Acta Oceanol. Sin., 2014, Vol. 33, No. 5, P. 103–112
DOI: 10.1007/s13131-014-0476-8
http://www.hyxb.org.cn
E-mail: [email protected]
An improved method of absolute calibration to satellite
altimeter: A case study in the Yellow Sea, China
LIU Yalong1,2,3, TANG Junwu1,2*, ZHU Jianhua2, LIN Mingsen3, ZHAI Wanlin2, CHEN Chuntao2
1 Ocean
University of China, Qingdao 266100, China
Ocean Technological Center, State Oceanic Administration, Tianjin 300112, China
3 National Satellite Ocean Application Service, State Oceanic Administration, Beijing 100081, China
2 National
Received 16 April 2013; accepted 8 October 2013
©The Chinese Society of Oceanography and Springer-Verlag Berlin Heidelberg 2014
Abstract
An improved absolute calibration technology based on indirect measurements was developed through two
probative experiments, the performance of which was evaluated by applying the approach to in situ sea surface height (SSH) at the Tianheng Island (tidal gauge) and the satellite nadir (GPS buoy). Using Geoid/MSS
(mean sea surface) data, which accounted for a constant offset between nadir and onshore tidal gauge water
levels, and TMD (tidal model driver), which canceled out the time-varying offsets, nadir SSH (sea surface
height) could be indirectly acquired at an onshore tidal gauge instead of from direct offshore observation.
The approach extrapolated the onshore SSH out to the offshore nadir with an accuracy of (1.88±0.20) cm
and a standard deviation of 3.3 cm, which suggested that the approach presented was feasible in absolute
altimeter calibration/validation (Cal/Val), and the approach enormously facilitated the obtaining SSH from
the offshore nadir.
Key words: radar altimeter, absolute calibration, Yellow Sea
Citation: Liu Yalong, Tang Junwu, Zhu Jianhua, Lin Mingsen, Zhai Wanlin, Chen Chuntao. 2014. An improved method of absolute
calibration to satellite altimeter: A case study in the Yellow Sea, China. Acta Oceanologica Sinica, 33(5): 103–112, doi: 10.1007/s13131014-0476-8
1 Introduction
Global Changes, especially destructive climate and disastrous environmental events that have arisen in recent years,
have increasingly attracted people’s attention. As a sensitive
indicator of changes in the climate and environment, sea level
variability plays a crucial role in the global change study (Ablain
et al., 2009; Beckley et al., 2010; Lubin and Massom, 2005), and
reflects the global change quantitatively in a relatively short
timescale (Cazenave and Nerem, 2004; Griffies and Bryan, 1997;
Nerem et al., 1999). Consequently, the evolution of climate and
environment (e.g., global circulation, meso-scale eddies, El
Niño and La Niña phenomen) can be monitored through sea
level observation (Bonnefond et al., 2003b; Traon and Dibarboure, 2004). Space-Borne radar altimeters, compared with tidal
gauges and GPS buoys, have obtained sea surface height (SSH)
globally and continuously for decades, and have become a critical approach to the global change study by acquiring a sea level
anomaly (SLA) (Bonnefond et al., 2010; Cazenave and Nerem,
2004; Watson, 2005).
It is true that altimeters not only provide a new perspective
that had not happened before in oceanography, but also facilitate geophysics and geodesy researches (Deng et al., 2001;
Evensen and Van Leeuwen, 1994; Skagseth et al., 2004; Strub and
James, 2000). Space-borne altimetry is qualified as a standard
tool for oceanography (Chelton et al., 2001). Nevertheless, the
accuracy of altimetry restricts its application: for instance, geotropic current studies require centimeter-level precision, and
annual sea level variability needs 2 mm or better (Bonnefond
et al., 2003b; Chelton et al., 2001; Cheng et al., 2010; Evans et
al., 2005). Long-term sea level change observations, however,
necessitate the high consistency of multi-altimeters (Ablain et
al., 2010; Beckley et al., 2010).
Calibration for altimeters, is one of the most critical issues,
which promotes quality, enhances accuracy, and extends the
applications of altimetry data (Beckley et al., 2010; Bonnefond
et al., 2003b; Haines et al., 2010; Nerem et al., 2010). There has
been a variety of Cal/Val activities conducted to qualify the geophysical data record (GDR) data. The T/P (Topex/Poseidon) altimeter was calibrated by Christensen et al. (1994), Haines et al.
(2003), and Ménard et al. (1994) at harvest and Lampedusa calibration sites, respectively. With regard to Jason-1 and Jason-2 altimeters, calibration activities were conducted at the Harvest oil
platform (Haines et al., 2010; Haines et al., 2003), the Bass Strait
(Watson et al., 2003; Watson et al., 2011; Watson et al., 2004; Watson, 2005), and Corsica (Bonnefond et al., 2010; Bonnefond et
al., 2003a; Bonnefond et al., 2003b; Bonnefond et al., 2011). The
calibration activities mentioned above were achieved by elaborate technology at the dedicated calibration sites. Although it
is vital and necessary to calibrate altimeters at dedicated sites,
problems remain as described below. It is expensive to construct
a dedicated calibration site for altimetry, along with taking a
long time to select a site. Furthermore, the geographically correlated errors derived from orbit accuracy cannot be accounted
for at a few sites (Jayles et al., 2010; Labroue et al., 2004; Watson,
2005). Bonnefond et al. (2010) suggested that both dedicated
sites and tidal gauges should collaborate to characterize the er-
Foundation item: The Marine Public Welfare Projects of China under contract No. 201105032; the National High-Tech Project of China under contract No. 2008AA09A403.
*Corresponding author, E-mail: [email protected]
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rors/bias in an absolute sense.
More importance should be attached to calibrating altimetry data by tidal gauges due to the number of available, widely
distributed tidal gauges and convenient maintenance to devices compared with GPS buoys (Bonnefond et al., 2010). In addition, they provide continuous measurements for long-term
Cal/Val, which the direct GPS buoy solution cannot. However,
agreement between tidal gauges and GPS buoys is the key to
Cal/Val activities. Continued efforts have been made in improving precision. For example, Christensen et al.(1994) used GPSbased estimates to assess the Harvest tidal gauge system with a
uncertainty of 1.5 cm, while Watson (2005) reported a subcentimeter level.
There are challenges when calibrating altimeters using
tidal gauges including extrapolating water level from the tidal
benchmark to the nadir, and the knowledge of geoid undulation
(White et al., 1994). In this study, a solution with tidal gauge,
GPS buoy, well-developed tidal model data, and 1'×1' Geoid/
MSS data was presented to calibrate not only for China HY-2 in
a truly absolute sense, but also for Jason-2 and subsequent altimetry missions (e.g., Saral 2013, Sentinel-3 2014, Jason-3 2014,
Jason-CS 2017, SWOT 2020).
gauge, with a nominal accuracy of ±0.1% in full scale and a resolution of 1 mm, was deployed at the test field along the bank
(Fig. 1a). A buoy with a TOPCON NET-G3A GPS receiver (precision: 5 mm+0.5×10−6 m) equipped inside was deployed near the
bank (Fig. 1b). Water levels were observed by tidal gauge and
GPS buoy simultaneously. The location where the GPS buoy
was deployed was approximately 100 m away from the gauge.
Within this distance, the water levels difference caused by either Geoid/MSS or tide was negligible.
The second experiment was implemented at Tianheng Island where the tidal gauge was installed; the GPS buoy was
placed 15 km away from the tidal gauge (Fig. 1). The second
observation was designed to resolve the issue of extrapolating
water levels from the tidal gauge out to the nadir. The distance
between the tidal gauge and GPS buoy was considered to minimize the baseline length for GPS buoy processing and maximize
the distance from nadir to land (to circumvent the contamination from land) (Watson, 2005). In addition, two reference stations were built on the bank for GPS data processing, and the
data from the reference stations and GPS buoy were processed
by a number of routines in the GPS analysis suite “GAMIT”.
2.2 Method
In this study, a solution, which transfers water levels from
the tidal gauge to the comparison point (nadir), was presented
based on Geoid/MSS undulation data and tidal model harmonic data. The technique-derived water levels (by extrapolation at
the nadir) were regarded as indirect measurements (Bonnefond
et al., 2011). Examples include National Centre for Space Studies (CNES) calibration activities (Bonnefond et al., 2003a), the
United Kingdom project (Dong et al., 2002; Woodworth et al.,
2004), and the GAVDOS calibration project (Erricos and Stelios,
2004). Bonnefond et al., (2011) and Watson (2005) pointed out
that this indirect method provided continuous estimates of water levels at the nadir beyond the intensive calibration phase.
While the tidal differences between the two sites, for which the
water levels observed at the tidal gauge can be transferred to the
nadir, were predicted by standard tidal prediction procedures
(Watson et al., 2004; Watson, 2005).
Differences between tidal gauge measurements and GPS
2 Site configuration and calibration methodology
2.1 Test field and instruments
Two probative experiments were conducted by the NOTC
(National Ocean Technological Center) to verify the feasibility of extrapolation of the tidal level from the tidal benchmark
out to the nadir of the altimeter. There were path-breaking experiments nationally aimed for the altimetry calibration. One
was designed to demonstrate the consistency between a GPS
buoy and tidal gauge at the same place on 18 October 2012, and
another was intended to connect the water level measured by
the gauge and GPS buoy at two sites with a distance of approximately 15 km, from 24 October 2012 to 20 November 2012.
In the first experiment, the in situ water levels jointly measured by the tidal gauge and the GPS buoy at Shazikou lasting
for approximately 5 h were used to ensure the homogeneity between the time series (Fig. 1). The Valeport 740 automatic tidal
36°30'N
a
120°30'E
120°40′
120°50′
121°00′
121°10′ E
Tianheng Island
nadir
15 k
m
2
“HY
-2” P
ass-5
Shazikou
47
Qingdao
b
36°N
c
China
ass-1
36°10′
Huanghai Sea
-2” P
“HY
36°20′
N
Fig.1. Test locations and field instruments.
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LIU Yalong et al. Acta Oceanol. Sin., 2014, Vol. 33, No. 5, P. 103–112
buoy observations consisted of the discrepancy between the
tidal level and Geoid/MSS undulation at two separate sites
(about 15 km away). Therefore, water levels at the nadir can be
expressed as:
H na =H ga +'H ,
(1)
where Hga is the water level measured at the tidal gauge, and 'H
is the difference between two sites, which includes the Geoid/
MSS undulation and tidal level:
'H ='H Geoid/MSS +'H ti .
(2)
Then, inserting Eq. (2) into Eq. (1), Hna can be rewritten as:
H na =H ga +'H Geoid/MSS +'H ti .
(3)
Meanwhile, the Hna can also be acquired by the GPS buoy as:
H na =H GPS .
(4)
The aim of this study was to obtain Hna through Hg, 'HGeoid/MSS
and 'Hti. Compared with HGPS, the error, consequently, was determined by
H er =H ga +'H Geoid/MSS +'H ti H GPS .
(5)
With reference to 'HGeoid/MSS, the 1'×1' global Geoid/MSS data
were used by means of interpolation (Pavlis et al., 2008), and the
tidal difference, 'Hti, was calculated based on (1/30)º × (1/30)º
tidal model data, which can be expressed as:
'H ti =H ti L1,T H ti L2,T .
(6)
L1 and L2 denote different locations; T is time; and Hti is the
tidal level predicted by a set of procedures:
H ti L,T = ¦ f j H Lj cos V jT +V0 u K Lj ,
(7)
j
where HLj is the amplitude of tidal constituents at the tidal
gauge and nadir in this study, and j indicates different tidal constituents (e.g., M2, S2, K1, O1); KLj is the lag phase of tidal constituents; f is the nodal correction factor for the constituents; V0+u
represents the initial phase of tidal constituents with reference
to Greenwich; and ıj is the frequency of tidal constituents.
Then 'Hti is calculated through a standard harmonic tidal
analysis procedure, the OTIS (Oregon State University tidal inversion software) (http://www.coas.oregonstate.edu/research/
po/research/tide/index.html) (Egbert et al., 1994; Egbert and
Erofeeva, 2002). The solution consists of regional and global
tidal constituents with different spatial resolutions, which have
assimilated various altimetry data (Topex/Poseidon, Topex Tandem, ERS, and GFO) and in situ data (e.g., tidal gauges, shipborne ADCP).
3 Data
The data used in this study consisted of two categories, including field observation data (from the tidal gauge and GPS
buoy) and model data (geoid, MSS and Tidal model data). In the
first verification experiment (Shazikou, Fig. 1), the GPS buoy
was tethered approximately 100 m from the tidal gauge, and the
bias derived from this distance is detailed in Section 4.
In the second tidal level difference extrapolation experiment
(Tianheng Island and nadir; Fig. 1), field data were intended to
derive the SSH differences between the two sites, whereas the
model data were aimed to offset the differences.
With regard to the model data, two reference ellipsoids were
utilized, comprising the WGS84 ellipsoid and Topex/Poseidon
ellipsoid (Table 1). All model data referred to the WGS84 ellipsoid with the exception of the DTU MSS model data, which related to the Topex/Poseidon ellipsoid.
3.1 In situ data
Two groups of data were obtained from each experiment,
consisting of water levels independently measured by a tidal
gauge and GPS buoy, respectively. The tidal gauge typically
measured water level at 1min interval, while the kinematic GPS
was processed at 1 Hz with short baseline distance. GAMIT software was used for the GPS data processing. Limitations of the
GPS water level were depended on the location, surrounding
environment, and number of satellites (less than the minimum
of five prevented reliable and robust solution) (Watson, 2005).
Both high-rate GPS data and low-rate tidal gauge data
(logged every 1 min, with each value an average of 60, 1-second
samples) required filtering to eliminate waves, swell effects, and
noise (Bonnefond et al., 2003a; Ménard et al., 1994). Meanwhile,
all data were transformed to universal time.
Due to the sampling frequency discrepancy between the kinematic GPS and tidal gauge, the gauge data were resampled
identically with GPS 1 Hz data by an ordinary interpolation.
3.2 Model data
Water levels observed at Tianheng Island and the nadir, were
intended to calculate differences between the levels. Nevertheless, the model data were used to account for the differences.
The Geoid/MSS model data were used to compensate for the
constant difference. The disagreement between geoid and MSS
is that the latter takes into account the mean dynamic topography (MDT):
H mss =H Ge +H MDT ,
(8)
where Hmss is the mean sea surface height, HGe indicates geoid
Table 1. The parameters of two relevant ellipsoids
Ellipsoid
Equatorial Radius (a)/m
Polar radius (b)/m
Reciprocal Flattening (1/f)
Eccentricity(e)
T/P
6 378 136.3
6 356 751.600 563
298.257
0.081 819 221 456
WGS84
6 378 137.0
6 356 752.314 245
298.257 223 56
0.081 819 190 843
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LIU Yalong et al. Acta Oceanol. Sin., 2014, Vol. 33, No. 5, P. 103–112
undulation, and HMDT means mean dynamic topography. MSS
and geoid data were used to analyze the influence of MDT.
Ideally, MSS and geoid data should cancel out the constant
difference. Nevertheless, the tidal model data aimed to explain
the remainders, 'Hti.
3.2.1 Geoid data
The geoid undulation values referring to WGS 84 were calculated from the EGM2008 official earth gravitational model,
which was released by the US National Geospatial-Intelligence
Agency (NGA) EGM Development Team (NGA, 2008). This gravitational model is complete to a spherical harmonic with degree
and order 2159, and contains additional coefficients extending
to degree 2190 and order 2159 (Pavlis et al., 2008). At present,
the NGA provides two grids of precomputed geoid undulations:
one at 1 min × 1 min resolution and another at 2.5 min × 2.5 min
resolution. The former are used to determine the difference in
levels in this study.
The widely used EGM 2008 geoid undulation, according to
Pavlis et al. (2008), compared within dependent data, introduced a RMSE of 5.2 cm and a slope of 0.3 seconds, while the
previous version, EGM96 had a RMSE of 20.0 cm. The difference between EGM96 and EGM2008 was large in the Yellow Sea,
especially at the second experiment site. It was approximately
7 m for EGM2008, but more than 8 m for EGM96 (Fig. 2). Although both are used worldwide as geoid models, the EGM2008
exceeded EGM96 in precision and resolution.
3.2.3 Tidal model data
In order to extrapolate the water levels from the tidal gauge
to the nadir GPS buoy, the geoid and MSS data were expected to
account for a constant difference, whereas the tidal model data
were used to account for the time-varying periodic difference,
'Hti (Eq. (6)).
The TMD (tidal model driver) is the current version of a
global model of ocean tides, which best fits, in a least-squares
sense, the Laplace tidal equations and along-track averaged
data from TOPEX/Poseidon and Jason (on TOPEX/POSEIDON
tracks since 2002) obtained with Oregon State University Tidal
3.4 – 4.74
4.74 – 5.96
5.96 – 7.15
7.15 – 8.28
8.28 – 9.41
>9.41
< −1.14
−1.14 – −0.96
−0.96 – −0.78
−0.78 – −0.61
−0.61 – −0.41
> −0.41
Tianheng Island
nadir
Difference/m
b
EGM2008/m
a
3.2.2 MSS data
MSS model data, derived from The Danish Technical University’s DTU10 MSS, were developed through the averaging of
satellite altimetry (Andersen, 2010), and improved based on the
DNSC08MSS global mean sea surface (Andersen and Knudsen,
2008). Compared with its previous version, DNSC08MSS, this
MSS data extended altimetry data from 12 a (1993–2004) to 17
a (1993–2009). In addition, the altimetry data (with which the
MSS was derived) were refined in orbit, wet troposphere, ocean
tide, and sea state bias (Andersen, 2010).
The difference between MSS data and EGM2008 was the
mean dynamic topography according to Eq. (8), which gave rise
to tens of centimeters difference as a result (left panel of Fig. 3).
The right panel of Fig. 3 presents the MSS distribution similar to
the geoid undulation (Fig. 2)
Shazikou
Tianheng Island
nadir
Shazikou
Fig.2. Geoid undulation of EGM2008 (a) and difference between EGM2008 and EGM96 geoids (b).
Tianheng Island
nadir
MDT/m
<0.53
0.53 – 0.56
0.56 – 0.59
0.59 – 0.62
0.62 – 0.66
0.66 – 0.70
0.70 – 0.74
>0.74
Shazikou
<5.22
5.22 – 6.35
6.35 – 7.42
7.42 – 8.46
8.46 – 9.59
>9.60
MSS/s
b
a
Tianheng Island
nadir
Shazikou
Fig.3. Time-averaged mean dynamic topography and mean sea surface with reference to ellipsoid (T/P).
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LIU Yalong et al. Acta Oceanol. Sin., 2014, Vol. 33, No. 5, P. 103–112
Inversion Software (OTIS). The methods used to compute the
model are described in detail by Egbert et al. (1994) and further
by Egbert and Erofeeva (2002).
The model provided in the China’s sea as complex amplitudes comprised eight major harmonic constituents (M2, S2, N2,
K2, K1, O1, P1, Q1), on a 901 × 1 201, (1/30)° resolution grid (for
versions 6.* and later), which has been improved continuously.
The latest version of each model is of better quality compared
with the earlier versions since: (1) it assimilates a longer satellite
time series; (2) more data sites are included in assimilation; and
(3) bathymetry has improved from version to version. According to VOLKOV/OCE/ORST (http://volkov.oce.orst.edu/tides/
global.html), 55 coastal tidal gauges and six “true shallow” tidal
gauges were used to validate (but not assimilate) the model.
Tidal levels in the study region could then be predicted based
on the TMD model mentioned above, which were referenced to
the mean sea level. In order to avoid introducing unnecessary
errors in datum transformation, it was preferable to calculate
tidal difference between two sites instead of acquiring absolute
tidal levels at each site.
(Fig. 1; 36º5ƍ48.48ƎN, 120º32ƍ1.32ƎE) were analyzed to check the
agreement of SSH derived from different instruments at the
same site. Second, the offsets from Geoid/MSS and time-varying tidal difference were calculated to cancel out Hss,d at the two
sites (Fig. 2; approximately 15 km apart) and to demonstrate the
accuracy of the extrapolation.
In short, the objective of the first field experiment was to
ascertain whether the observations from the two instruments
were identical. Based on the first experiment, the second one
demonstrated agreement of SSH observation from two the instruments with a relatively large distance.
During the second field experiment, there was one point
available from the “Jason-2” observation, however, the absolute
bias of “Jason-2” could not be determined statistically due to
the small sample size.
4.1 In situ SSH mutual verification between gauge and buoy
As the SSH was measured by two instruments, the agreement between the two observations needed to be ascertained.
Although the first experiment at Shazikou lasted approximately
5 hours, 154 min was taken to acquire concurrent SSH (Fig. 4).
The 1 Hz raw SSHs measured by the GPS buoy were smoothed
to filter out surface waves, swell, and high-frequency noise. The
lag length of 3 min was adopted in smoothing (Fig. 4a, green
line), which was longer than waves and swell periods, but was
also short enough to avoid over-smoothing (a change in the
tendency of raw data).
For SSH, the two ways agreed well one another except approximately 11 min of data since 15:50 that showed a relatively
large difference, corresponding to degraded GPS satellite coverage. Figure 4a presents the SSH from the two instruments, and
Fig. 4b is the Hss,d time series in which the green lines denote the
mean value and dash lines denote the 95% confidence interval
of Hss,d. The gray patches from Figs 4a and b are SSH with a large
discrepancy. The degraded Hss,d showed a mean value of 4.72
cm, in light of the short distance between the gauge and buoy
(approximately 100 m), however, a difference this large was unacceptable. After the degraded data were removed, the Hss,d presented a mean value of 1.07 cm and standard deviation of 1.19
cm, which was comparable with the result from Watson (2005).
4 Results
The determination of absolute bias of altimeters requires
accurate in situ SSH (nadir). As described above, a long-term
and continuous observation of water levels far away from the
continent (to avoid waveform contamination from the land)
poses a challenge to the GPS buoys, which require calm weather condition and sea state, and guarding. Extrapolating the SSH
from the onshore tidal gauge out to the offshore nadir provides
a feasible solution. However, this solution entails high precision
from both measurements. In order to verify the feasibility and
accuracy of the solution, the differences (SSH difference, hereinafter referred to as Hss,d) between the two instruments (same
instruments for both field experiments) should be estimated at
the same site (for consistency) and then at different sites (for
extrapolation):
H ss,d =H ss,gb H ss,tg .
(9)
First, the Hss,d between the two measurements at Shazikou
8. 0
0.08
GPS buoy: 1 Hz raw SSH
tidal gauge: 1 min averaged SSH
GPS buoy: filtered SSH
7. 5
0.06
b
SSH difference/m
a
7. 0
SSH/m
Hss, g−Hss, b
mean value
95% confidence interval
6. 5
6. 0
0.04
0.02
0
−0.02
5. 5
−0.04
5. 0
14:24
14:52
15:21
Time
15:50
16:19
16:48
14:24
14:52
Fig.4. SSH from tidal gauge and buoy (a) and Hss,d (b).
15:21
Time
15:50
16:19
16:48
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LIU Yalong et al. Acta Oceanol. Sin., 2014, Vol. 33, No. 5, P. 103–112
Table 2. Statistics of SSH difference (m) between gauge and
buoy
Statistics
Dataset1)
Dataset2)
Dataset3)
Mean
Std
−0.010 7
0.047 2
−0.006 5
0.011 9
0.013 4
0.019 1
Upper bound
0.014 3
0.081 1
0.050 8
Lower bound
−0.034 2
0.026 5
−0.033 8
1)
Notes: SSH difference without the area covered by the gray
patch (Fig. 4); 2) gray patch covered data; 3) entire difference data.
4.2 Extrapolation of water level from bank to nadir
There were many factors contributing to the Hss,d that derived from onshore SSH minus offshore (nadir) SSH, but two
were dominant in determine the difference: Geoid/MSS difference (constant) and tidal offsets (time-varying). The Hss,d was
then canceled out by the two factors based on geoid undulation/MSS data and TMD data, respectively.
GPS buoy 1 Hz raw SSH from the synchronization experiment (Tianheng Island and nadir), was smoothed based on robust local regression using weighted linear least squares and a
second degree polynomial model (Cohen, 1999) to average out
surface waves, swell, and random noise with a length of 3 min.
10. 5
Nevertheless, it was not necessary to apply smooth technology
to the tidal gauge 1-min SSH due to the fact that the SSH records
were averaged values from the integrated water level at 1-min
intervals.
All data sets (in situ and model data) referred to the WGS84
ellipsoid, except MSS data that referred to the Topex/Poseidon
ellipsoid (Table 1). The impact from datum transfer was also
analyzed later.
4.2.1 Difference accounted for by constant offset
Geoid undulation or MSS data were not equivalent between
Tianheng Island (tidal gauge) and the nadir (GPS buoy), which
gave rise to differences of SSH (Fig. 2): geoid undulation did not
take mean MDT (dynamic topography) into account, and MSS
data included both geoid undulation and MDT. The influence of
MDT in extrapolating water level out to the offshore site could
be evaluated by comparing the difference between the two
model data with constant offset.
The mean SSH difference was regarded as a constant offset,
which was not equivalent in the whole Hss,d time series, and
presented a larger difference between peak and trough than
in other positions (Fig. 5). The mean Hss,d had a mean value of
0.425 2 m, which was theoretically supposed to be aligned to
zero (Fig. 6).
GPS buoy: 1 Hz raw SSH
GPS buoy: filtered SSH
tidal gauge: 1 min averaged SSH
10. 0
9. 5
SSH/m
9. 0
8. 5
8. 0
7. 5
7. 0
09:36
12:00
14:24
16:48
Time
19:12
21:36
Fig.5. SSH observed by the GPS buoy and tidal gauge.
0.6
Hss, b−Hss, g
Hss, b−Hss, g+ΔHg
Hss, b−Hss, g+ΔHm
SSH difference/m
0.5
0.4
0.3
0.2
0.425 2 m
0.090 2 m
0.1
0
−0.1
−0.2
09:36
0.018 8 m
12:00
14:24
Time
16:48
19:12
Fig.6. SSH difference and geoid/MSS offsets.
21:36
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LIU Yalong et al. Acta Oceanol. Sin., 2014, Vol. 33, No. 5, P. 103–112
The mean SSHDiff (mean difference between Hs,b and Hss,g)
was 42.52 cm (Fig. 6). After shifting a mean value of 33.5 cm (the
difference of the geoid between the Tianheng Island and the
nadir; green dashed line in Fig. 6), there was still 9.02 cm to be
canceled out, resulting in 21.21% a relative deviation. Regarding MSS data, after compensating for the mean value of 44.4 cm
(difference of MSS between the two sites), there was an overestimate of 1.88 cm (red dashed line in Fig. 6), reaching a relative
deviation of 4.42%. Compared with the geoid model, the MSS
offset was more effective, and the MSS had higher performance
due to consideration of the MDT.
4.2.2 Difference accounted for by periodic time-varying offsets
After removing the constant offset, the remaining time series
fluctuated with time, which were determined by tides (Figs 7
and 8). Namely, the differences between tides at the two sites led
to time varying offsets (Fig. 7, green patch). The major tidal constituents, which were used to estimate the tidal level as well as
time-varying Hss,d, could be extracted by applying the standard
harmonic tidal analysis to the water level data, provided that
there were long-term observed water level time series available.
However, with tens of hours' time series length, the eight major
tidal constituents could not be identified. Therefore, the TMD
solution was preferable to solve the periodic offsets (Eqs (6) and
(7)). The eight major tidal constituents were extracted from the
TMD data to calculate tidal offsets (Table 3). Compared with
the onshore tidal level, the nadir tidal level presented an approximate 2° phase lag in the M2 dominant constituents, which
showed an amplitude of 0.069 6 m (Table 3). Except for M2, S2
and N2 constituents, the remaining constituents had a subtle
influence at the two sites, since the amplitudes of constituents
were at the millimeter level or smaller.
Based on the eight major tidal constituents (amplitude and
phase lag), Hss,d could be offset by predicting water levels using
the TMD solution (Fig. 8). Afterwards, the differences between
SSHDiff and the predicted tidal differences could be determined,
shown as the blue line in Fig. 8. Note that before the differences
were obtained, the mean Hss,d should be aligned, to 0 which the
predicted water levels were referred.
The results suggested that the predicted data agreed with
Hss,d except local parts (Fig. 8). The predicted data fits well the
shape feature of fluctuating Hss,d time series, as expected. The
time−varying differences
tidal gauge: plus canstant offset (0.425 2 m)
GPS buoy: filtered SSH
tidal gauge: 1 min averaged SSH
10.0
9.5
SSH/m
9.0
8.5
8.0
7.5
7.0
09:36
12:00
14:24
Time
16:48
19:12
21:36
Fig.7. Theoretical time varying differences.
Hss,d
predicted tidal level
residual
0.15
0.10
Hss,d/m
0.05
0
−0.05
−0.10
−0.15
09:36
12:00
14:24
16:48
Time
19:12
21:36
Fig.8. Hss,d and predicted tidal level.
Table 3. Major tidal amplitude and phase lag of Hss,d (Hgps−Hg)
Constituent
M2
S2
N2
K2
K1
O1
P1
Q1
Amplitude/m
−0.069 6
−0.021 8
−0.012 1
−0.004 8
−0.005 9
−0.002 7
−0.001 1
−0.000 3
Phase lag/(º)
−1.889 9
−2.496 9
−1.866 1
−2.476 4
−0.014 7
−0.710 5
−0.930 9
−0.709 7
110
LIU Yalong et al. Acta Oceanol. Sin., 2014, Vol. 33, No. 5, P. 103–112
residual revealed normally distributed shape indicating that
there was no higher order nonlinear behavior in Hss,d (Watson,
2005) (Fig. 9). Meanwhile, the residual had a standard deviation
of 3.33 cm, while the Hss,d and the predicted water levels present standard deviations of 9.74 and 8.33 cm, respectively, demonstrating that the TMD solution provided accurate offsets to
explain the time-varying Hss,d.
1 500
σ =3.33 cm
Frequency
1 000
500
0
−0.08 −0.06 −0.04 −0.02
0
0.02 0.04
Residual/m
0.06
0.08
0.10
Fig.9. Histogram of residual and normal distribution
probability density function.
5 Discussion
According to the results, the SSH at the tidal gauge could
be extrapolated out to the nadir by taking the constant and
time-dependent differences into account. However, other factors were omitted, for instance, tidal model impact (TMD solution), currents, wind setup, atmospheric pressure, and ellipsoid
transformation (Crétaux et al., 2011; Watson, 2005). Concerning
tides, errors rose from imperfect representation by the model of
real ocean tides, due to inadequate knowledge of bathymetry,
inaccurate lateral boundary, and imprecise parameterizations
(e.g. bed friction). The bathymetry especially played an important role in tidal propagation (Benveniste and Vignudelli, 2009;
Lyard et al., 2006). For example, the M4 constituent influenced
by shallow water (Andersen, 2011), reached a significant am-
plitude (Rosmorduc et al., 2011). The research concluded that
this constituent had a significant amplitude about 5 to 10 cm in
some parts of the Atlantic Ocean, but a noise level in the deep
ocean (Lyard et al., 2006). In this study, the tidal difference was
calculated without including any long periods of tidal and the
shallow-water constituents, which may have led to a millimeter-level or larger error (although the short distance between
the two sites, the M4 constituent, for example, was sensitive to
the shallow water).
The currents also affected the Hss,d. Watson (2005) suggested
that the speed of along-shore currents, is typically less than
10 cm/s, inducing an approximate maximum difference of 6.5
mm with an offshore distance of 15 km (Fig. 10a; 'h is currentinduced bias, f is the Coriolis parameter, u is current speed, g
is gravity acceleration, and 'y is offshore distance). Concerning
the ellipsoid conversion, the transformation between WGS84
and T/P resulted bias is less than the millimeter-level due to
the subtle difference between latitudes of the two sites (Figs 1
and 10b). Meteorological factors (wind setup and atmospheric
pressure), had a negligible impact to Hss,d due to the close geographic location (Crétaux et al., 2011).
The extrap.olated SSH at the nadir was acquired by adding constant and time-dependent offsets (hereinafter referred
to as Hss,re). The ordinary least square regression analysis was
applied to Hss,b and Hss,re, which showed scale parameter, a ,
of 0.995 9±0.000 3 and intercept parameter, b, of (3.4±0.26) cm
(Fig. 11). The coefficient a appeared significantly close to one
(the theoretical slope of regression), while the intercept seemed
relatively large, most likely caused by incompletely filtered
noise. Moreover, the incompletely filtered noise induced a
ragged feature in either the filtered GPS buoy data or tidal gauge
data (Fig. 12), which theoretically vary smoothly. According to
the constant and time-varying offsets, the tidal gauge SSH was
extrapolated out to the nadir with a remaining offset of 1.88 cm
and standard deviation of 3.3 cm. It can be concluded that the
accuracy of this extrapolation was (1.88±0.20) cm (0.20 cm denotes the standard error related to sample number 831/3). Figure12 shows that the extrapolated the SSH (yellow line) dovetailed with Hss,b, which suggested that the extrapolation method
detailed in this study is feasible in acquiring the nadir SSH.
6 Conclusions
The absolute in situ calibration and validation (Cal/Val)
0.715
1.4
a
WGS84
T/P
WGS84−T/P
b
1.2
Earth Radius/106 m
∆h/cm
0.8
0.6
0.4
0.710
6.37
0.705
6.36
0.2
0.700
0
0
5
10
Speed /cm∙s−1
15
20
80°S
60°S
40°S
20°S
0°
Fig.10. Influence from currents and ellipsoid transformation.
20°N 40°N 60°N 80°N
Difference/m
∆h=−∆yfu/g
1.0
LIU Yalong et al. Acta Oceanol. Sin., 2014, Vol. 33, No. 5, P. 103–112
10.5
10.0
Hss,g/m
9.5
Hss,g=a×Hss,tc± b
a=0.995 9±0.000 3
b=0.034±0.002 6
9.0
SSH
fitted line
95% confidence interval
8.5
111
effects and noise. Second, long-term concurrent time series
(water levels from both the tidal gauge and GPS buoy) should be
observed to preclude necessity of determination of the marine
geoid and to simplify the solution involving just the geometrical
framework. Third, filtering algorithms entail more focus. In addition, contributions including wind setup, along/cross shore
currents, loading tides, solid tides, and other factors should be
taken into account.
8.0
7.5
7.0
7.5
8.0
8.5
Hss,tc/m
9.0
9.5
10.0
10.5
Fig.11. Regression analysis (Hss,b versus Hss,tc).
10.5
10.0
References
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tidal gauge: 1 min averaged SSH
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9.5
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8.5
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7.5
7.0
09:36
12:00
14:24
16:48
Time
19:12
Acknowledgements
The authors would like to thank all colleagues from the Remote Sensing Group the National Ocean Technological Center,
the State Oceanic Administrator, for their hard work in the two
experiments and proposals in data processing.
21:36
Fig.12. Hss,b versus Hss,tc.
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