Marine Gravity Recovery for the Malaysia Region from TOPEX,
ERS-1 and Geosat Satellites Radar Altimeter Data
Altimetry-derived Sea Surface Heights
The satellite serves as a stable platform from which a radar altimeter can measure the distance to the instantaneous ocean surface. The radar altimeter works by transmitting a short pulse (approximately 3 nanoseconds). By measuring the travel time of the pulse it is possible to obtain data on the shape of the sea surface. A simplified geometry of a radar altimeter can be written as follows:
hSSH = h-r (1)
where h
SSH is the sea surface height (SSH) above the reference ellipsoid, h is the satellite altitude above the reference ellipsoid, and
r is the distance from the footprint on the sea surface to the satellite.
The altimeter data is usually given as a set of geophysical data records (GDR) that include altimeter measurement information, as well as corrections that should be applied to the data. In mathematical terms the altimetric observation of the sea surface height can be described according to the following expression:
hSSH = N+ S+ e (2)
where N is the geoid height, S is the sea surface topography, and e is the error. The error term of equation (2) can be described as follows:
e = e0 + eT + eA + n (3)
where
e0 is the radial orbit error,
eT is the ocean tide residuals,
eA is additional errors (eg. Wet troposhperic correction), and n is the measurement noise.
The noise of the measurements is in the order of a few centimeter. The largest error component is the radial orbit error, which is split into an errors caused by geopotential errors and errors caused by initial state vectors errors, drag, and solar radiation pressure. The second largest error term is not related to the altimeter measurement itself. The tidal models which has an accuracy of around 10m were applied in order to convert an instantaneous sea surface height into a mean sea surfaces height.
New gridded data sets of mean sea surface (MSS) were produced on 30 45"grid using one-year mean sea surface height (SSH) data of Geosat, ERS-1 and Topex altimeter satellites (Yi, 1995). An inverted barometer correction and improved ocean tide correction (Cartwright and Ray, 1990) were applied to all SSH data used in this study. A gridding method of the least-squares collocation (LSC) based on the covariance function of the second-order Markov process was used. The residual SSH data were gridded after removing the reference geoid undulation implied by the JGM-3 potential coefficient model. The grided MSS values were computed by adding the reference geoid undulations back to the gridded residual SSH values.
Recovery of Gravity Anomalies form Altimeter
There are three well known techniques used for the recovery of free-air gravity anomalies from satellite altimeter. The first involves converting vertical deflections a attained from SSH that is widely used by geophysicists (Haagmans et alkalinity., 1995; Sanwell and Mcadoo, 1998), and the second is by LSC, which was developed by geodesists (Rapp, 1079; Basic and Rapp, 1992). The third approach which has been used in this study involves direct conversion of Geoid undulations to free-air anomalies using Fast Fourier Transform (FFT) technique (Kim, 1996). After reasonable approximations, a direct relation between geoid undulations and gravity anomalies is given as follows (Heiskeanen and Moritz, 1967).
| Dg= - |
¶
-----
¶R |
(Ng)- |
2
----
R |
(Ng) (4) |
Where T is the disturbing potential, N is the geoid undulation,
g is the normal gravity and R is the radius of the earth. The first term on the right hand side of equation (4) is called the gravity disturbance (dg) that can be directly calculated by the FFT technique using satellite altimeter SSH for N. Following Kim (1996), for T=gN on the x-y plane, dg is the calculated by the inverse Fourier transformation as:
Where T
Kl(z) is the 2-D Fourier transformation of T
xy at the wave numbers k and l. M and N is the data points in the x and y directions x is the coefficient of the Fourier wave number given as:
Since the data (T=
gN) are given on the sea surface with geographical coordinate system
(
j,
g) for the limited area, a planner and map projection approximation are necessary for actual computations.
