354
IEEE GEOSCIENCE AND REMOTE SENSING LETTERS, VOL. 8, NO. 2, MARCH 2011
Altimeter-Derived Ocean Wave Period
Using Genetic Algorithm
Remya Govindan, Raj Kumar, Sujit Basu, and Abhijit Sarkar
Abstract—It is known that wave period can be estimated
from altimeter measurements of wave height, wind speed, radar
backscatter cross section, etc., using empirical relationship. Of
late, the data adaptive approach of neural networks has been
used to derive wave period from altimeter data, and it has been
shown that the technique appears to be superior compared to the
empirical approaches. Another powerful data adaptive approach
of genetic algorithm has been advocated more recently in oceanographic studies. Although primarily used for forecasting time
series, the algorithm can be tuned to find a relationship between
input and output variables. In the present work, this algorithm has
been used to find estimates of wave period from altimeter-observed
parameters, and the performance of the algorithm has been found
to be quite satisfactory. It has been also found that the introduction
of wave age leads to significant enhancement of the accuracy of the
estimate.
Index Terms—Altimeter, genetic algorithm, wave age, wave
height, wave period, winds.
S
EA STATE can be fully described using 2-D wave spectra. However, there are only sparse observations of wave
spectra over the global oceans. There are very few wave rider
buoy observations. Spaceborne synthetic aperture radar (SAR)
is able to provide spectral information [2], [4], [12], [17], [18].
However, this information is largely confined to longer waves
with lower temporal resolution. In the absence of full spectral
information, one is forced to use integrated parameters like
wave height and wave period in various practical applications.
Measurements of significant wave height are routinely available
from satellite observations [8], [9], [23], [26]. However, these
measurements in isolation are not sufficient for a complete
description of wave state and, hence, are not useful for many
applications in offshore engineering and ship design. For example, to compute wave power, the knowledge of the wave period
is essential in addition to that of wave heights. In the past, wave
period information has been acquired from ship-based instruments, visual observations from ships, and wave rider buoys. A
wave period is computed from wave spectrum measurements in
a given frequency range. To generate wave climate or routine
monitoring of the period, a vast network of buoys at regular
spatial intervals is required. This is quite impractical due to
the need to carry out maintenance of buoys in the open ocean
areas such as the southern ocean region. The best alternative
Manuscript received July 1, 2010; revised September 1, 2010; accepted
September 6, 2010. Date of publication October 18, 2010; date of current
version February 25, 2011.
The authors are with the Oceanic Sciences Division, Meteorology
and Oceanography Group, Space Applications Centre (Indian Space Research Organisation), Ahmedabad 38001, India (e-mail: remyagovind@gmail.
com; [email protected]; [email protected]; sarkar.abhi@
gmail.com).
Digital Object Identifier 10.1109/LGRS.2010.2075911
for regular monitoring of such parameters is SAR onboard an
orbiting satellite. However, as mentioned earlier, SAR is able to
provide 2-D ocean surface wave spectra but only for the longer
waves. Altimeters are quite well known for their capability
to estimate significant wave height and surface wind speed in
addition to sea surface height. Although an altimeter is unable
to provide a direct estimate of wave period, an indirect estimate
computed from altimeter measurements is an attractive alternative. Accordingly, a number of studies have been undertaken to
derive wave period from altimeter measurements of backscatter
coefficient and significant wave height with increasing success
in terms of the degree of achieved accuracy, as found out by
comparing with colocated buoy measurements. The present
work is also an attempt in this direction. The distinguishing
feature of the present work is the use of a genetic algorithm
(GA), which allows one to obtain an explicit analytical equation
for estimating wave period from satellite observations of wind
speed and wave height. This equation is quite easy to use in
practice for computing wave period from altimeter data and
also provides more accurate estimate compared to the known
methods.
Long ago, Challenor and Srokosz [5] computed wave period
using satellite-derived wave height and backscattering coefficient (σo) with the method of spectral moments. Sarkar et al.
[25] found that wave period computed using this method is
highly underestimated for the Indian Ocean region. Hence,
they semiempirically tuned the coefficients by equating climatic wave period distributions with those computed by satellite altimeter observations using “spectral moments” method.
Hwang et al. [19] reported a semiempirical function for the
characteristic wave period for the swell-free Gulf of Mexico
region. Davies et al. [10], [11] were first to use the concept
of “wave age” in the computation of wave period with European Remote Sensing 1 (ERS-1) satellite altimeter data and
colocated buoy data. Kshatriya et al. [22] also made use of
this approach to compute wave period in the seas around India
with TOPEX/Poseidon altimeter data and buoy measurements.
They also brought out superiority of the wave age approach by
showing that the wave age approach helps capture the peak
wave period quite well. The RMS error in computed wave
period was found to be ±0.62 s. Gommenginger et al. [14] also
developed an empirical relationship using buoy and altimeter
data with better results mainly for wind sea-dominated regions.
Later, Cairs et al. [7] improved upon this relationship even for
moderate swell conditions. The utilization of a nonparametric
technique was demonstrated for the first time using a neural
network (NN) by Quiflen et al. [24]. Carter [6] has shown that a
wave period computed using an NN provides better comparison
with buoy data in comparison with the parametric model of
Cairs et al. [7]. In this letter, we advocate the modern powerful
1545-598X/$26.00 © 2010 IEEE
GOVINDAN et al.: ALTIMETER-DERIVED OCEAN WAVE PERIOD USING GENETIC ALGORITHM
approach of GA for computing a wave period using altimeter
data, and it is shown that GA is able to estimate a wave period
with better accuracy than earlier methods for all the cases.
I. WAVE P ERIOD E STIMATION M ETHODS
Wave period from the altimeter data (referred to as Ta ) can
be related to in situ measurements of wave period (Tz and
Tc ) as Ta = (Tz · Tc )1/2 , where Tz is the zero cross and Tc is
the crest period. It shows that Ta represents a measure of the
average wave period. Ta can be expressed in terms of altimetermeasured quantities with the assumption of a random Gaussian
process for the sea surface [5], [25] through
1/2
1/2
Hs σ o1/2 ) |R(0)|
(1)
T a = (π 2 /g)
o
where Hs is the significant wave height, σ is the radar backscattering coefficient, R(0) is the Fresnel reflection coefficient
in the altimeter frequency band, and g is the acceleration
due to gravity. Because of the inherent limitation in the theoretical formulations, as well as unrealistic representation of
wave periods, semiempirical approaches developed later led
to significant improvement in accuracy [10], [11], [22]. The
major improvement in the above approaches was the use of
the dimensionless parameter “wave age” defined as the ratio
of the phase speed of the waves of the dominant waves and the
surface wind speed. As the wave “ages,” it grows longer, and as
it grows longer, it moves faster. So, the “wave age” parameter
is higher for older waves. Wave age can be computed using the
ocean wave spectrum. Since the altimeter does not measure the
wave spectrum, it is convenient to use the “pseudo wave age”
parameter, developed by Fu and Glazman [13], rather than the
wave age. The pseudo wave age (ξ) can be expressed in terms
of significant wave height (Hs) and surface wind speed
0.31
.
(2)
ξ = 3.25 Hs2 g 2 /U 4
In the present study, the data from TOPEX/Poseidon radar
altimeter and in situ buoy data from the National Data Buoy
Center (NDBC) and National Institute of Ocean Technology
(NIOT) for the years 2004 and 2005 have been utilized.
For this purpose, the data for open ocean with depths of more
than 50 m and normal sea state conditions with wind speed
between 2–25 m/s have been used. The JASON satellite system
carrying a state-of-the-art altimeter sensor, launched on December 7, 2001, is providing wind and wave (besides sea level)
information over global oceans regularly. The data are being
made available every second, corresponding to approximately a
7-km resolution along satellite track. The spacing of the tracks
is nominally 316 km at the equator and much smaller at higher
latitudes. The revisit period of each track is close to ten days.
Radar altimeter data have been provided by the Delft Institute
for Earth-Oriented Space Research Radar Altimeter Database
System (http://rads.tudelft.nl/rads/rads.shtml). Altimeter data
used in this study contain wind speed and wave height. Wave
age was computed using (2).
II. WAVE P ERIOD E STIMATION BY GA
GA is a nonlinear data-fitting algorithm which has been
presented in sufficient detail by Alvarez et al. [3]. The algorithm
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is most often used for carrying out forecast of a given data
time series. However, it can also be used for finding relation
between input and output variables, similar to the NN. The two
algorithms are, however, totally different. In the case of NN,
the network structure is generally fixed beforehand, whereas
there is considerable flexibility in the case of GA, and in some
sense, the algorithm itself generates the structure. In a recent
study [1], GA has been used to find such a relation, and the
description of the algorithm is adopted from that study. Let us
assume that there exists a smooth mapping function f (.) that
explains the relationship between a desired variable x and a set
of independent variables (a, b, . . .) so that
x = f (a, b, . . .).
(3)
For the variable x, a set of candidate equations for f (.) is
randomly generated. An equation is stored in the computer as a
set of characters that define the independent variables a, b, . . .,
etc., in (3) and four elementary arithmetic operators (+, −,
×, and /). A criterion that measures how well an equation
string performs on a training set of data is its fitness to the
data, defined as the sum of the squared differences between
the data and the parameter (counterpart of data) derived from
the equation string. The strongest individuals (equations with
best fits) are then selected to exchange parts of the character
strings between them (reproduction and crossover), while individuals less fitted to the data are discarded. Finally, a small
percentage of the equation strings’ most basic elements, single
operators and variables, is mutated at random. The process is
repeated for a large number of times to improve the fitness of
the evolving population of equations. The fitness strength of the
best scoring equation is defined as
(4)
R2 = 1 − Δ2 /Σ(x0 − x0 2
where Δ2 = Σ(xc − x0 )2 , xc is the parameter value estimated
by the best scoring equation, x0 is the corresponding “true”
value of x, and x0 is the mean of the “true” values of x.
It is evident that a high value of R2 signifies a very robust
relationship. In our case, the desired variable x is the wave
period, while the independent variables a, b, . . ., etc., are the
wind speed and wave height. In some cases, wave age parameter
has also been added in order to achieve enhanced performance.
To develop the algorithm for wave period estimation, three
hourly wind speed, wave height, and wave period data of
124 NDBC buoys, covering the North American coasts, Gulf
of Mexico, Alaska, and the Hawaii island area, have been
considered. The data are for the year 2004. In Fig. 1, the
locations of the moored buoys deployed by the NIOT in the
Indian Ocean are also shown. Data from these buoys have
been used to validate our estimation algorithm in the Indian
Ocean. The training data set from the NDBC buoys is selected
within the global data set randomly, and it is made sure that
there are sufficient data points in the wave period bin of 0.5 s.
Out of a total number of 184 185 data points, approximately
half has been utilized for the training, and the training process has been repeated many times to improve the fitness of
the evolving equation (GA-1). The validation of the equation
has been performed using the remaining part of the data set,
which has not been used during the training. After validating
the estimated wave period with measurements, the equation
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IEEE GEOSCIENCE AND REMOTE SENSING LETTERS, VOL. 8, NO. 2, MARCH 2011
TABLE II
WAVE PERIOD RMS ERROR FOR DIFFERENT ξ RANGES
Fig. 1. Locations of the buoys used in the study.
TABLE I
COMPARISON OF BUOY WAVE PERIOD WITH GA, AND NN-1 MODEL
FOR THE G LOBAL D ATA . RMSE IN T HIS AND S UBSEQUENT
TABLES MEANS ROOT MEAN SQUARE ERROR
generated for wave period estimation has been utilized to
estimate wave period based on TOPEX/Poseidon radar
altimeter-derived parameters. The estimation was done using
the altimeter data of 2004 as well as 2005. Wave period estimated using altimeter data has been compared with independent
buoy observations. For this purpose, buoy observations within
100 km of altimeter passes and within 1 h of the altimeter
observations have been used. As discussed above, wave height
is a combination of wind-dominated waves and swells, and
these have different characteristics. To study the impact of
different types of waves on wave period estimation, we have
also included wave age (ξ) in the global data set, and the above
process has been repeated again to generate new equation based
on wind, wave height, and ξ (GA-2).
Fig. 2.
Global wave period comparison for different wave age ranges.
TABLE III
STATISTICS OF DIFFERENT WAVE PERIOD ESTIMATION MODELS
FOR G ULF OF M EXICO AND H AWAII A REA
III. R ESULTS AND D ISCUSSION
Wave periods estimated from satellite altimeter data using
different methods have been compared with wave periods from
independent buoy data. As depicted in Table I, the wave period
estimated using GA-1 shows root mean square (RMS) error
of 0.83 s with a correlation of 0.86. It has been discussed by
Kshatriya et al. [22] that ξ has significant impact on the wave
period estimation. For this purpose, we have again computed
altimeter wave period using GA-2 which includes ξ, and it has
been observed that the inclusion of ξ reduces the RMS error
to 0.76 s which means that ξ has a significant impact on wave
period. These two methods have also been compared with the
NN-1 model developed by Quilfen et al. [24], and it has been
found that GA-2 is superior to the NN method in [24].
Since in the ocean, both sea and swell waves are present with
different characteristics, the analysis was also performed separately for different types of waves using the wave age criteria.
Total data set was divided according to ξ in the range of 0–1,
1–2, 2–3, 3–4, and more than 4. It has been observed that for sea
waves (low ξ), wave period estimation is significantly improved
with an RMS error of 0.55 s for waves with very low ξ (Table II
and Fig. 2). The estimation accuracy reduces with increasing
ξ, which clearly shows that wave period estimation algorithm
should be different for swell and sea-dominated waves.
The difference in wave period algorithms for different wave
conditions has been further demonstrated with the data of the
Gulf of Mexico region, which is dominated mostly by sea
waves, and with the data of Hawaii region having mixed wind
seas and swell trains. For the Gulf of Mexico also, wave periods
were estimated using GA-1 and GA-2, and validation was
performed with the data of 12 buoys in this region (Table III).
For this region, with 685 data points, almost all the waves
are lying in a ξ range of 1–3, and very few wave periods
GOVINDAN et al.: ALTIMETER-DERIVED OCEAN WAVE PERIOD USING GENETIC ALGORITHM
357
were compared with the estimated wave periods, where it could
be seen that the wave periods estimated without using ξ do not
cover the low wave period region, whereas with the inclusion of
ξ, estimated wave period covered the full range of wave heights.
However, for the economy of space, the corresponding figure is
not shown.
IV. C ONCLUSION
Fig. 3.
Comparison of estimated and buoy wave period in the Gulf of Mexico.
Fig. 4.
Comparison of estimated and buoy wave periods in the Indian Ocean.
TABLE IV
STATISTICS OF DIFFERENT WAVE PERIOD ESTIMATION
MODELS FOR THE INDIAN OCEAN REGION
This letter demonstrates the power of the modern data adaptive approach known as GA for the estimation of ocean wave
period from spaceborne altimeter data. The major advantage
of this technique over other data adaptive approaches lies in
the fact that one obtains an explicit analytical equation for
estimating wave period. An interesting outcome of the investigation is that there are different processes involved in wind seaand swell-dominated regions leading to separate algorithms
for wind sea and swell regions. This difference in resulting
algorithms should instigate researchers to pursue investigations
for a better understanding of sea and swell waves. With the
launch of Ka-band SARAL/ALTIKA altimeter constellations
and wide swath altimeter, these techniques may prove to be
beneficial to provide accurate wave period from satellite data. In
future work, we would like to incorporate additional constraints
coupled with improved technique for better estimation of wave
period for global oceans.
A PPENDIX
We list below the global equations for wave period estimation
using two different GA models (GA-1 and GA-2)
GA−1 : Tz = (((wh+((((wh/ (ws/(8.01)))−ws) /(4.38))
+ (5.09))) /ws) ∗ ws)
GA−2 : Tz = (((ξ −(5.78)) / (ξ +(ws/ (wh
∗ ((ws/wh)+wh)))))+(wh+(5.70))) .
greater than 8 s are encountered. This clearly shows a wind
sea-dominated region. The difference in the RMS error of 0.76
and 0.66 s for GA-1 and GA-2, respectively, once again proves
the significance of ξ in the estimation of wave period for the
wind sea condition. In Fig. 3, we graphically show the result
of estimation as a scatter plot. In the Hawaii region, however,
inclusion of ξ does not improve the wave period estimation, and
here, both algorithms provide similar results with RMS error of
0.59 and 0.6 s. It was observed that in this region, steady trade
winds give rise to mixed wind seas and swell.
In the next experiment, nine deep ocean buoys in the Indian
Ocean region with mixed sea state conditions are selected to
evaluate the performance of the GA models once again. In the
Indian ocean data set, almost all the ξ values lie between one
and four. There are very few values outside this range. This
clearly shows a mixed wave condition. The wave periods computed by applying different GA models on satellite altimeter
data were compared with colocated buoy data, and it was found
that the inclusion of ξ leads only to a marginal improvement
in the result. The result of estimating wave periods is shown in
Fig. 4 as a scatter plot. (Table IV).
Finally, the importance of ξ in wave period estimation is also
observed in a different manner. For this, buoy wave periods
Here, Tz denotes the wave period, while wh and ws denote
the wave height and wind speed, respectively. As mentioned
previously, ξ denotes wave age.
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