moments directional distribution S(q ).
a , b , a and b [Kuik 1988], given by, 6. Using the vehicles mean velocity, and the
1 1 2 2
mean direction obtained from (26) or (28),
(cid:230)(cid:230) b (cid:246)(cid:246)
q ==nat --1 (cid:231)(cid:231) 1 (cid:247)(cid:247) (26) apply the Doppler equation to determine the
(cid:231)(cid:231) (cid:247)(cid:247)
m ŁŁ a łł frequency shift.
1
s 2 ==2 [[1 -- [[a cos (q ) ++b nis (q ) ]]]] (27) 7. Return to 3 and complete the process until
q 1 m 1 m frequencies converge.
q == 1 nat --1 (cid:231)(cid:230)(cid:231)(cid:230) b2 (cid:247)(cid:246)(cid:247)(cid:246) (28) The corrections to the estimation of S(q ) using the
(cid:231)(cid:231) (cid:247)(cid:247)
m 2 ŁŁ a łł Doppler equation are quite small for slow moving
2
vehicles. Considering, for example, the NPS
1 [[ [[ ]]]]
s 2 == 1 -- a cos (2 q ) ++b nis (2 q ) . (29) xineohP VUA hcihw sah a mumixam drawrof
q 2 2 m 2 m
velocity of 1.5 m/s, the error associated with not
Again, this method fails to identify bimodal
using the Doppler equation are approximately – 1
distributions but it is useful to determine a base
sec in identification of wave periods between 4 and
direction so that a control command could be
40 . sdnoces Similarly, the erro r in direction
determined. More on this approach will be
estimation is approximately 5-7 degrees. When an
discussed later in this chapter.
AUV goes into a station-keeping mode, where
vehicle velocities are significantly reduced, the
4
modifications required due to the Doppler shift are obtained during this offshore exercise. As seen in
negligible. Figure 3, the bi-modal properties of the seaway are
captured, as well as an estimate of the significant
Experimental Results Of Wave Directional wave height. The ability to estimate the dual
Estimation Using The NPS Phoenix AUV directions is due to the use of the MEM algorithm
employed. Figure 4 presents the associated
During November 1998, the NPS Center spectrum plots for this energy estimate. The ability
for AUV Research, under the direction of Professor of an AUV to estimate currents is shown in Figure
Anthony Healey, participated in the ONR 5. Using a triangular, time based run; the current
sponsored AUVFEST '98. This AUV technology can be determined using set and drift calculations
demonstrator was held in the Gulf of Mexico, south from the error in final position as well as the
of Gulfport, MS. A complete description of the heading error on each leg. This information can be
exercise can be found in [Bunce 1998]. feed directly into the vehicle’s navigation process
The Phoenix AUV was used during this exercise as to account for the offset due to current thereby
a mobile sensor to gather oceanographic data. increasing navigation accuracy. Short-term
Using the concepts presented in the previous averages from each of the three legs are in general
section, the vehicle conducted 27 short-term agreement with the overall average determined
sampling missions. The products that were from the navigational drift.
obtained included directional energy plots, During experimental validation of a new
directional spectrum plots and mean current Disturbance Compensation Controller [Riedel
estimations. 1999], wave direction estimates were obtained in
The key to the ability of Phoenix to obtain Monterey Bay, see Figure 6. Again the Phoenix
data capable of producing this information is the was able to record data which can be used to
combined ADV/RDI/MotionPak/Directional Gyro determine directionality spectra, Figure 7.
sensor suite. The physical layout of the NPS
Phoenix AUV is shown in Figure 1. Detailed Conclusions
description of the vehicle can be found in [Marco
1996] and [Brutzman 1997]. In addition an online This paper has presented the techniques
description can be found on the NPS center for currently employed for the determination of wave
AUV Research web site at directional estimations from standard wave
http://www.cs.nps.navy.mil/research/auv/auvstats. following buoys. It has extended this analysis for
With these sensors, accurate three-dimensional use in determining directional estimates from data
fluid velocities, expressed in global quantities, were gathered by an Autonomous Underwater Vehicle.
capable of being obtained in post-processing. Since Using this data gathered, an approach was
the RDI/ADV sensors are collocated, little vehicle presented which will allow the deployed vehicle to
induced motion remains after processing the data. obtain information about the directionality of its
The data obtained validated the concept of working environment thereby allowing the vehicle
obtaining tactical oceanographic data from an to have information available to make decisions
underwater vehicle. During the collection of the regarding mission execution. Actual experimental
data, remnants of Hurricane Mitch were still results using the NPS Phoenix AUV have been
present in the Gulf, providing an excellent source of presented. It was shown that tactical oceanographic
ground swell. In addition, there was a significant data is obtainable from a moving AUV. The
wind generated wave component in a different vehicle in this manner becomes an intelligent,
direction than the swell component, resulting in a mobile off-board sensor, thus presenting the
multi-modal spectrum, Figure 2. argument for AUV deployment with operational
The results presented in Figures 3-5 teelf.stinu
provide a sample of the oceanographic data
5
ST725 SONAR
RDI DOPPLER ST1000 SONAR
SONAR
SonTek ADV
DEPTH CELL
TRANSDUCER
FIN SERVO (8)
BOW LEAK
DETECTOR
BOW LATERAL
THRUSTER
ADV PROCESSOR
BOW VERTICAL MOTION PAK
THRUSTER
DC POWER SUPPLIES
COMPUTER POWER
AND RELAYS
SUPPLY (2)
MOTOR SERVO 12 VOLT BATTERY (2)
CONTROLLERS (6) FOR COMPUTERS
FreeWave RADIO
COMM. LINK GESPAC COMPUTER
WITH SIGNAL BOARDS
QNX PENTIUM
COMPUTER
12 VOLT BATTERY (2)
STERN VERTICAL FOR GYROS/MOTORS
THRUSTER
FREE GYRO
FREE GYRO
POWER SUPPLY
STERN LEAK
STERN LATERAL DETECTOR
THRUSTER
REAR SCREW MOTOR (2)
REAR SCREW (2)
CONTROL FINS (8)
Drawn by D. Marco 1999
Figure 1 Physical Layout of Phoenix AUV
Figure 2 Phoenix Being Launched from the R/V Gyre
6
Figure 3 Sample Direction Energy Plot From Phoenix Data, Nov. 4, 1998 (Run # 9), Gulf of Mexico,
AUVFEST ‘98
0.04 0.1
16 sec 10.7 sec s8e c
0.04
q) 0.02 q) 0.02 q) 0.05
S( S( S(
0 0 0
0 60 120 180 240 300 360 0 60 120 180 240 300 360 0 60 120 180 240 300 360
20 1x0 -3
6.s4e c 5.33 sec 4.57 sec
0.04 0.04
q) 10 q) 0.02 )q 0.02
S( S( S(
0 0 0
0 60 120 180 240 300 360 0 60 120 180 240 300 360 0 60 120 180 240 300 360
1x0 -3
0.2 0.04 20
s4e c 3.56 sec 3.s2e c
q) 0.1 q) 0.02 q) 10
S( S( S(
0 0 0
0 60 120 180 240 300 360 0 60 120 180 240 300 360 0 60 120 180 240 300 360
Figure 4 Sample Direction Spectrum Plots From Phoenix Data, Nov. 4, 1998 (Run # 9), Gulf of Mexico,
AUVFEST ‘98
7
Phoenix Survey Track
350
300
250
200
150
position (m) 100 Surface Current (at 3’
depth): 0.38 m/s from 080 o
50
0
-50
-200 -150 -100 -50 0 50 100 150 200
position (m)
Figure 5 Short-term Current Estimation from Phoenix, November 8, 1999, (Run # 2), Gulf of Mexico,
AUVFEST ‘98
Figure 6 Sample Direction Energy Plot From Phoenix Data, June1 5,1 999( Run # 4), Monterey Bay
8
0.2
3s2e c
q) 0.1
S(
0
0 60 120 180 240 300 360
0.2 0.3 0.2
1s6e c 10.s7e c 8s ec
)q 0.1 )q 0.15 q) 0.1
S( S( S(
0 0 0
0 60 120 180 240 300 360 0 60 120 180 240 300 360 0 60 120 180 240 300 360
0.2 0.1 0.2
4.6ces 5.3s3e c 4.5s7e c
)q 0.1 )q 0.05 q) 0.1
S( S( S(
0 0 0
0 60 120 180 240 300 360 0 60 120 180 240 300 360 0 60 120 180 240 300 360
0.2 0.2 0.2
4s ec 3.5s6e c 2.3ces
)q 0.1 )q 0.1 q) 0.1
S( S( S(
0 0 0
0 60 120 180 240 300 360 0 60 120 180 240 300 360 0 60 120 180 240 300 360
Figure 7 Sample Direction Spectrum Plots From Phoenix Data, June 15,1999 (Run # 4 ),M ontereyB ay
Oceanic Engineering, v. OE-8, October 1983,
Acknowledgements pp.254-271.
Krogstad, H.E., Gordon R.L., and Miller M.C.,
The authors wish to thank the Office of Naval
“High-Resolution Directional Wave Spectra From
Research for financial support of the NPS AUV
Horizontally-Mounted Acoustic Doppler Current
program under Work Request N0001498WR30175
Meters ,” Journal of Atmospheric and Oceanic
Technology, v 5, pp. 340-352.
References
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Allender, J., “The WADIC Project: A
L.H., “A Method for Routine Analysis of Pitch-
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and-Roll Buoy Data,” Journal of Physical
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Oceanography, v 18, 1988.
1989, pp. 505-536.
Long, R.B., “The Statistical Evaluation Of
Brutzman, D., Healey, T., Marco, D., and McGhee,
Directional Spectrum Estimates Derived From
B., “The Phoenix Autonomous Underwater
Pitch/Roll Buoy Data,” Journal of Physical
Vehicle,” AI-Based Mobile Robots , IFAA/TIM
Oceanography, v 10, 1980.
Press, 1997.
Longuet-Higgins, M.S., Cartwright, D.E. and
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