CHAPTER 1 INTRODUCTION
1.2 Objectives and Outline
The main objective of this dissertation is the improvement of global and local marine gravity anomaly derivations from multi-satellite altimeter data. Early such works, can be found in Hwang and Parsons (1995), Andersen et al. (1996), Sandwell and Smith (1997), Hwang et al. (1998). Based on this objective, there are several main issues to be studied in this dissertation:
Chapter 1 introduces the background, development and the latest missions of Satellite Altimetry. The objectives and thesis outline are also given in Chapter 1.
In Chapter 2, an overview about marine gravity field from satellite altimetry is described. Four existing computation methods developed with different altimeter data types to compute gravity anomaly are reviewed and the basic concept of altimeter data and observations are introduced, in preparation for demonstrating the database of multi-satellite altimeter used in this study. Multi-satellite altimeter data processing such as the choice of model and the geophysical corrections are also discussed in Chapter 2.
Global model of mean sea surface height (MSSH) and gravity anomaly on a 2 minute by 2 minute grid will be determined in Chapter 3. Comparisons of the global MSSH model with the T/P and the ERS-1 MSSH are carried out and discussed. The RMS differences between the predicted and shipborne gravity anomalies are computed in 12 areas of the world’s oceans.
In Chapter 4, issues concerning gravity anomaly recovery over shallow waters are investigated and tests are carried out . A method will be introduced that uses to detect outliers in altimeter data and to filter the non-repeated mission data. Several
examples from repeated and non-repeated mission data will be tested. Two methods with three kinds of data types of gravity anomaly computations were introduced and compared in the East China Sea (ECS) and the Taiwan Strait (TS) in order to find the best parameters and method with a new data type -differenced height.
Chapter 5 compares global gravity anomaly models and tide models to analyze the errors in altimeter-gravity conversion over shallow waters. Case studies and analysis will be performed over the ECS and the TS. The best method given in section 4.3 will demonstrate the case of using land data in enhancing the accuracy of altimeter-derived gravity anomalies.
In Chapter 6, conclusions and recommendations are given.
CHAPTER 2
MARINE GRAVITY ANOMALIES RECOVERY BY SATELLITE ALTIMETER DATA
2.1 Introduction
With the advent of satellite altimetry, the applications of satellite altimeter data have been extensively investigated. One of the major applications is to recover gravity information from satellite altimetry data. Several methods for gravity derivation from altimetry exist, e.g., least-squares collocation (LSC), inverse Vening Meinesz Formula, FFT with Deflection of Vertical (DOV), the inverse Stokes integral. This chapter will discuss these methods.
In this study, we use multi-satellite altimeter data – Seasat, Geosat Exact Repeat Mission (Geosat/ERM), Geosat Geodetic Mission (Geosat/GM), ERS-1 35-day repeat mission (ERS-1/35-daay), ERS-1/GM and TOPEX/POSEIDON (T/P)- in the gravity and mean sea surface height derivation. With altimeter data from such a variety of satellite missions, a good data management system and data processing is important and will be discussed in this chapter.
2.2 Marine Gravity Field from Altimetry
2.2.1 Overview
A review of altimeter contribution to gravity field modeling is given below.
Satellite altimetry will keep its role in gravity even if there are improvements through
the gravity satellite missions. Altimetry is able to map the mean sea surface and gravity anomaly with a spatial resolution of 2’x2’ or higher (Hwang et al., 2002). A 2’x2’-resolution corresponds to a harmonic degree beyond 2000. The gravity field models of GOCE will not go beyond degree 300. Thus the high frequency information of the marine gravity filed will be mainly based on satellite altimetry. However, due to the low density of shipboard and seafloor gravity measurements, satellite altimetry provides the most valuable data sets for the recovery of the marine gravity field.
Since the advent of satellite altimetry, investigators created numerous local and global marine gravity field models using a variety of successful techniques. According to Fu and Cazenave (2001), the first regional (Haxby et al., 1983), and global (Haxby, 1987) color portrayals, created from the 1978 Seasat data, demonstrated the promising potential of satellite altimetry for the global recovery of the marine gravity field. The results Haxby are based on the planar spectral method of using two-dimensional fast Fourier transform (FFT) to convert the altimeter-derived sea-surface slopes to gravity anomalies on flat-earth domains. In an alternate study, a global simultaneous recovery of the sea-surface height and the marine gravity anomaly field was developed from the Seasat data (Rapp, 1983) using the least-squares collocation technique.
The planar spectral method and the least-squares collocation are the most widely used tools in the short-wavelength marine gravity recovery. The major advantage of the least-squares collocation is that randomly spaced heterogeneous data can be combined and gravity anomalies can be derived in grid or discrete. Besides, this method has the capability to give accuracy estimates for the computed gravity anomalies. As such, the accuracy of the result depends on the accuracy of the statistical information used. However, the LSC is numerically cumbersome and needs
hand, the spectral method has great simplicity and computational efficiency when compared with any least-squares techniques.
Up to 2007, all marine areas within the area 82°S to 82°N and 0°to360°E longitude have been covered with sufficient altimetric observations to derive the global marine gravity field on a 2 minute by 2 minute resolution, corresponding to 3.6 by 3.6 km at the Equator. Numerous local and global marine gravity anomalies have been created using a variety of successful techniques (e.g., Haxby (1987), Sandwell (1992), Tscherning et al. (1993), Hwang et al. (1998)). Numerous comparisons between marine observations and altimetry-derived gravity anomalies have been presented, e.g., Hwang and Parsons (1995), Sandwell and Smith (1997), Andersen and Knudsen (1998), Hwang et al. (1998). Using the high-density data collected from multi-satellite missions, the precision of the derived global gravity fields is reported to range from 3 to 14 mgal (Hwang et al,, 2003) based on the comparisons between shipborne and altimeter-derived results.
Detailed knowledge of the gravity anomalies are used for a variety of purposes, such as the guidance of aircrafts, and spacecrafts over geophysical exploration purposes, over bathymetry, and understanding of the tectonics, territorial claims.
2.2.2 Remove-restore Technique
The gravity derivations in this work are all based on the remove-restore procedure. In this procedure, a reference gravity field is needed. The choice of reference field has been somewhat arbitrary in the literature, eg., Sandwell and Smith (1997), Hwang (1989) and Rapp and Basic (1992) chose to use degree 70, 180 and 360 fields, respectively (note that the gravity models are also different). Wang’s (1993)
theory suggests the use of a reference field of the highest degree, provided that the geopotential coefficients are properly scaled by the factor Sn given by
n
where Cn and εn are the degree variance and the error degree variance of the chosen reference field. Wang’s theory was tested by Hwang & Parsons (1996) and, in the case of OSU91A (Rapp, Wang and Pavlis, 1991), the scaling factors improve slightly the accuracy of the computed gravity anomalies. In Hwang et al. (1998), EGM96 model was used to see whether the scaling factor Sn is necessary. The results show that the use of Sn does not increase the accuracy of the computed gravity anomalies in the two test areas, the Reykjanes Ridge and the South China Sea. This is due to the fact that EGM96’s high-degree coefficients are substantially improved compared to OSU91A, since Sn has a larger effect on the high-degree coefficients than on the low-degree ones.
2.2.3 Altimeter Data Types
It has been shown by, e.g., Hwang and Parsons (1995), Sandwell and Smith (1997), that use of geoid gradients for derivation of gravity anomaly from altimetry is more stable than doing so using geoidal heights and can reduce the effect of long wavelength errors in altimeter data. A typical long wavelength error is orbit error.
Another advantage of using geoid gradients is that we do not need to adjust the sea-surface height as in Knudsen (1987). In the following three kinds of altimeter data types are introduced and will be used for predicting gravity anomaly.
Taking the first horizontal derivatives of the altimeter-sensed sea surface heights along-track yields the negative deflections of the vertical at the geoid. Along-track DOV is defined as
where h is geoidal height obtained from subtracting dynamic ocean topography from sea surface height (SSH), and s is the along-track distance. In using DOV satisfactory result can be obtained without crossover adjustment of SSH, and this is especially advantageous in the case of using multi-satellite altimeter data. The problem with Equation (2.2) is that DOV can only be approximately determined because along-track geoidal heights are given on discrete points.
A data type similar to along-track DOV is differenced height defined as
i i
i h h
d = +1− (2.3)
where i is index. Using differenced height has the same advantage as using along-track DOV in terms of mitigating long wavelength errors in altimeter data. We go one step forwards by using “height slope” defined as
i using height slope is similar to that of using DOV and differenced height in terms of
error reduction.
2.3 Methods of Marine Gravity Anomalies From Altimetry
Seasat, Geosat, ERS-1/ERS-2 and TOPEX/POSEIDON satellite altimetry missions have collected a vast amount of data over ocean areas. These data enable us to determine the marine gravity field with unprecedented resolution and accuracy.
Practical computations of marine gravity anomalies and geoid heights from satellite altimetry data have been carried out for more than two decades; e.g., see Koch (1970), Balmino et al. (1987), Basic and Rapp (1992), Hwang (1989), Rapp (1985), Zhang and Blais (1993) and Zhang and Sideris (1995). Here we introduce four methods of gravity anomalies from satellite altimetry. The overview below only provides a brief outline and the reader is referred to the cited literature for the exact details of the theories and computational procedures.
2.3.1 Method 1: Least Squares Collocation
The application of conventional (space domain) LSC in physical geodesy has been discussed in detail by Moritz (1980). Its practical applications in gravity field modeling can be found in, e.g., Tscherning (1974), Rapp (1985) and Basic and Rapp (1992). A fast frequency domain LSC method was studied by Eren (1980). Examples of the use of LSC to calculate gravity anomalies are the works by Rapp (1979,1985), Hwang (1989), and Rapp & Basic (1992), who all used altimeter data alone. The LSC method needs more computer time but has the capability to combine heterogeneous data and to give accuracy estimates for the computed gravity anomalies. The conventional LSC method and derived formulae by using of different altimeter data types will be introduced in this section
Using geoidal heights as observations, the prediction of gravity anomalies evaluated by the LSC method can be expressed as:
∆ =g C∆gh(C D h+ )−1 (2.5)
where C∆gh is the covariance matrix between gravity anomalies g∆ and h, C and D are the observations and error covariance matrices which are geoidal heights observed by altimetry.
If geoid heights h’ and shipborne gravity anomalies ∆g’ are used simultaneously, the LSC formula reads
To use differenced height for gravity estimation, again one may employ LSC.
First, the covariance function between two differenced heights is
The covariance function between gravity anomaly and differenced height is
(
∆g,di)
=cov(
∆g,hi+1−hi)
cov
=cov
(
∆g,hi+1)
−cov(
∆g,hi)
(2.9)The needed covariance functions of using height slope are then
(
i j)
The spectral characteristics of height slope are the same as DOV and gravity anomaly as they are all the first spatial derivatives of earth’s disturbing potential.
With differenced height or height slope, gravity anomaly can be computed using the standard LSC formula
where vector l contains differenced heights or height slopes, C and l C are the n
signal and noise parts of the covariance matrices of l, and C is the covariance sl matrix of gravity anomaly and differenced height or height slope. Differenced height or height slope can also be used for computing geoidal undulation: one simply replaces C by the covariance matrix of geoid and differenced height or height slope sl
in Equation (2.12). Furthermore, for two consecutive differenced heights along the same satellite pass, a correlation of -0.5 exits and must be taken into account the C n
matrix in Equation (2.12).
Next formula uses along-track DOV defined in (2.2.3) and the LSC method for gravity anomaly derivation (Hwang and Parsons, 1995). For this method, the covariance function between two along-track DOV is needed and is computed by
C and are the covariance functions of longitudinal and transverse DOV components, respectively and αpq is the azimuth from p to q. The covariance function between gravity anomaly and along-track DOV is computed by
g l QP
g C
C∆ ε =cos(αεQ −α ) ∆ (2.14)
where Cl∆g is the covariance function between longitudinal component of DOV and gravity anomaly. Cll, CmmandCl∆gare isotropic functions depending on spherical
distance only. With these covariance functions, gravity anomaly can be computed by LSC as in Equation (2.12) using along-track DOV for l and covariance matrices computed with Cεε andC∆gε for C and l C . For the detail of this method, see sl
Hwang and Parsons (1995).
2.3.2 Method 2: Inverse Vening Meinesz Formula
This method employs the inverse Vening Meinesz formula (Hwang, 1998) to compute gravity anomaly. The inverse Vening Meinesz formula reads:
(
q qp q qp)
q the kernel function defined by⎟⎟⎠
where ψpqis the spherical distance between the computation point p and the moving point q of the integration, see also Figure 2.1. In the practical computation, the 1D
FFT method is used to implement the spherical integral in Equation (2.15). For the 1D FFT computation, the two DOV components ξqandηq are prepared on two regular grids. We use LSC to obtain ξqandηq on the grids along-track DOV by LSC, see also Hwang (1998).
Figure 2.1: Geometry for the inverse Vening-Meinesz formula
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
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