Journal of Oceanography, Vol. 59, pp. 235 to 244, 2003
A Spectral Approach for Determining Altimeter Wind
Speed Model Functions
D ONGLIANG ZHAO1,2* and YOSHIAKI TOBA 1,3
1
Earth Observation Research Center, National Space Development Agency of Japan,
Tokyo 104-6023, Japan
2
Physical Oceanography Laboratory, Ocean University of China, Qingdao 266003, China
3
Research Institute for Environmental Sciences and Public Health of Iwate Prefecture,
Morioka 020-0852, Japan
(Received 8 January 2002; in revised form 7 October 2002; accepted 10 October 2002)
We propose a new analytical algorithm for the estimation of wind speeds from altimeter data using the mean square slope of the ocean surface, which is obtained by
integration of a widely accepted wind-wave spectrum including the gravity-capillary
wave range. It indicates that the normalized radar cross section depends not only on
the wind speed but also on the wave age. The wave state effect on the altimeter radar
return becomes remarkable with increasing wind speed and cannot be neglected at
high wind speeds. A relationship between wave age and nondimensional wave height
based on buoy observational data is applied to compute the wave age using the significant wave height of ocean waves, which could be simultaneously obtained from
altimeter data. Comparison with actual data shows that this new algorithm produces
more reliable wind speeds than do empirical algorithms.
1. Introduction
The remote sensing of sea surface winds with an altimeter is based on specular returns from the sea surface.
An altimeter is an active satellite sensor that transmits
microwaves (Ku band of 13.5 GHz or C band of 5.2 GHz)
to the sea surface in the nadir direction. At small incidence angles, the specular reflection is dominant at the
sea surface (Valenzuela, 1978; Donelan and Pierson, 1987;
Ebuchi et al., 1987, 1993). If the sea surface is smooth,
more signals will return to the altimeter, which means
that the normalized radar cross section (NRCS) σ0 is large.
If the sea surface is rough, the signals will be reflected in
various directions, and many of them will not return to
the altimeter, so the NRCS will be small. The altimeter
NRCS is thus determined by the sea surface roughness:
the rougher the sea surface, the lower the expected NRCS.
NRCS can therefore be considered as a function of the
statistical moments of the sea surface elevations and
slopes. The dominant parameter of this function is the
mean square slope (MSS) of the sea surface, which is
mainly determined by short wind waves in addition to
surface currents and films. The wind waves are certainly
generated by the sea surface wind. Therefore, the NRCS
Keywords:
⋅ Altimeter,
⋅ wind speed model
function,
⋅ wind waves,
⋅ wave age.
is related to the sea surface wind through the wind wave,
but not directly (Fig. 1).
In application, however, researchers usually correlate NRCS with the wind speed directly due to the difficulty of quantitatively estimating the sea surface roughness. The common approach in empirical algorithms is to
compare the altimeter measurements of σ 0 with the coincident observations of wind speed by using in situ measurements, scatterometer, or numerical weather prediction
model data (Ebuchi et al., 1992b; Young, 1993; Glazman
and Greysukh, 1993; Lefevre et al., 1994). The accuracy
of the model functions depends on the method used for
their calibration, including the space chosen for the minimization process, the quantity of data and the prescribed
form of the function, if any (Lefevre et al., 1994). The
most well known empirical wind speed functions proposed
so far are those of Brown (Brown, 1979; Brown et al.,
1981), CM (Chelton and McCabe, 1985), SB (Goldhirsh
and Dobson, 1985), CW (Chelton and Wentz, 1986), and
MCW (Witter and Chelton, 1991). These empirical algorithms have been used daily to derive wind speed from
altimeter observations.
Another method is to calculate MSS by integrating
the wind-wave spectra (Wu, 1992; Apel, 1994; Hwang et
al., 1998; Elfouhaily et al., 1998; Liu et al., 2000). Although these algorithms have some theoretical background for retrieving the wind speeds from altimeter data,
* Corresponding author. E-mail: [email protected]
Copyright © The Oceanographic Society of Japan.
235
2.
Radar Cross
Section of
Altimeter
Sea-Surface Wind
Sea-Surface
Roughness
Sea-Surface
Wind Waves
Surface
Current
Surface
Films
Fig. 1. Schematic relationship of normalized radar cross section, sea surface roughness, wind waves and sea surface
wind speed.
they correlate NRCS with wind speed just as the empirical algorithms do. Thus, the influence of wave states or
tilting effects of long waves on radar returns have usually been completely neglected.
Although various empirical algorithms produce comparable results in the parameter range for which they are
developed, they yield very different results when extrapolated to wind speeds above 20 m/s. In fact, due to their in
situ data set limitation, most of them cannot be applied to
high wind speeds (Young, 1993). There is evidence to
suggest that altimeter measurement is impacted by additional, non-wind-dependent factors, which can include
surface current and atmospheric stability (Vandemark et
al., 1997). However, the largest contamination factor appears to be the degree of sea state development (Glazman
and Pilorz, 1990; Lefevre et al., 1994). This is referred to
as wave age and is normally characterized by the phase
speed of the dominant wind wave divided by the wind
speed at a height of 10 m. The wave-dependent algorithms
were first proposed by Glazman and Pilorz (1990) and
refined by Glazman and Greysukh (1993). Based on altimeter data, Ebuchi et al. (1992b) and Ebuchi (1999)
found a significant influence of the fetch on the growth
of wind wave and therefore on the estimation of wind
speed from NRCS.
In this paper we calculate MSS by integrating a
widely accepted wind-wave spectrum including the gravity-capillary range. The result is found to agree well with
the optical measurements in the field (Cox and Munk,
1954a, b). Thus, a new algorithm depending not only on
wind speed but also on wave age is proposed that gives
more reliable wind speeds than the empirical algorithms
mentioned above.
236
D. Zhao and Y. Toba
Relationship between Mean Square Slope and
Radar Backscatter Cross Section
At small incidence angles (<15° for satellite
altimetry) the backscattering is primarily due to specular
reflections. The NRCS due to specular reflection can be
expressed (Valenzuela, 1978; Donelan and Pierson, 1987;
Apel, 1994) as,
(
σ 0 (θ ) = π R(0) sec 4 θp η x , η y
2
)
(1)
where θ is the radar incidence angle and |R(0)| 2 is the
Fresnel reflection coefficient for normal incidence. The
term p(η x, ηy) is the probability density function (PDF)
of the ocean surface slope at the specular points, where η
is the surface elevation, and η x, ηy are the slope components for upwind and crosswind direction, respectively.
From linear wave theory, the surface height, slope, and
curvature should be Gaussian according to the central
limit theorem of probability. The slope density function
is
(
)
p ηx , ηy =
 1  η 2 η y2  
1
exp −  2x + 2  
2πs x s y
sy  
 2  s x

(2 )
where sx2 and s y2 are the components of the mean square
slope in the upwind and crosswind directions, respectively. For an isotropic rough surface of Gaussian distribution, Eqs. (1) and (2) give us
σ 0 (θ ) =
R(0)
s
2
2
 tan 2 θ 
sec 4 θ exp − 2 
s 

(3)
where s2 = sx2 + s y2 is the total mean square slope of the
sea surface. For the altimeter, the incidence angle θ = 0,
Eq. (3) is simplified to
σ0 =
R(0)
s2
2
.
( 4)
Theoretical computations (Skirta et al., 1993) indicate that
the Fresnel reflection coefficient would decrease by only
about 1.5% between nadir and 14° off-nadir in the frequency interval of 10 to 36 GHz. We will assume it to be
constant. Therefore, σ0 will be completely determined by
the mean square slope of the sea surface.
Many efforts have been undertaken to investigate the
MSS by measurements (Cox and Munk, 1954a, b; Hughes
et al., 1977; Tang and Shemdin, 1983; Hwang and
Shemdin, 1988; Apel, 1994; Hwang et al., 1998; Liu et
al., 2000). Shaw and Churnside (1997) published recent
measurements of the sea surface slope statistics utilizing
scanning laser glint measurements; they also reviewed
other optical experiments. However, the classic sun glitter photographs of Cox and Munk (1954a, 1954b, 1956)
remain the most complete measurements of the slope
probability density function as a function of wind speed.
Their three papers describe the results of their analysis
of airborne sun glitter photographs, and are hereafter referred to simply as CXMK.
CXMK found that the azimuthally averaged MSS
was approximately linearly related to U12.5, the wind speed
(m/s) they measured at a height of 12.5 m
s 2 = 0.003 + 0.00512U12.5 .
(5)
Wu (1990) reviewed and analyzed a variety of data and
concluded that the CXMK data were better represented
by two logarithmic curves,
0.90 + 1.20 ln U10
s2 = 
−0.84 + 6.0 ln U10
U10 < 7 m / s
U10 > 7 m / s
(6 )
where U10 is the wind speed at 10-m height.
MSS has also been determined by airborne microwave measurements (Masko et al., 1986; Jackson et al.,
1992; Walsh et al., 1998a, b). Taking into account the
offset due to the fact that the sea surface appears smoother
at microwave wavelengths than at optical wavelengths,
the microwave MSS is comparable to the optical MSS.
The same authors also confirmed the work of CXMK,
stating that the MSS is a function of wind speed. Walsh
et al. (1998a) used the backscattered power falloff to compute MSS with an incidence angle measured by an airborne 36-GHz scanning radar altimeter. They indicated
that their data fit best to
s 2 = 0.00417U10 .
(7)
The MSS has also been derived from the ocean surface spectra (Donelan and Pierson, 1987; Wu, 1992; Apel,
1994; Hwang et al., 1998; Liu et al., 2000), which is defined as (Phillips, 1977)
r
r
s 2 = ∫ F k k 2 dk
()
(8)
r
where F( k ) is the
r directional wave number spectrum of
wind wave and k is the wave number vector. Based on
Phillips’ equilibrium spectrum and their model of the gravity-capillary wave spectrum (Liu and Yan, 1995), Liu et
al. (2000) calculated MSS and they found
s 2 = 0.0103 + 0.0092 ln U10

kd2
2.1 
+0.000012U10
ln
4 + 1
2
.
5
×
10


( 9)
where kd is called the cutoff wave number because only
the ocean waves of wavelengths longer than the radar
wavelength contribute to the measured MSS. This can be
determined by the radar frequency. For example, the values of kd are 730, 314, and 209 m –1 for Ka, Ku, and X
bands, respectively.
From the above analysis we can see that all of the
previous studies indicated that MSS was merely a function of wind speed, and was independent of wave states.
If we want to consider the influence of wave states in the
altimeter wind speed algorithms, the key is to obtain MSS
which should include the parameters describing the development degree of ocean waves, such as wave age. The
next section calculates the MSS by integrating the wave
spectrum. The result is compared to the optical observation data.
3. Determination of Mean Square Slope
In order to calculate MSS, we must first select a wave
spectrum. Although there is controversy regarding the
proper spectral form, it is appropriate for this study that
the wind-wave spectrum in the equilibrium range, except
the gravity-capillary range, varies as k–3.5 or ω–4 (Toba,
1973; Forristall, 1981; Kahma, 1981; Phillips, 1985;
Donelan et al., 1985; Banner et al., 1989), where k is the
wave number and ω is the angular frequency. It is also
clear that a k–3.5 dependence cannot be extended to arbitrarily large k, since the maximum wave steepness is limited (Plant, 1982; Jackson et al., 1992). According to the
studies of Kitaigorodskii (1983), Jähne and Riemer
(1990), and Jackson et al. (1992), we can assume that the
wave number spectra vary with k –4 at higher wave numbers. Taking k1 as the wave number that separates the gravity waves from the gravity-capillary waves, the wave
number spectrum can be expressed as
 1 −1 / 2 u k −7 / 2 f θ
( ) kp < k < k1
r  αg
∗
Ψ k =  21
 αg∗−1 / 2 u∗ k11 / 2 k −4 f (θ ) k1 < k < kd
2
()
(10)
where α is Toba’s constant (after Phillips, 1985) with a
value between (6~12) × 10–2, g∗ = g + γsk 2 and γ s = Γ/ ρw,
g is the acceleration of gravity, Γ is surface tension, and
ρw is water density. f(θ ) is the angular spreading funcπ
tion, which satisfies ∫−π f (θ )dθ = 1, and kp is the peak
wave number of wind wave. kd is the cutoff wave number
A Spectral Approach for Determining Altimeter Wind Speed Model Functions
237
at high wave numbers, introduced here for the convenience of discussion.
The first part of the spectrum in Eq. (10) is cited
directly from the results of Toba (1973) and Phillips
(1985), and is believed to have been confirmed by many
theoretical and laboratory or field studies. The second part
is obtained by the continuity at k1 after the fluctuation of
the spectrum with k–4 is given. Instead of gravitational
acceleration g, g∗ is used in the gravity-capillary spectrum, following Toba (1973) and Mitsuyasu and Honda
(1974), in order to consider the influences of water surface tension at high wave numbers.
According to Eq. (8), MSS can be calculated as follows
(
s 2 = αg −1 / 2 u∗ k11 / 2 − kp1/ 2
)
kd
1
1
+ αu∗ k11 / 2γ s−1 / 2 ∫
dk
k1
2
2
k a + k2
where a =
g / γ s . Using the integral formula
dx
∫
(11)
x a2 + x 2
=−
1 a + a2 + x 2
ln
+ constant,
a
x
spectrum, because the fourth moment would become infinite in general. Therefore, our result is more reasonable
in physical sense.
Parameters α, k1 in Eqs. (12) and (13) are controversial. A detailed discussion of these issues is beyond the
scope of this article. Phillips (1985) pointed out that the
field observations of α give values between 0.06 and 0.12;
we simply adopt the value of α as 0.08 in this study. α =
0.096 is the value of the self-similarity ω –4 equilibrium
range spectral form, which is consistent with the 3/2power law (Toba, 1972) with a factor B = 0.062 (Joseph
et al., 1981). In fact, different values of α do not substantially change the properties of MSS in this study; they
just change the overall level of MSS. For k1, we take a
value of 9kp following Kitaigorodskii (1983), Donelan and
Pierson (1987), Jackson et al. (1992) and Hwang et al.
(1996). Analysis shows that MSS would by 7% as k1 fluctuates by 10%.
The field observations of MSS are usually related to
the local wind speed U10 at 10 m above sea level instead
of wave parameters. For comparison with observational
data, Eq. (12) is transformed by rewriting the peak wave
number as a function of wind speed and wave age. The
wave age, β, which represents the sea maturity or the
present wave status under wind action, is commonly defined as
s 2 = αg −1 / 2 u∗
[(k
1/ 2
1
)
− kp1/ 2 + 0.5k11 / 2
 a + a2 + k 2
a + a 2 + kd2
1
⋅ ln
− ln

k1
kd

(14)
β = cp / U10
MSS can be expressed as

 .


(12)
where cp = g/ ωp is the phase speed of waves corresponding to the peak frequency. Generally, the wave age varies
between 0 and somewhat greater than 1 according to the
wave status of wind waves. For a fully developed wind
sea and in the cases of swells, β will be greater than 1.
With the dispersion relation ω 2 = gk, the peak wave
number can be written as a function of wave age as
kp = g / ( βU10 ) .
(15)
2
We also note that, by introducing g∗, the mean square
slope still converges when the cutoff wave number becomes infinite

a + a 2 + k12
s 2 = αg −1 / 2 u∗  k11 / 2 − kp1/ 2 + 0.5k11 / 2 ln
k1


(
)

. (13)


Equation (13) represents the total MSS that includes
all slopes generated by wind waves and capillary waves.
It is worthwhile to note that MSS is the fourth moment of
the wave spectrum and cannot be theoretically determined
using the wave spectra that have been proposed. The cutoff wave number had to be introduced into the studies
related to calculation of the fourth moment of wind-wave
238
D. Zhao and Y. Toba
Substituting Eq. (15) and k1 = 9kp into Eq. (12), MSS can
finally be expressed as
s 2 = αCD1/ 2 β −1



 
(16)

 a + a 2 + 81g 2 / βU 4
( 10 )
a + a 2 + kd2
3
⋅2 +  ln
ln
−
2
kd
2
9g / ( βU10 )


where CD = u∗2/U 102 is the drag coefficient of the sea surface, which is used to convert u∗ to U10. Here we follow
β = 0.4
S2
Cox and Munk (1954)
Tang and Shemdin (1983)
Hwang and Shemdin (1988)
S2
1.0
2.0
15 m/s
Cox and Munk (1954)
Tang and Shemdin (1983)
Hwang and Shemdin (1988)
Hughes et al. (1977)
3 m/s
U = 1 m/s
U (m/s)
β
Fig. 2. Mean square slope as a function of wind speed (solid
lines) with wave age as a parameter, together with field
optical measurements. The family of solid curves from top
to bottom corresponds to the wave ages from 0.4 to 2.0 in
steps of 0.2, calculated by Eq. (16).
Fig. 3. Mean square slope as a function of wave age (solid
lines) with wind speed as a parameter, together with field
optical measurements. The family of solid curves from bottom to top corresponds to wind speed from 1 m/s to 15 m/s
in 2 m/s steps, calculated by Eq. (16).
the relationship given by (Wu, 1980, 1988):
[
(
)
]
 1 / κ ln C1/ 2U z / ν + 5.5
(
)
D
10
CD = 
−3
(0.8 + 0.065U10 ) × 10
−2
U10 < 2.4 m / s
U10 > 2.4 m / s
(17)
where z = 10 m is the standard anemometer height, ν is
the kinematic viscosity of air, and κ = 0.4 is the von
Kármán constant.
Equation (16) shows clearly that slopes associated
with the gravity-capillary range (second part) are much
more sensitive to the local wind than the slopes associated with the gravity-wave range (first part). This is consistent with the common concepts of radar remote sensing, in which contributions of the gravity-wave range to
roughness elements on the ocean surface are usually neglected, and only the contributions of short-scale slopes
generated by the gravity-capillary range are considered
as a function of local wind. It is also obvious from Eq.
(16) that MSS depends on both the wind speed and wave
age. The dependence on wave age reflects the fact that
the presence of surface gravity waves will modify the
physical roughness and the boundary conditions on the
water surface.
Because of the difficulties in resolving the mean
square slope by theoretical development, direct wave
measurements in the field and laboratory remain one of
the most important resources for understanding the characteristics of short waves. However, due to the difficulties involved in making in situ measurements in the ocean,
only a handful of field programs have been reported over
a span of 40 years (CXMK; Hughes et al., 1977; Tang
and Shemdin, 1983; Hwang and Shemdin, 1988). When
we compare our results with actual MSS observations,
we should bear in mind that not all the slopes can be detected by the sensors used. For the optical measurements
mentioned above, in which the detectable wave number
of short gravity-capillary waves exceeds that of microwave radar sensors, kd can be set to 1250 m–1 (Liu et al.,
1997). The results of MSS varying with wind speed and
wave age are depicted in Figs. 2 and 3, along with the
optical measurements mentioned above.
Figure 2 shows the MSS calculated by Eq. (16) as a
function of wind speed with the wave age as a parameter,
together with the optical measurements in the field by
CXMK, Hughes et al. (1977), Tang and Shemdin (1983)
and Hwang and Shemdin (1988). The family of curves
from the upper to lower corresponds to wave ages from
0.4 to 2.0 in steps of 0.2. We should not expect any one
of these lines in Fig. 2 to agree with all observations, since
the observed values are distributed over a large span of
wave states. The curves in the figure indicate that the in-
A Spectral Approach for Determining Altimeter Wind Speed Model Functions
239
β = 0.45
Beta = 0.4 - 0.5
Beta = 0.5 - 0.6
Beta = 0.6 - 0.7
Beta = 0.7 - 0.8
Beta = 0.8 - 0.9
Beta = 0.9 - 1.0
Beta = 1.0 - 2.0
Beta > 2.0
Brown et al. (1981)
Goldhirsh and Dobson (1985)
Chelton and McCabe (1985)
Witter and Chelton (1991)
0.55
0.65
S2
0.95
σ0 (dB)
0.75
0.85
1.45
β = 2.0
3.35
1.0
0.6
U (m/s)
U (m/s)
Fig. 4. MSS calculated by Eq. (16) versus wind speed and wave
age (solid lines) together with the measurements of Cox
and Munk (1954a). The measured data are divided into eight
subgroups according to wave age. The family of solid curves
from top to bottom corresponds to average wave age of each
subgroup (0.45, 0.55, 0.65, 0.75, 0.85, 0.95, 1.45, and 3.35).
Fig. 5. Comparison of the proposed analytical algorithm (Eq.
(18), thin solid lines) with four previously proposed empirical algorithms. The family of thin solid curves from
bottom to top corresponds to wave age from 0.6 to 2.0 in
steps of 0.2.
fluence of wave age is pronounced at higher wind speeds,
and declines rapidly with decreasing wind speeds. The
wave-age dependence is thus difficult to detect at low
wind speeds. Unfortunately, most of the experimental
studies are limited to low wind speeds; for example,
Hwang and Shemdin’s data were obtained at less than 7
m/s, because of operational difficulties. At a given wind
speed, a greater wave age produces a smaller MSS, which
means a smoother sea for an older wave state in the sense
of MSS. This situation is shown clearly in Fig. 3, in which
MSS has been drawn as a function of wave age. Although
all of the observational data suggest a decrease of MSS
with increasing wave age in general, our results are closer
to those of CXMK. It is worth pointing out that CXMK’s
classical measurements over a large span of wind speeds
and wave ages using the sun’s natural glint still constitute an extraordinarily useful data set which has been cited
by numerous studies even now (Walsh et al., 1998a, b).
Our formula is further compared with CXMK’s data in
Fig. 4. It can be seen that they are generally consistent
with one another.
σ0 =
4.
A New Analytical Algorithm Depending on Wind
Speed and Wave Age
Based on Eqs. (4) and (16), we can obtain a new algorithm for altimeter data immediately,
240
D. Zhao and Y. Toba
R(0)
α
2
βCD−1/ 2
4

2
2
2
2
3 a + a + kd
3 a + a + 81g / ( βU10 )

ln
⋅ 2 + ln
−
2

2
2
kd
9g / ( βU10 )

−1

 .


(18)
This is the relationship between the radar cross section
of the altimeter and parameters related to the ocean surface including the wind speed and wave age. In comparing this analytical algorithm with the empirical ones, we
confront the problem of how to determine the value of
|R(0)| 2, although its magnitude does not affect the substantial relations between σ 0 and U 10. In fact, we do not
have enough knowledge to determine the Fresnel reflection coefficient completely. A possible practical method
may be to fit the proposed analytical algorithm according to the sea truth data. Analysis in the next section shows
that, if the constant α is set to 0.08, the derived NRCS
from Eq. (18) is closest to the satellite NRCS on the condition that |R(0)| 2 = 0.3. |R(0)| 2 is thus taken as 0.3
(Jackson et al., 1992; Walsh et al., 1998a, b).
The altimeter returns consist of specular reflections
from small facets distributed over the illuminated area.
Only those facets with a radius of curvature exceeding
the radar wavelength can contribute to the altimeter returns. It may be supposed that the ocean waves in whose
wavelength shorter than the radar wavelength could not
contribute to specular returns (Brown, 1978). Since the
working frequency of altimeters is usually 13.5 GHz, we
set the cutoff wave number kd = 314 m–1.
Figure 5 shows our results (solid lines) along with
the empirical model functions of Brown (Brown et al.,
1981), SB (Goldhirsh and Dobson, 1985), CM (Chelton
and McCabe, 1985), and MCW (Witter and Chelton,
1991). The family of solid curves from bottom to top corresponds to wave ages from 0.6 to 2.0 in steps of 0.2. For
the same σ0, the analytical algorithm shows that larger
wave ages are associated with higher wind speeds. The
effect of wave age on the derivation of wind speeds becomes weaker as wind speeds decrease. This can partly
explain the fact that, although large discrepancies occur
at high wind speeds, the empirical algorithms give almost
the same results at lower wind speeds (Fig. 5).
5. Comparison with Actual Data
We propose an alternative approach where NRCS is
a function of both the wind speed and wave age. Another
advantage of this new algorithm is that it puts no limitation on the value of NRCS and wind speed. In principle,
it can be applied to any high wind speed we may encounter. For application of this algorithm, however, the wave
age required is not available directly from the altimeter
data, which contains information only on the significant
wave height (SWH) of ocean waves. Therefore, we must
find a method to convert wave age into SWH and some
other available parameters.
The Japan Meteorological Agency (JMA) has been
operating ocean data buoys near Japan since 1972. The
buoys are 10 m in diameter, weighing 48 tons. They measure eleven variables every 3 hours, and at 1 hour intervals when wind speed is greater than 16.3 m/s. The data
from the buoys are published every year as “Data from
Ocean Data Buoy Stations”. This publication gives detailed explanations about the buoys and stations.
This study uses three of the variables, viz., the wind
speed U10, the significant wave height of ocean waves H s
and wave period Ts, from three buoys denoted by B21002,
B21004 and B22001. The wind speed and direction are
measured by a three-cup anemometer and a wind vane,
which are installed on the top of the mast at 7.5 m above
the sea surface. The wind speed is converted to that at a
height of 10 m by multiplying by a factor of 1.03, which
is determined by assuming the logarithmic law of wind
profile for neutral stratification of the air-sea interface
with a value of the drag coefficient of 1.6 × 10–3 (Ebuchi
and Kawamura, 1994). The significant wave period Ts is
converted to the peak frequency of wind-wave spectra
Fig. 6. Wave age versus nondimensional wave height based on
buoy observational data. Equation (20) is plotted as solid
line.
ωp by using ωp = 2 π/(1.05Ts). The wave age can then be
estimated by
β = g / U10ω p .
(19)
Another parameter that can be used to describe the wave
state is the nondimensional wave height gHs/U 102. All of
the data available from 1997 to 2000 are plotted in Fig. 6
without any discrimination. It is obvious that these data
include various wave states, such as active wind waves,
swell and other complicated situations that occur in the
ocean under natural conditions. Nevertheless, the wave
age calculated from these various wave states still correlates very well with nondimensional wave height. With
the method of least squares, it is found that this relationship can be expressed as
 gH 
β = 3.31 2s 
 U10 
0.6
(20)
with a correlation coefficient greater than 0.9. Equation
(20) corresponds approximately to the 3/2-power law
(Toba, 1972; Ebuchi et al., 1992a). It is noted that Eq.
(20) is applicable to any wave state in natural conditions,
no matter whether it is a wind wave or not. In this way,
the wave age can be computed through the SWH data
obtained simultaneously from the altimeter. Equation (18)
can thus be regarded as a function of wind speed and SWH
instead of wave age. Because the wind-wave spectral form
is used in our derivation of this new algorithm, the maximum value of 1.4 for wave age is adopted in the following discussion. On deriving wind speeds from this new
A Spectral Approach for Determining Altimeter Wind Speed Model Functions
241
Fig. 7. Scatter plot comparisons of wind speeds derived from various algorithms and buoys. (a) New algorithm, (b) MCW,
(c) CM, (d) SB, (e) LF.
algorithm, the NRCS space is chosen for the minimization process, that is, the wind speed is obtained by changing wind speed until the calculated NRCS most closely
approaches to the satellite NRCS, given SWH.
A set of collocated JMA buoy and TOPEX/
POSEIDON (T/P) altimeter data has been obtained from
December 1992 to June 1994. T/P parameters were extracted when the satellite footprint was within a 0.6-degree box centered on the location of two JMA buoys denoted by B21004 and B22001 and when the time difference between the satellite pass and the buoy measurement was less than 1 hour.
We applied this new algorithm as well as the empirical CM, MCW, SB, LF (Lefevre et al., 1994) algorithms
to the buoy/altimeter data set. The values of Ku-band σ0
are adjusted by subtracting 0.7 dB from the value contained in the Geophysical Data Records (GDRs) to obtain wind speed (Ebuchi and Kawamura, 1994). It is found
that the new algorithm gives the best results when the
242
D. Zhao and Y. Toba
constant |R(0)|2 in Eq. (18) is taken as 0.3 on the condition α = 0.08. The results are shown in Fig. 7, and the
root mean square differences (RMSD) for each algorithm
are shown in Table 1.
It is clear that the new algorithm yields the best estimation of wind speed compared with other empirical algorithms as a whole. Although the SB algorithm gives
the lowest RMSD with buoy data among these algorithms,
the wind speeds derived are systematically overestimated
at low wind speeds, and underestimated at high wind
speeds (Ebuchi et al., 1992b). The same behavior is displayed by the LF algorithm with higher RMSD. By contrast, CM algorithm, with the highest RMSD, gives wind
speeds higher than those measured by buoys at high wind
speeds. The MCW algorithm, which is in tabular form,
operates well either at low wind speeds or at high wind
speeds, with higher RMSD than that of our new algorithm. It is noted that MCW algorithm cannot be validated for much higher wind speeds, because the highest
Table 1. Summary of RMSD for various algorithms.
Algorithm
CM
MCW
SB
LF
NEW
RMSD
2.86
1.62
1.56
1.84
1.59
wind speed on which this algorithm operates is 20 m/s.
By contrast, the proposed algorithm can in principle give
reliable results at any high wind speed we may encounter. It is shown that, with the consideration of NRCS and
SWH together, the accuracy when deriving wind speeds
by this new algorithm can be significantly improved.
6. Concluding Remarks
Calculating MSS through the integration of a widely
accepted wind-wave spectrum including the gravity-capillary range has allowed us to propose a new analytical
algorithm for deriving wind speeds from altimeter data
(Eq. (18)), which explicitly depends on wind speed and
wave age. Comparison with actual observational data
shows this new algorithm can give us more reliable wind
speeds than previous empirical algorithms. We conclude
that the information of wave age or significant wave
height could improve the ability for inferring wind speeds
from altimeter observations, especially at high wind
speeds.
Acknowledgements
One of the authors (D. Zhao) expresses his thanks to
Japan Society for the Promotion of Science for providing
him with the postdoctoral fellowship. Thanks are also
extended to Dr. G. Chen who provided the TOPEX/
POSEIDON altimeter data. This work was begun at Earth
Observation Research Center of NASDA, and finally finished at Ocean University of China under National Natural Science Fund of China (40276005, 40076003), Natural Science Fund of Shandong Province (Z2002E01) and
863 project (2001AA630307).
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