CHAPTER 2 MARINE GRAVITY ANOMALIES FROM SATELLITE
2.3 Methods of Marine Gravity Anomalies from Altimetry
2.3.3 Method 3: Fourier Transform with Deflection of Vertical
In this approach, complete grids of east and north deflections of vertical are computed and the conversion is achieved using the relationship between gravity anomalies and deflections of vertical formulated via Laplace’s equation (e.g. Haxby et al., 1983; Sandwell, 1992; Sandwell and Smith, 1997).
The geoid height h and gravity anomaly Δg cane derived from the disturbing potential T by simple operations.. To a first approximation, the geoid height is related
to the disturbing potential by Bruns’ formula,
T
h
γ
≅ 1 (2.17)
where γ is the normal gravity of the earth. The gravity anomaly is the vertical derivative of the potential,
the east component of deflection of vertical is the slope of the geoidal height in the x-direction,
and the north component of deflection of vertical is the slope of the geoidal height in the y-direction,
These quantities are related to Laplace’s equation in rectangular coordinates:
2 0
gradient and the sum of x and y derivatives of the east and north deflection of vertical
In the frequency domain, Equation (2..2) becomes
{ }
γ( { }
η{ }
ξ)
where u and v are spatial frequencies corresponding to x and y.
2.3.4 Method 4: Inverse Stokes Integral
In this case marine geoid heights from altimetry are converted to gravity anomalies using the inverse of Stokes integral formulas. According to Heiskanen and Moritz (1967), the Stokes’ integral equation can be expressed as:
∫∫
∆gravity anomaly on a surface of the sphere of radius R; S(ψ) is the Stokes kernel
By the planar approximation, Equation (2.25) can be expressed as the convolution between gravity anomaly and Stokes function in Equation (2.28). That is (Wang, 1999): The corresponding frequency-domain version of Equation (2.28) is
) The above equation can be used to recover gravity anomaly from geoidal heights.
That is,
∆g =γF−1
(
ωN~(u,v))
(2.30)where ω= u2 +v2 , such a transformation from geoidal heights into free-air gravity anomalies uses a differentiation operator, which enhances the high frequencies and it is sensitive to noise (Andersen and Knudsen, 1998).
2.4 Radar Altimeter Data
2.4.1 Altimetry Data and Observations
During the last two decades altimetry has been available from the following satellites listed in Table2.1. Altimetry increased vastly in accuracy from meters to centimeters (Wunch and Zlotniki, 1984, Fu and Cazenave, 2001), and has opened for a whole new suite of scientific problems that can now be addressed using altimetry.
One of these is the ability to perform a high-resolution mapping of the Earth’s marine gravity field. Table 2.1 shows the specifications for selected satellite missions.
Table 2.1: Specifications for satellite missions.
Satellite Mission
After the altimetric range observations have been corrected for orbital, range and geophysical corrections (Fu and Cazenave, 2001), they provide mean sea surface height for gravity derivation. The surface height can be described according to the following expression in its most simple form:
h = N + ξ + e (2.31)
where
N is the geoid height above the reference ellipsoid, ξ is the sea surface topography
e is the error (treated below).
In geodesy the geoid N or the geoid slope is the important signal. In oceanography the sea surface topography ξ is of prime interest.
The geoid N can be described in terms of a long wavelength geoid NREF, geoid contributions from nearby terrain NDTM and residuals ΔN to this. Similarly, the sea surface topography can be described in terms of a mean dynamic topography (ξMDT) and a time varying or dynamic sea surface topography (ξ(t)). Some minor contributions hs to sea level is also seen from aliased barotropic motion and atmospheric pressure loading. Therefore, sea surface height can then be written as (ref.
Figure 2.1):
h = N + N + ΔN + ξ +ξ(t) + h (2.32)
The magnitudes of these contributions are :
The geoid NREF +/- 100 meters Terrain effect NDTM +/- 30 centimeters Residual geoid ΔN +/- 2 meters Mean dynamic topography ξMDT +/- 1.5 meter Dynamic topography ξ(t) +/- 1 meters.
Aliased barotropic motion and
atmospheric pressure loading hs +/- 10 centimeters
In order to enhance the signal to noise of the residual geoid height ∆N used for geoid and gravity field modeling in this study, as many contributors to sea level variation as possible should be modeled and removed.
Figure 2.2: Schematic illustration of the measurement principle (from http://www.aviso.oceanobs.com/html/alti/principe_uk.html).
2.4.2 Time-averaging of SSH
The repeated ground tracks of altimetric satellites do not exactly coincide with each other, and the separations of them is about 1 or 2 km. In order to reduce the anomalous temporal changes of SSH caused by some significant oceanographic phenomena, such as EL Nino or La Nina occurred during particular seasons or years, the altimetric SSH data from repeat missions is time-averaged for all available cycles and the mean tracks are obtained.
Mean track is derived from a selected reference tracks and the related collinear tracks. After the reference tracks are determined, the SSH at each point of the collinear tracks corresponding to the point of the reference track can be computed.
Altimeter data from repeated missions used in this study are also processed to get the mean sea surface, more details can be found in Hsu (1997).
2.5 Multi-Satellite Altimeter Data Processing
2.5.1 Multi-Satellite Altimetry Data Base
The satellite altimeter data used in this study were from the National Chiao Tung University (NCTU) altimeter data base. Table 2.2 lists the altimeter data used in computing the global MSSH and gravity anomaly grids. The data are from five satellite missions and span more than 20 years. The Seasat data are from the Ohio State University and are edited by Liang (1983), who also have crossover adjusted the Seasat orbits. The Geosat data are from National Oceanic and Atmospheric Administration and contain the latest JGM3 orbits and geophysical correction models (NOAA, 1997). In the Geosat/GM JGM3 GDRs, the wet and dry troposphere corrections are based on the models of National Centers for Environmental Prediction (NCEP)/National Center for Atmospheric Research (NCAR) (Kalnay et al., 1996）and NASA Water Vapor Project (NVAP) (Randel et al., 1996), and the ionosphere correction is adopted from the IRI95 model (Bilitza, 1997). The sea state bias, which introduces an error to the measured range, is also recomputed and is more accurate than the previous version of GDRs. The ERS-1 and ERS-2 data are from Centre ERS d'Archivage et de Traitement (CERSAT)/France and their orbits have been adjusted to
the T/P orbits by Le Traon and Ogor (1998). Finally, the T/P data are provided by Archiving, Validation, and Interpretation of Satellite Oceanographic Data (AVISO) (1996) and should have the best point data quality among all altimeter data, due to the low altimeter noise and the state-of-the-art orbit and geophysical correction models.
The Geosat/GM and ERM GDRs from NOAA contain both raw measurements at 10 samples per second (10 Hz) and the one-second averaged SSH. To increase spatial resolution, we re-processed the raw data to obtain SSHs at 2 samples per second.
When re-sampling, the 10-Hz SSHs were first approximated by a second-degree polynomial and the desired 2 per second (2 Hz) SSHs were then computed from the solve-for polynomial coefficients. Pope’s (1976) tau-test procedure was used to screen any erroneous raw data. Among these data sets, Geosat/GM and ERS-1/GM have very high 2-D spatial density and will contribute most to the high-frequency parts of MSSH and gravity fields. In one test over the SCS we used separately the new JGM3 and the old T2 versions of Geosat/GM altimeter data to predict gravity anomalies and it was found the root mean square (RMS) differences between the predicted and shipborne gravity anomalies are 10.65 and 9.77 mgals, respectively. Thus the JGM3 version indeed outperforms the T2 version.
Table 2.2: Satellite altimeter missions and data used for the global computation Mission Repeat
2.5.2 Averaging SSH to Reduce Variability and Noise
The altimeter data from the repeat missions (Geosat/ERM, ERS-1/35d, ERS-2/35d and T/P) were averaged to reduce time variability and data noise. When averaging, Pope’s (1976) tau-test procedure was also employed to eliminate erroneous observations. But it turns out that the Geosat observed SSHs behave erratically (for example, large jump of SSHs in along-track observations) and Pope’s method failed to detect outlier SSHs in several occasions. Thus for Geosat/ERM a modified averaging/outlier rejection procedure was used. In this new procedure, at any locations SSHs from repeat cycles were first sorted to find the median value. Then, the difference between individual SSHs and the median were computed. Any SSH with difference larger than 0.45 m is flagged as an outlier and removed. (0.45 is based on three times point standard deviation of Geosat/ERM, see below). The desired MSSH is finally computed from the cleaned SSH by simple averaging. Table 2.3 lists the statistics associated with the averaged and non-averaged SSHs. For the repeat missions in Table 2.3, we computed the point standard deviation (SD) of SSH as
1 SSH and n is the number of points. According to the statistical theory, the SD of averaged height h(i,j) is
n
Thus the accuracy increases with number of repeat cycles. A point SD can be expressed as
σi: instrument error (random part + time-dependent part)
σo: orbit error (random part + non-geographical correlated error)
σg: errors in geophysical correction models (random + systematic errors) σs: sea surface variability (excluding tidal variation)
In Equation (2.35) it is assumed that the involving factors are uncorrelated. Thus, a point SD contains both random noises and variabilities arising from a variety of sources. Figure 2.3 shows the point SDs derived by averaging Geosat/ERM, ERS-1/35d and T/P. (ERS-2/35d SD is close to ERS-1/35d SD, so it is not shown here). Clearly the SDs from the three repeat missions have the same patterns of distribution. Over oceanic areas of high variability such as the Kuroshio Extension, the Gulf Stream, the Brazil Current, the Agulhas Current and the Antarctic Circumpolar Currents, sea surface variability contributes most to SD. SD is also relatively high in the tropic and the western Pacific areas, where mesoscale eddies are very active. Clearly, the pattern of sea surface variability is very stable over the past two decades, as the Geosat/ERM-derived SDs in the 1980s and the ERS-1 and T/P–derived SDs in the 1990s show very consistent signature. In the polar regions
samples. Over shallow waters, tide model error becomes dominant in SD and is particularly pronounced in the continental shelves of the western Pacific, the northern Europe and the eastern Australia. In the immediate vicinity of coasts, the interference of altimeter waveforms by landmass further increases SD. Over the deep, quiet oceans, SD is in general very small and here along-track MSSH will be best determined. Note that in Table 2.3 the SD of Geosat/GM is simply the SD of the 2-Hz SSHs as derived from the fitting of the 10-Hz SSHs, so it does not represent the noise level of 2-Hz SSHs.
Table 2.3: Statistics of SSHs from seven satellite altimeter missions
Mission No. of
Geosat/GM no 15708 25530238 151044 0.141
Geosat/ERM 68 488 1991672 4798 0.026
ERS1-/35d 26 1002 1677190 4805 0.023
ERS-1/GM no 9532 14702377 44928 －
ERS-2/35d 37 1002 1141786 4815 0.022
T/P 239 254 553525 1387 0.009
aOver the area 25º S to 15º S and 235º E -245º E where there is no land
bThe SD of Geosat/GM is the SD of 2-Hz SSHs from fitting the 10-Hz SSHs
2.5.3 Choice of ocean tide model
Ocean tide creates a deviation of the instantaneous sea surface from the mean sea surface. There are now more than 10 global ocean tide models (Shum et al., 1997) available for correcting tidal effect in altimetry. Table 2.4, partly from Matsumoto et al. (2000), shows the RMS collinear differences of T/P and the errors in along-track DOV using NAO99b (Matsumoto et al., 2000), CSR4.0 (Eanes, 1999) and GOT99.2b
Figure 2.3: Estimated standard deviations of point SSH of Geosat/ERM (top), ERS-1 (center) and TOPEX/POSEIDON (bottom)
(Matsumoto et al., 2001) tide models. Both NAO99b and GOT99.2b are based on hydrodynamic solutions and a further enhancement by assimilating T/P altimetry data into the solutions. The CSR tide models (3.0 and 4.0 versions) use the orthotide approach to model the residual tides of some preliminary hydrodynamic oceans tide
models using T/P altimeter data. From Table 2.4, it seems that NAO99b is the best
model among the three. Using T/P SSHs, Chen (2001) found that the NAO99b tide model yields the smallest RMS crossover differences of SSH compared to the CSR4.0 and GOT99.2b tide models. Over shallow waters, all the collinear differences exceed 10 cm, which translate to a 48-µrad (10-6 radian) error in DOV of Geosat/GM. Even in the deep oceans, the collinear difference-implied DOV errors are still very large.
Using available resources here, we conducted a test over the SCS to compare the accuracies of predicted gravity anomalies using the CSR3.0, CSR4.0 and GOT99.2b tide models. As shown in Table 2.5, the NAO99b tide model produces the best accuracy in gravity anomaly. However, as seen in Table 2.5, the differences in accuracy are very close. This should be partly due to the fact that DOV is insensitive to long wavelength tide model error. For Geosat/ERM, ERS-1/35d, ERS-2/35d and T/P, the use of CSR3.0, CRS4.0 and NAO99b will probably not make too much difference because of the reduction of tide model error by averaging data from repeat cycles.
Table 2.4: RMS collinear differences (in cm) of T/P (Matsumoto et al. 2000) and corresponding error (in µrad) in along-track DOV of Geosat/GM using different tide models
0 < Ha < 0.2 0.2 < H < 1 1 < H Tide model
Difference Error Difference Error Difference Error
NAO99b 11.20 45 6.98 28 8.56 35
CSR4.0 15.77 64 7.37 30 8.55 35
GOT99.2b 13.99 57 7.37 30 8.65 35
aH: ocean depth in km
Table 2.5: RMS differences (in mgals) between shipborne and altimeter-derived gravity anomalies with different tide models
Tide model ERS-1/GM Geosat/GM (JGM3)
NAO99b 11.90 9.77
CSR3.0 11.93 Not available
CSR4.0 Not available 9.78
CHAPTER 3
GLOBAL MODELS OF MEAN SEA SURFACE AND GRAVITY ANOMALY
3.1 Introduction
Satellite altimetry has begun a new era in earth sciences research. Mean sea surface height (MSSH) and marine gravity anomaly are the most two important products of satellite altimetry for geodetic and geophysical applications. MSSH is useful in numerous applications such as global tide modeling, sea level change study, reduction of altimeter observations to reference tracks, and bathymetry prediction.
Applications of marine gravity have been illustrated in, e.g., Sandwell and Smith (1997). Ignoring the sea surface topography (SST), marine gravity anomaly is basically equivalent to MSSH, and the two have a simple, linear relationship in the spectral domain. Current global MSSH models have been largely constructed by direct gridding of altimeter observed sea surface heights (SSHs), e.g., the MSSH models of Kort-og Matrikelstyrelsen (KMS) (Andersen and Knudsen, 1998), National Aeronautics and Space Administration (NASA)/Goddard Space Flight Center (GSFC) (Wang, 2000) and Collecte Localisation Satellites (CLS) (Hernandez and Schaeffer, 2000). A somewhat unconventional approach was employed by Yi (1995), who used a combined SSH and geoid gradient from multi-satellite missions to construct the OSU95 MSSH model, see also Rapp and Yi (1997). When using the direct gridding method, SSHs must be carefully crossover adjusted to remove inconsistency of SSHs at crossover points. Insufficient or improper crossover points
will easily lead to artifacts in the resulting field such as track pattern and extremely large signature. With MSSH computed, marine gravity anomaly can be obtained by a simple conversion in the spectral domain, see, e.g., Schwarz et al. (1990) for a complete derivation of the spectral conversion between MSSH and gravity anomaly (assuming that SST is removed).
As discussed in (2.2.3), using deflections of the vertical (DOV) as the data type is one of the choices in gravity anomaly recovery from satellite altimetry data. The major argument of using DOV is that DOV is less contaminated by long wavelength errors than SSH, and using DOV requires no crossover adjustment; see, e.g., Sandwell and Smith (1997), Hwang et al. (1998), and Andersen and Knudsen (1998). With DOV from altimetry, it is possible to compute MSSH using the deflection-geoid formula derived by Hwang (1998). Taking the advantage of DOV, in this chapter we will compute simultaneously a global MSSH grid and a global gravity anomaly grid by the deflection-geoid and the inverse Vening Meinesz formulae from multi-satellite altimetry data. In parallel to this study is the latest models of ocean tide and other geophysical corrections.
3.2 Forming north and east components of DOV
3.2.1 Computing along-track DOV
The methods for computing MSSH and gravity anomaly in this paper will use DOV as the data type. By definition, an along-track DOV is the geoid gradient with an opposite sign:
s g N
∂
−∂
= (3.1)
where N is the geoid, which is a surface function, and s is the along-track distance. By this definition one would need to first construct a surface of the geoid and then perform directional derivative along s to get DOV. Following this concept, we first fit a cubic spline (De Boor, 1978) to the along-track geoidal heights from altimetry. Then the along-track derivative is obtained by differentiating the spline. The actual numerical computations were done by the International Mathematical and Statistical Library (IMSL) routines. It turns that such a procedure results in very noisy DOV, which should be due to the interpolation error in fitting the spline. The interpolation error is particularly large when point intervals along a track segment are not uniform.
Although this approach seems rigorous, it did not produce good results. A better result is obtained by simply approximating DOV by the slope of two successive geoidal heights:
where d is the point spacing. The geographic location of g is the mean location of the two geoidal heights. The estimated standard deviation of g is simply
g d obtain geoidal height from SSH, both the time-dependent and quasi-time independent
SST values should be removed. In this paper, the time-dependent SST is reduced by filtering, and for the quasi time-independent SST we adopt the model of Levitus et al.
(1997), which is available at NOAA’s Ocean Climate Laboratory (see http://www.nodc.noaa.gov/OC5/dyn.html). The Levitus SST values from NOAA are given as monthly averages on a 1º×1º grid for 12 months. We averaged the monthly values to get the needed quasi time-independent SST. Figure 3.1 shows the averaged Levitus SST, which looks very similar to the 1982 version of Levitus SST (cf: Hwang, 1997, Fig. 3). The divided difference method (Gerald and Wheatley, 1994) is then used to interpolate the needed SST value from this SST grid at any altimeter data point.
Figure 3.1: Quasi time-independent sea surface topography from Levitus et al. (1997), contour interval is 10 cm.
3.2.2 Removing Outliers and Gridding DOV
Because of the use of 1D FFT algorithm in the computations of MSSH and gravity anomaly (see below), the north and east components of DOV need to be formed on regular grids with constant intervals in latitude and longitude. Before
forming the regular grids, erroneous along-track DOV must be removed. Again we adopt Pope’s tau-test method to remove possible outliers. Within a 4′×4′ cell, any along-track DOV, εi, can form an observation equation as (cf: Heiskanen and Moritz, 1985, p. 187)
ξ and are the north and east components. After least-squares estimating ξandη, all residuals in the 4′×4′ cell can be determined using Equation (3.4). A DOV is flagged as an outlier and is removed if its residual satisfies the condition:
) tau-value at degree of freedom of m; see Koch (1987, p.336) and Pope (1976) for the methods of computing σviand τc(m). Table 3.1 shows the ratios between removed and raw DOV in 12 selected areas. The removed DOV are largely from the non-repeat missions. In general, the removal ratios are relatively high in shallow waters and in higher latitudes, e.g., Ross Sea, and low in the open ocean, e.g., the Reykjanes Ridge and the East Pacific Rise. One problem with the above procedure of outlier removal is that, in areas with sparse data such as coastal regions and polar regions, there are not enough data points to produce large degree of freedom to make the result of the tau-test reliable, leading to undetected/improperly detected outliers.
After removing outliers the north and east components are then computed on a regular grid by the method of LSC (Moritz, 1980):
l
Table 3.1: Test areas and ratios of removed outliers
Area Geographic boundaries
where vector l contains along-tack DOV, vector s contains north (ξ) and east (η)components, Csl, Cll and Cn are the covariance matrices for s and l, l and l, and the
where vector l contains along-tack DOV, vector s contains north (ξ) and east (η)components, Csl, Cll and Cn are the covariance matrices for s and l, l and l, and the