A Low-Cost GPS-Based Wave Height and
Direction Sensor for Marine Safety
Masatoshi Harigae, Isao Yamaguchi, Tokio Kasai and Hirotaka Igawa, Japan Aerospace Exploration Agency (JAXA)
Hiroto Nakanishi and Takahiro Murayama, Japan Weather Association (JWA)
Yasunori Iwanaka, Zeni Lite Buoy Co., Ltd.
Hirotaka Suko, Furuno Electric Co., Ltd.
Hiroto Nakanishi is a technical planning manager at the
Japan Weather Association. He has about 15years
experience of air pollution data and weather data
collection systems.
BIOGRAPHY
Masatoshi Harigae is the director at the Advanced Control
Research Group of the Institute of Space Technology and
Aeronautics of JAXA, and has over 15 years experience
in the development of GPS/INS hybrid navigation
systems and their application to the air traffic system. He
is also engaged in GPS technology transfer to commercial
fields. He received B.S., M.S. and Ph.D. degrees in
aerospace engineering from the University of Tokyo.
Takahiro Murayama is an engineer at the Applied
Meteorology Research Section, Research Department of
JWA. He is involved in the development of a sea wave
observation system with buoys and the atmospheric and
sea wave numerical simulations. He received a B.E.
degree in Marine Science and Technology from Tokai
University.
Isao Yamaguchi is a senior researcher at the Structure
Research Group of the Institute of Space Technology and
Aeronautics of JAXA. His research interests are dynamics,
system identification, modeling and control of spacecraft
and large space structures. He received B.S., M.S. and the
Ph.D. degrees in aeronautical engineering in 1981, 1983
and 1997, respectively from the University of Tokyo. He
is a member of SICE, JSME, The Japan Society for
Aeronautical and Space Sciences, AIAA and IEEE.
Yasunori Iwanaka is a chief of technical development
department of Zeni Lite Buoy Co., Ltd. He has about 5
years experience of marine observation monitoring
systems. He has a B.S. in electronics engineering from
Osaka Institute of Technology.
Hirotaka Suko is a software engineer at Furuno Electric
Co., Ltd., and has over 10 years experience in the
development of GPS receivers for navigation, time
synchronization and terrain remote monitoring. He has a
B.S. in management engineering from Setsunan
University.
Tokio Kasai is a senior researcher at the Structure
Research Group of the Institute of Space Technology and
Aeronautics of JAXA. His research interests include
structural dynamics, system identification and modeling
of flexible spacecraft. He received B.S. and M.S. degrees
in mechanical engineering from Waseda University.
ABSTRACT
Predicting the development, decay and propagation of
ocean waves is essential to allow maritime operations to
be carried out safely and economically. A low cost wave
height and direction sensor that can be deployed in the
open sea is expected to improve the accuracy of wave
prediction. We propose a new method for measuring wave
height and direction by installing a point-positioning GPS
receiver on a buoy placed in the open sea. The essential
idea of measuring wave height with centimeter accuracy
depends on the fact that the most of the spectrum of GPS
Hirotaka Igawa is a researcher at the Structure Research
Group of the Institute of Space Technology and
Aeronautics of JAXA. His research interests are structural
dynamics, modeling of lightweight structures and smart
structures. He received B.S., M.S. and Ph.D. degrees in
mechanical engineering science in 1992, 1994 and 1997
from Tokyo Institute of Technology.
ION GNSS 17th International Technical Meeting of the
Satellite Division, 21-24 Sept. 2004, Long Beach, CA
1234
for waves that are several meters high. The advantages of
this sensor are that there are no restrictions as to its
location and it is easy to handle. It is therefore usually
deployed on drifting buoys. The sensor cost, especially
for 3-axis accelerometers, is generally high.
point-positioning error exists in the frequency domain
below 0.01 Hz, but the frequency spectrum of ocean
waves is in a separate band at around 0.1 Hz. Therefore, a
sophisticated high-pass filter can extract the movements
of a GPS equipped buoy excited by ocean waves with
minimum effect of GPS point-positioning error. The
three-dimensional GPS positioning data also enables us to
measure wave direction. Our proposed system has been
already adopted by the Japan Meteorological Agency as
the world’s first GPS-based wave height and direction
measuring system. This paper presents the concept of the
point-positioning GPS-based wave sensor, the algorithms
for filtering and extracting wave data, and the results of
field trials carried out in the open sea.
Another approach to measuring wave height and direction
is based on kinematic GPS. Kinematic GPS can achieve
centimeter-level positioning of a floating buoy in
horizontal and vertical directions and measure wave
height and direction. Kinematic GPS has the advantage of
being able to measure long-period waves with small
accelerations, such as tsunami and tidal waves. However
the operational range of kinematic GPS is presently
limited to about 20 km from the shore even using
dual-frequency GPS receivers, and the system is complex
and costs are very high.
INTRODUCTION
The current prediction of ocean waves based on CFD
(Computational Fluid Dynamics) does not satisfy the
requirements of maritime navigation because of the poor
accuracy of ocean wave models. A low cost wave height
and direction sensor that can be deployed in the open sea
is expected to provide data that will allow the prediction
models to be improved.
We propose a novel method that uses only a low-cost
point-positioning GPS receiver (consumer car navigator
class is acceptable) installed on a buoy that extracts wave
height with an accuracy of several centimeters and
direction with an accuracy of 5 degrees. This method does
not require a pair of GPS receivers, and so can be
deployed even in the middle of the Pacific Ocean. The
essential idea of measuring wave height with centimeter
accuracy depends on the GPS measurement characteristic
that the most of the frequency spectrum of GPS pointpositioning error exists below 0.01 Hz, while the
frequency spectrum of ocean waves is in a separate band
at around 0.1 Hz. Therefore, a sophisticated high-pass
filter can extract the movement of a GPS-equipped buoy
excited by ocean waves with minimum influence of GPS
point-positioning errors (U.S. Patent Application No.
P7153-2069-030397).
While there are several ways to measure wave height and
direction, ultrasonic sensors are widely used as the most
reliable sensors, and Japan has constructed an observation
network of such sensors. Ultrasonic sensors measure
wave height with a design resolution of several
centimeters by measuring the distance to the sea surface
from an observation device installed on the seabed by the
emission of ultrasonic waves. There is also a wave sensor
with canted ultrasonic beams that measures wave
direction. However, the measurable distance from the sea
bed to the sea surface is limited to around 50 m, so the
sensors cannot be deployed in the open ocean but only in
littoral areas. It should be also noted that initial
installation and maintenance costs are usually very high.
This new GPS-based system does not require a
complicated algorithm to extract wave data, but simply
applies a high-pass filter to point-positioning data.
Therefore, the algorithm can execute in the navigation
processing core of a GPS receiver. We have developed a
GPS receiver board with a wave height and direction
measurement function. This transmits only analyzed
ocean wave data to a land station via communication
satellite, rather than raw GPS position data that must be
processed separately. This reduces the volume of data that
has to be communicated, and means our method also has
low operating costs as well as low acquisition cost..
An accelerometer installed on a floating buoy can
measure wave height by detecting the vertical motions of
the buoy. It is easy to expand its function to measure wave
direction by using 3-axis accelerometers directionally
stabilized by gyros or an oil pod. Horizontal and vertical
displacements can be obtained by double integration of
three acceleration signal components. The vertical
component of displacement gives wave height, and
correlation analysis between the horizontal and vertical
components gives wave direction. Measurement accuracy
is 2-10% of wave height, which is several ten centimeters
ION GNSS 17th International Technical Meeting of the
Satellite Division, 21-24 Sept. 2004, Long Beach, CA
Our proposed system has been already adopted by the
Japan Meteorological Agency as the world’s first
1235
GPS-based wave height and direction measuring system.
The system provides valuable ocean wave data
automatically via the Internet for 24 hours continuously. It
is also planned to retrofit our system to add a wave height
and direction measurement capability to existing buoys.
z
x
This paper presents the concept of the point-positioning
GPS-based wave sensor, the algorithms of filtering and
extracting wave data, and the results of experiments
carried out in the open sea.
WAVE FUNDAMENTALS 1)
Movement of the ocean surface is classified as various
kinds of waves according to time and space scales. Ocean
waves, which we treat here, consist of wind waves and
swells. The first step of a wind wave is called a capillary
wave and its restitution force is surface tension. Capillary
waves develop into ripples with 0.3 sec period by the
effect of gravity. Ripples develop further with the aid of
wind forces, and their periods reach 10-15 sec and they
can sometimes grow to 10 m in height. Wind waves that
develop further and move away from their origin are
called swells. Swells are divided into three categories
according to wavelength and period: short, middle and
long swells. Short swells refer to those with a wavelength
below 100 m and a period less than 8 sec. Middle swells
have wavelengths 100-200 m and periods 8-12 sec. Long
swells have wavelengths over 200m, sometimes reaching
400 m, with periods of up to 20 sec.
Among the movements of the ocean surface, ocean waves,
whose restitution force is gravity, have the most energy. It
is therefore important for maritime safety to predict these
gravitational waves. The motion of gravitational waves is
described as follows:
cosh k ( z 0 + h)
x = x0 + a
cos(kx 0 − ωt ),
sinh kh
y = y0 ,
z = z0 + a
Fig. 1 Trajectory of water particle by gravitational wave
floating buoy
Fig. 2 Trajectory of water particle by surface wave
cosh k ( z 0 + h)
,
sinh kh
sinh k ( z 0 + h)
Az ( z 0 ) = a
.
sinh kh
Ax ( z 0 ) = a
(1)
sinh k ( z 0 + h)
sin( kx 0 − ωt )
sinh kh
Fig. 1 shows the motion of gravitational waves according
to equation (2). Because Ax > Az > 0, a particle of water
travels along a horizontal elliptical. When we consider
surface waves where h is much greater than the
wavelength (~1/k), the following approximation can be
made:
where k is the wave number, h is the distance from mean
water level to the bottom and ω is the wave angular
frequency. When we consider x0 = 0, equation (1)
becomes
x = Ax ( z 0 ) cos ωt ,
z = z 0 − Az ( z 0 ) sin ωt
Ax = Az ≈ ae kz 0
(2)
Then, a particle of water travels in a circular trajectory
with a radius indicated by equation (3). Fig. 2 shows the
trajectories of water particles in a surface wave.
where
ION GNSS 17th International Technical Meeting of the
Satellite Division, 21-24 Sept. 2004, Long Beach, CA
(3)
1236
Table 1 Expected accuracy of GPS navigation
accuracy (3σ)
algorithm
note
horizontal
vertical
point-positioning
10 m
20 m
differential GPS
1m
2m
kinematic GPS
5 cm
10 cm
< 20km*
* Baseline length
Table 2 Expected characteristics of GPS system errors
error sources
range (1σ)
time constant
ephemeris
~3m
~ 1 hr
satellite clock
~3m
~ 5 min
ionosphere
~9m
~ 10 min
troposphere
~2m
~ 10 min
multipath
~3m
~ 100 sec
receiver noise
~1m
white noise
GPS point positioning navigation
120
118
116
x 10
4
power spectrum
114
GPS altitude (m)
power spectrum of GPS positioning error
band of ocean waves
9
112
110
108
3
2
1
0 -4
10
106
10
-3
10
104
-1
10
frequency (Hz)
10
0
10
1
Fig. 4 Power spectrum of GPS position error
102
0
500
1000
1500
2000
2500
time (sec)
3000
3500
4000
GPS altitude filtered by HPF (m)
100
-2
Fig. 3 Altitude given by GPS point-positioning
navigation (experiment)
If the movement of a floating buoy corresponds to that of
a water particle in an ocean wave, a GPS receiver fixed to
the buoy would measure rotational motion with a period
of 0.1-20 sec (0.05-10 Hz) as shown in Fig. 2.
1
0
-1
-2
0
500
1000
1500
2000
time (sec)
2500
3000
3500
Fig.5 GPS altitude filtered by high-pass filter with 0.02
Hz cut-off frequency
CHARACTERISTICS OF GPS MEASUREMENT
moves slowly on account of GPS system errors. This
fluctuation indicates the error of GPS navigation by the
point-positioning algorithm. However, it should be noted
that the time constant of the fluctuation is on the order of
from one hundred seconds to several ten minutes order,
that is, less than 0.01 Hz. This is because the GPS system
errors that affect the positioning error have relatively long
time constants.
The accuracy of GPS positioning depends on the
navigation algorithm, such as the point-positioning
algorithm, the differential GPS algorithm and the
kinematic GPS algorithm. Table 1 summarizes the
expected accuracies of these navigation algorithms.
According to the table, only the kinematic GPS method
satisfies the centimeter-level accuracy required to sense
buoy movement because it eliminates almost all GPS
error sources and uses a carrier phase with low receiver
noise. On the other hand, the point-positioning algorithm
uses pseudorange with larger receiver noise and suffers
from the effects of GPS system errors such as atmospheric
errors, ephemeris errors, etc. Fig.3 shows a typical
point-positioning result over one hour of measurements
from a regular GPS receiver set on the roof of a building.
Although the GPS antenna is fixed, the positioning result
ION GNSS 17th International Technical Meeting of the
Satellite Division, 21-24 Sept. 2004, Long Beach, CA
2
Table 2 summarizes the GPS system errors and their
expected time constants. As shown in the table, with the
exception of receiver noise, the principal GPS system
errors have time constants greater than 100 sec. Therefore,
the fluctuation of GPS altitude error shown in Fig. 3
moves gradually. Fig. 4 shows the power spectrum of the
positioning data of Fig. 3, which is the power spectrum of
the error of GPS navigation by the point-positioning
1237
algorithm. As predicted by Table 2, almost all of the
power of the GPS positioning error exists in a band less
than 0.01 Hz. On the other hand, buoy movement excited
by ocean waves is rotational with a period of 0.1-20 sec
(0.05-10 Hz), which is also shown in Fig. 4. Therefore, a
suitably designed high-pass filter can extract the
movement of a GPS equipped buoy excited by ocean
waves
with
minimum
influence
from
GPS
point-positioning errors. This is the key principle of
measuring ocean waves by GPS point-positioning.
H ( z) =
H ( e jω T ) =
θ (ω ) = −τ c ω
θ (ω )
ω
= −τ c
τ p (ω ) = −
τ g (ω ) = −
(4)
(10)
This means the delay of the envelope of waves coincides
with that of each wave. This is an important property that
our high-pass filter should have to reconstruct the
waveforms correctly.
Cut-off frequency fc
Fig. 6 shows the amplitude characteristic of a high-pass
filter. The cut-off frequency is the width of the window
open to ocean waves without any reduction of amplitude.
As described in the previous section, the longest period of
ocean waves is 20 sec. In this research, we keep some
margin and set the cut-off frequency to 0.03 Hz.
(5)
∞
∑ x(kT )h(nT − kT )
k = −∞
where h(nT) is called the impulse response. Through the
z-transform, equation (5) becomes
ION GNSS 17th International Technical Meeting of the
Satellite Division, 21-24 Sept. 2004, Long Beach, CA
dθ (ω )
dω
(9)
= −τ c
k = −∞
Y ( z) = X ( z)H ( z)
(8)
In this case, the phase delay and group delay become the
same form.
∑
=
(7)
Linear phase characteristic
When the phase characteristic is expressed as below, the
time invariant system has a linear phase characteristic.
 ∞

y (nT ) = ℜ 
x(kT )δ (nT − kT )
 k = −∞

∑ x(kT )ℜ[δ (nT − kT )]
− jωnT
When designing a high-pass filter for measuring ocean
waves, we considered the following.
where δ is the unit impulse function. We construct the
high-pass filter by the linear time invariant system. Then,
the output of the filter is expressed as follows:
∞
∑ h(nT )e
We can then design the impulse response, which
determines the parameters of the digital high-pass filter,
by using filter characteristics such as amplitude
characteristic, phase characteristic, phase delay and group
delay.
k = −∞
=
∞
n = −∞
By limiting the bandwidth of the GPS point-positioning
data to cut out positioning error, a high pass filter can
extract the form of ocean waves correctly with minimum
effect of GPS system errors. The filter’s input data are the
sequence of sampled positions of the GPS antenna on a
buoy, and are generally described as follows:
∞
−n
When we replace the variable z by e − jωT , we obtain the
frequency response of the transfer function H:
DESIGN OF HIGH PASS FILTER 3)
∑ x(kT )δ (nT − kT )
∑ h(nT ) z
n = −∞
To demonstrate this, we applied a simple high-pass filter
(2nd-order Butterworth) with a cut-off frequency of 0.02
Hz to the data of Fig. 3, and obtained the result shown in
Fig. 5. This reduced the fluctuation to 8 cm (1σ). When
the high-pass filter is adopted, the mean value of the
antenna’s altitude becomes zero; however this is not a
problem since we do not have to measure the altitude of
the buoy, only track its rotational movement. This result
suggested that our idea would work well for measuring
ocean waves with centimeter accuracy.
x(nT ) =
∞
Edge frequency of blocking region fa
This edge frequency of blocking region is the maximum
frequency at which the effects of GPS system errors are
blocked completely. GPS system errors exist in the region
below 0.01 Hz as shown in Fig. 4. Therefore, we set the
(6)
1238
1.0
50
Ac
Gain (dB)
Gain
0
-50
-100
-150
Aa
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0
0.1
0.2
0.3
0.4
0.5
0.6
Frequency (Hz)
0.7
0.8
0.9
1
Phase (deg)
10000
0
0
fa
fc
0
-10000
-20000
-30000
Frequency
Fig.6 Amplitude characteristic of linear phase high-pass
filter
Fig. 7a Frequency characteristics of HPF in linear scale
50
Gain (dB)
edge frequency to 0.01 Hz.
Ripple of pass region Ac
As described below, we can not design a filter with ideal
characteristics because of limitations of the order of filter
parameters. Consequently, there is a ripple in the filter
pass band. We set the ripple to less than 0.08 dB (= 1.009),
which means that we will have a 9 cm height sensing
error for a 10 m high wave.
0
-50
-100
-150
Phase (deg)
5000
0
-5000
-10000
-15000
-20000
10 -4
Suppressed gain Aa
There is a positioning error of several meters due to GPS
system errors in the region below 0.01 Hz as shown in
Table 2. We suppress this error to the order of centimeters
by setting the suppressed gain to 40 dB.
10 -3
10 -2
Frequency (Hz)
10 -1
10 0
Fig. 7b Frequency characteristics of HPF in log scale
motor
DC gain
The DC gain is set to zero because absolute altitude data
from GPS are useless for measuring wave data correctly.
1.7 m
GPS antenna
(pendulous)
rotational arm
counter weight
GPS receiver
We realize the above performance by designing a linear
phase FIR (Finite Impulse Response) filter whose number
of the impulse response h(nT) in equation (6) is limited by
N ( 0 ≤ n ≤ N − 1 ). There are several methods to
determine the appropriate impulse response according to
the design conditions, such as the window function
method and the Remez method. We adopted the linear
programming method to realize arbitrary amplitude
characteristics. Fig. 7 shows the design result of our
high-pass filter. The order N of the filter is 241. This filter
satisfies all of the design conditions.
Fig. 8 Schematic view and photograph of test apparatus
ION GNSS 17th International Technical Meeting of the
Satellite Division, 21-24 Sept. 2004, Long Beach, CA
1239
GPS raw
East (m)
non jump
East (m)
To prove the concept, we carried out a laboratory test in
which our high-pass filter was applied to observed GPS
data. Fig. 8 shows the test apparatus that consists of a
wave simulator and a GPS receiver. The wave simulator
has a rotating arm to which a GPS antenna is fixed. The
rotating arm simulates the motion of a buoy floating in the
ocean as shown in Fig. 2. The diameter of rotation is set
by an arm length (170 cm), which simulates wave height,
and rotation speed can be controlled by varying the speed
of the motor. A commercial GPS receiver produced by
Furuno Electric Co. Ltd was used. This has 11 channels to
track the C/A code and the carrier phase, and carries out
point-positioning navigation using all GPS satellites in
view. The output rate of navigation data is 5 Hz.
4
2
0
-2
-4
4
2
0
-2
-4
filtered
East (m)
PROOF OF CONCEPT BY LABORATORY TEST
2
1
0
-1
-2
0
200
400
600
800
1000
1200
0
200
400
600
800
1000
1200
0
200
400
600
time (sec)
800
1000
1200
GPS raw
North (m)
non jump
North (m)
In the experiment, we collected GPS point-positioning
data for 20 min, which is widely used as a sensing period
for ocean waves. We set the period of rotation to 11 sec
and the direction of the arm to 114 deg from the true north.
Fig. 9 depicts the GPS data in vertical and horizontal
planes. The origin of the graph is the initial GPS position.
As shown in the figure, the point-positioning data drifted
gradually on account of GPS system errors although the
wave simulator did not change its position. It should be
also noted that there is a jump in the GPS data around
6,550 sec UTC. This kind of position jump is often seen
in GPS receivers, and is due to satellite change, carrier
phase lock-loss, etc.
4
2
0
-2
-4
4
2
0
-2
-4
filtered
North (m)
Fig. 10a Result of processed GPS data in east axis
2
1
0
-1
-2
0
200
400
600
800
1000
1200
0
200
400
600
800
1000
1200
0
200
400
600
time (sec)
800
1000
1200
GPS raw
Height (m)
4
2
0
-2
-4
non jumped
Height (m)
4
2
0
-2
-4
filtered
Height (m)
Fig. 10b Result of processed GPS data in north axis
2
1
0
-1
-2
0
200
400
600
800
1000
1200
0
200
400
600
800
1000
1200
0
200
400
800
1000
1200
600
time (sec)
Fig. 10c Result of processed GPS data in height axis
Table 3 Statistics of wave data derived from GPS
parameter
experiment result
true value*
wave height
170 cm
167.5 cm
wave period
11 sec
10.83 sec
wave direction
114 deg
115.7 deg
* True values are measured by other means such as
measure, clock and magnetic compass.
ION GNSS 17th International Technical Meeting of the
Satellite Division, 21-24 Sept. 2004, Long Beach, CA
1240
Num. of GPS
mean sea level
7
6
5
5600
2
5800
6000
6200
6400
6600
6800
7000
H1
H2
Height (m)
1
0
-1
T1
T2
-2
-3
5600
5800
6000
6200
6400
UTC time in day (sec)
6600
6800
7000
Fig. 11 Definitions by zero up cross method
1.4
estimated period is 0.17 sec, and the error of the estimated
direction is 1.7 deg. These results confirmed our proposed
concept for measuring wave height and direction using a
low-cost point-positioning GPS receiver.
1.2
1
North (m)
0.8
PARAMETERS OF OCEAN WAVES
0.6
0.4
An actual ocean wave is a superposition of various wind
waves and swells. The waveform is not a simple sine
curve given by the previous ground test. Therefore, we
must define the key ocean wave characteristics - height,
period and direction - before analyzing the GPS data. The
zero up cross method is widely used to define wave height
and period. Fig. 11 shows an example of an observed
waveform. In this method, the waveform is delimited at
the up crossing point of a water surface, and each interval
is considered as a single wave. The height of a single
wave is the distance between the top and the bottom of
the wave. The period is defined to be a length of the
interval.
0.2
0
-0.2
-1.5
-1
-0.5
East (m)
0
0.5
1
Fig. 9 Position data of GPS installed on wave simulator
We performed the following process to extract the buoy
motion (in this case, the rotational motion of the GPS
antenna on the wave simulator). The first step is to change
the coordinate system from latitude, longitude and height
to an east, north and height local frame. The origin of the
local frame is the initial position from GPS because
absolute position is not necessary to obtain the rotational
motion of the buoy. The second step is to remove jumps
in the GPS position data. As previously described, jumps
are often seen, and their higher harmonics become a
problem when using a high-pass filter. We therefore
remove jumps as a second step using a simple algorithm,
where we remove the offset between the positions before
and after each jump. To detect jumps, we check for
extremely large changes in position as well as satellite
changes for navigation. After the second step, we apply
the high-pass filter to the pre-processed GPS data. Fig. 10
shows the position results of the above three steps in east,
north and height axes respectively. You can see the lower
graph, which shows the output of the third step indicates
the antenna’s rotational motion precisely. Table 3
summarizes the statistics of the processed data. The mean
wave height (diameter of rotation) is 1.675 m, a deviation
of only 2.5 cm from the true value. The error of the
ION GNSS 17th International Technical Meeting of the
Satellite Division, 21-24 Sept. 2004, Long Beach, CA
In the analysis, we delimit the waveforms and make a
group of single waves. The number of waves is defined as
the result of counting single waves in the group during the
observation period. The wave height and period are
usually defined in the concept of significant wave, 1/10
maximum wave and maximum wave. The wave height
and period of the significant wave are the mean of those
of waves extracted by the criterion of wave heights from
maximum to one third ranked in descending order. The
1/10 maximum wave depends on the criterion from
maximum to one tenth ranked in descending order. The
maximum wave is a single wave that has the maximum
height.
There are many definitions for wave direction but most of
these depend on the spectral analysis of ocean waves. We
define the movements of a buoy in the east, north and
height axes as xe, xn and xh respectively. The correlation
functions of these are defined as follows:
1241
1
T →∞ T
Aij (τ ) = lim
∫
T /2
−T / 2
1
a 0 + (a1 cos θ + b1 sin θ )
2
+ (a 2 cos 2θ + b2 sin 2θ ) + L
xi (t ) x j (t + τ )dt , i, j = e, n, h (11)
G (ω , θ ) ≈
The cross spectra are derived from the Fourier transform
of equation (11)
C ij (ω ) = K ij (ω ) − jQij (ω )
=
∫
∞
−∞
Aij (τ )e jωτ dτ
The Fourier parameters in equation (16) and the cross
spectra in equation (14) have the following relationships:
(12)
a1 =
Because the movement of the buoy tracks a circle as
shown in Fig. 2, the phase difference between xh and xe, xh
and xn is 90 deg, and the phase difference between xe and
xn is 0 deg or 180 deg. We then have the following
equation:
K he = K hn = Qen = 0
b1 =
b2 =
(13)
(17)
2 K en
φe + φ n
We performed an ocean field test using a
point-positioning GPS receiver installed on a buoy in the
Kitan Strait between the islands of Honshu and Shikoku
from June 17 to June 23, 2004. Fig. 12 shows a map
indicating the Kitan Strait and a photograph of the strait
and the buoy. In addition to the GPS receiver, the buoy
was equipped with an accelerometer based wave sensor,
and an ultrasonic wave sensor was located on the seabed
in the vicinity but about 5 km away from the buoy. Data
from these sensors were used as reference data to evaluate
the GPS-based wave sensing system.
C hn = − jQhn
(14)
C nn = φ n
C hh = φ h
where φe, φn, φh denote the power spectrum of the wave in
each axis, and φh in particular is the power spectrum of
the wave φ.
The spectral resolution of ocean wave energy can be
expressed by multiplication of the power spectrum and its
direction distribution function:
S (ω , θ ) = φ (ω )G (ω , θ )
φh
Qhn
FIELD EXPERIMENT RESUTLS
C he = − jQhe
C ee = φ e
Qhe
φh
φ − φn
a2 = e
φe + φ n
After substitution of equation (13), equation (12) becomes
as follows:
C en = K en
(16)
Then, we can calculate the direction distribution function
(15)
from the cross spectra. The wave direction θ is defined
where θ is the direction of the wave. Each function has
following relationships:
as the direction in which G(ω, θ) has the greatest value
( ∂G / ∂θ = 0 ).
2π
φ (ω ) = ∫ S (ω , θ )dθ
θ = tan −1
0
2π
∫ G(ω ,θ )dθ =1
0
(18)
We usually calculate the wave direction at the frequency
ω that has the first, second and third greatest power
spectra.
According to Longuett-Higgins, Cartwright and Smith
(1963)5), the direction distribution function can be
expanded as a Fourier series:
ION GNSS 17th International Technical Meeting of the
Satellite Division, 21-24 Sept. 2004, Long Beach, CA
Q
b1
= tan −1 hn
a1
Qhe
1242
14
accelerometer
ultrarsonic
GPS
12
wave period (sec)
10
Kitan Strait
8
6
4
Buoy
2
Ultrasonic sensor
0
00:00 12:00 00:00 12:00 00:00 12:00 00:00 12:00 00:00 12:00 00:00 12:00 00:00 12:00 00:00
06/17
Fig.13b
06/18
06/19
06/20
06/21
date/time (UTC)
06/22
06/23
Significant wave period observed at the Kitan
Strait
350
Fig. 12 Kitan Strait and buoy equipped with GPS receiver
accelerometer
ultrarsonic
GPS
300
250
7
200
direction (deg)
accelerometer
ultrarsonic
GPS
6
5
150
wave height (m)
100
4
50
3
0
00:00 12:00 00:00 12:00 00:00 12:00 00:00 12:00 00:00 12:00 00:00 12:00 00:00 12:00 00:00
2
06/17
06/18
06/19
06/20
06/21
date/time (UTC)
06/22
06/23
Fig. 13c Wave direction at frequency that has maximum
power spectrum, observed at the Kitan Strait
1
0
00:00 12:00 00:00 12:00 00:00 12:00 00:00 12:00 00:00 12:00 00:00 12:00 00:00 12:00 00:00
06/17
06/18
06/19
06/20
06/21
date/time (UTC)
06/22
06/23
In the analysis, we used 20 min GPS point-positioning
data each hour during the experiment. Fig. 13 shows the
results of the analysis, which show the height and period
of significant waves and wave direction at the frequency
with the biggest power spectrum. During the experiment,
the sixth typhoon of the season passed close to the Kitan
Strait as shown in Fig. 14. Our system therefore
experienced ocean waves from small wind waves to long
swells. When the typhoon passed, our system observed
that the significant wave height reached 7m where the
maximum wave height was over 10 m. Fig. 15 shows the
reconstructed trajectory of the buoy movements in the
east, north and height axes when the maximum wave
height was observed.
Fig. 13a Significant wave height observed at the Kitan
Strait
ION GNSS 17th International Technical Meeting of the
Satellite Division, 21-24 Sept. 2004, Long Beach, CA
1243
Because the GPS wave sensor and accelerometer wave
sensor were installed on the same buoy, the GPS derived
data coincide very well with those of the accelerometer.
The differences between the two systems are summarized
in Table 4. Table 4a shows the statistical result for all data
during the experiment and Table 4b indicates the results
corresponding to significant wave heights. Table 4b is
indicated because the error of the accelerometer based
wave sensor is proportional to wave height. The
accelerometer sensor has a guaranteed 10 % accuracy in
this case, which means a 70 cm error is expected for a 7 m
significant wave height. The differences shown in Table
4b coincide well with the accelerometer error
specifications, which confirms the correctness of the GPS
wave sensor. On the other hand, the data from the
ultrasonic wave sensor shows smaller values. This is
because the ultrasonic sensor was located apart from the
buoy in the neighborhood of the coast that may have
reduced the energy of the waves. Wave direction obtained
from GPS corresponds well to the ultrasonic sensor in Fig.
13, where the resolution of the wave direction data is 10
deg and 22.5 deg for the GPS and ultrasonic sensor
respectively. However, the accelerometer wave sensor
data differs from these. It may be difficult to keep the
3-axis accelerometers horizontal, and in that case that the
data of the accelerometer in horizontal plane would be
unreliable. This would cause the drift seen in the
accelerometer wave direction data. From this ocean
experiment, we can conclude that the concept of GPS
wave sensor is verified and that the GPS wave sensor can
replace existing accelerometer based and ultrasonic based
wave sensors.
buoy movement H (m) buoy movement N (m) buoy movement E (m)
Fig. 14 Trajectory of sixth typhoon (2004) during
experiment
4
0
-4
-8
8
4
0
-4
-8
8
maximum wave height 10.1 m
4
0
-4
-8
0
1:4
2:4
0
2:0
2:4
0
2:2
2:4
0
2:4
2:4
0
0
0
0
0
0
0
3:0
3:2
3:4
4:0
4:2
4:4
5:0
2:4
2:4
2:4
2:4
2:4
2:4
2:4
UTC time (June 21, 2004)
Fig. 15 Reconstructed waveform by GPS
Table 4a Differences between the GPS and
accelerometer wave sensors
GPS BASED WAVE SENSOR BOARD
data
wave height
wave period
The new GPS based system does not require
complicated processing to extract wave data, but
simply applies a high-pass filter to point-positioning
data. Therefore, the algorithm can be executed in the
navigation processing core of any GPS receiver. We
developed a GPS receiver board shown in Fig. 16,
which includes wave height and direction measurement
functions. This GPS board enables us to reduce the
amount of data transmitted from sensors to a land
station via communication satellite because it is not
necessary to transmit raw GPS data for processing,
only analyzed ocean wave data is transmitted. This
means that our method reduces not only the price of a
sensor system but also the operating cost.
ION GNSS 17th International Technical Meeting of the
Satellite Division, 21-24 Sept. 2004, Long Beach, CA
8
mean
8.8 cm
0.6 sec
1σ
27.6 cm
0.8 sec
rms
28.9 cm
1.0 sec
Table 4b Wave height differences between the GPS
and
accelerometer
wave
sensors
corresponding to significant wave height
height range
mean
1σ
rms
0-1 m
-4.3 cm
6.8 cm
8.1 cm
1-3 m
11.4 cm
20.0 cm
22.9 cm
3-5 m
43.6 cm
16.7 cm
46.8 cm
>5 m
50.8 cm
76.7 cm
92.1 cm
Note: The accelerometer sensor has a guaranteed 10 %
accuracy in wave height.
1244
CONCLUSIONS
We propose a new method to measure wave height and
direction by installing the point-positioning GPS receiver
on buoys placed in the open sea. The essential principle of
measuring wave height with centimeter accuracy depends
on the characteristic of GPS measurement errors which
allows them to be removed by a sophisticated high-pass
filter without affecting wave information.
Laboratory tests confirm the validity of proposed method
and the GPS based wave sensor shows an accuracy of
several centimeters in wave height and less than 1 sec in
wave period. Wave direction can be also measured with
an accuracy of several degrees. Experiments at sea verify
the feasibility of our proposed system, and the experiment
data indicate the GPS based wave sensor can replace
existing wave sensors.
Fig.16 GPS board including wave sensing function
ACKNOWLEDGMENT
We would like to appreciate the great assistance of Kinki
Trunk Roads Investigation Office, Kinki Regional
Development Bureau of the Ministry of Land,
Infrastructure and Transport for field experiments at the
Kitan Strait.
REFERENCES
1)
2)
3)
4)
5)
Tatsuo Tokuoka, Theory of Wave Motion, Science Co.
Ltd., Tokyo, 1984 (in Japanese).
J. Lighthill, Waves in Fluids, Cambridge University
Press, Cambridge, 1978.
Hiroshi Ochi, Digital Signal Processing Studied by
Simulation, CQ Press Co. Ltd., Tokyo, 2001 (in
Japanese).
J. H. McClellan, T. W. Parks and L. R. Rabiner, “A
Computer Program for Designing Optimum FIR
Linear Phase Filters”, IEEE Trans, Audio Electro.,
Vol. 21, pp. 506-526, 1973.
M. Longuett-Higgins, D. E. Cartwright and N. D.
Smith, “Observations of the Directional Spectrum of
Sea Waves Using the Motions of a Floating Buoy”,
Ocean Wave Spectra, U. S. Naval Oceanographic
Office, 1963.
ION GNSS 17th International Technical Meeting of the
Satellite Division, 21-24 Sept. 2004, Long Beach, CA
1245