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Research Article
 

Determination of Gravity Anomalies over the Arabian Sea



Ahmed Abdo Ali, Liu Jing Nan and Jiang Wei Peng
 
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ABSTRACT

Gravity anomalies derived from satellite altimetry are progressively taking the place of ship-borne gravity for many marine geo-scientific investigations this paper compares the marine gravity anomalies-derived from multi-mission satellite altimetry-with gravity anomalies implied by the EGM96 global geopotential model. The results show some significant differences among these gravity data sources. Ocean satellite altimetry implied free-air gravity anomalies have had the shortest wavelength removed during the processing to generate the optimal solution between multiple radar altimeter missions. Compute gravity anomalies by LSC using along-track,differeced geoidal heights and height slopes. Gravity Anomalies over the Arabian Sea (latitudes: 0-25 °N and longitudes: 35-70 °E) is computed by using along track deflections of vertical and grid along track deflections of vertical using Shepard`s method of gridding procedure and finally the gravity anomalies were computed using the inverse Vening-Meinesz formula we used altimeter data from Topex/Poseidon, Jason1, Geosat, ERS1 and ERS2 missions.

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  How to cite this article:

Ahmed Abdo Ali, Liu Jing Nan and Jiang Wei Peng , 2007. Determination of Gravity Anomalies over the Arabian Sea. Journal of Applied Sciences, 7: 877-882.

DOI: 10.3923/jas.2007.877.882

URL: https://scialert.net/abstract/?doi=jas.2007.877.882

INTRODUCTION

The main goal of satellite altimetry is the study and observation of the processes and properties of the marine environment, so that when utilizing altimetry data the monitoring of phenomena like the mean sea level variations and changes, the ice transfer, the wind speed, the wave height and the water temperature would be feasible. It is possible, through various techniques, to derive information about the marine gravity field and the ocean tides as well (Emadi et al., 2003).

Altimetric measurements have a vital importance for geodesy, since we are able to provide a direct measurement of the main estimation quantity of geodesy i.e., geoid heights. Since the altimetric observations, the Sea Surface Heights (SSHs), correspond to the separation of the sea surface from the reference ellipsoid is very close to geoid undulations. The difference between Mean sea surface height and geoid is called Sea Surface Topography (SST). The deflections of the vertical are calculated directly by differentiating the sea surface height observations in the along track direction, or by differentiating the geoid undulations in the regular grid and then the free-air gravity anomalies, the difference between the magnitude of the actual gravity (W0) on the geoid and the magnitude of the normal gravity (γ0) on the ellipsoid’ were computed using the inverse Vening-Meinesz formula.

2-D of deflections of vertical components and gravity anomaly with satellite altimetry data: Deflection of vertical is defined as the spatial angel between the normal gravity vector on the reference ellipsoid and the actual gravity vector on the geoid (θ).

It can also be interpreted as the maximum slope of the geoid with respect to the reference ellipsoid at the point of interest.

Image for - Determination of Gravity Anomalies over the Arabian Sea
(1)

By using gradients of the sea surface height many of the long wavelength altimetry error sources are limited. An obvious advantage is that the task of having to perform a crossover reduction can be avoided. The deflections of the vertical are calculated directly by differentiating the sea surface height observations in the along track direction and then computing the deflections of the vertical (ξ, η) from equation (1)At the crossover location where ascending and descending ground tracks intersects from either the same of different satellites a much more stable determination of the slopes can be obtained.

Vertical deflections from along-track slopes: Taking the first horizontal derivatives of the altimeter-sensed sea surface heights along-track yields the negative deflections of the vertical at the geoid. These are interpolated onto a regular grid and can be converted to gravity anomalies using the inverse Vening-Meinesz formula.

The derivative of the geoid height N with respect to time t along the ascending profile is (Smith and Sandwell, 1994):

Image for - Determination of Gravity Anomalies over the Arabian Sea
(2)

and along the descending profile is:

Image for - Determination of Gravity Anomalies over the Arabian Sea
(3)

Where

n λ are the geodetic latitude and geodetic longitude of data points, respectively. At the crossover point the following relationships are accurate to better than 0.1%.

Image for - Determination of Gravity Anomalies over the Arabian Sea
(4)

The geoid gradient (deflection of the vertical) is obtained by solving (1) using (4).Then we can write:

Image for - Determination of Gravity Anomalies over the Arabian Sea
(5)

Image for - Determination of Gravity Anomalies over the Arabian Sea
(6)

When two or more satellites with different orbital inclinations are available, the situation is slightly more complex but more stable.

Gridding the deflection of vertical components: The computed the deflections of vertical components were discrete values, which were randomly distributed over Arabian sea and that the distances between them were not uniform. As required by FFT techniques that the input data should be uniformly spaced on a regular grid (Dadzie, 2005).

We used Shepard’s method of gridding procedure to interpolate values of deflection of vertical components at locations where no data existed, using a grid spacing of 2.5’ arc-minute in both longitude and latitude directions. the computational steps as follow:

Knowing the coordinates of the discrete crossover point positions (φi, λi) in a spherical coordinate system(φ, λ ) and the corresponding values of the components of the deflection of vertical, fi = f (φi, λi) (I = 1, 2,..., N) the differences in longitude (Δλi) and latitude (Δφi) between the coordinates of each of the discrete crossover points (φii) within a gird cell and the coordinates (φ0, λ0) of the grid node ( i.e., the centre point of the grid cell where Z value is to be interpolated ) are computed;

Image for - Determination of Gravity Anomalies over the Arabian Sea
(7)

Where ri is the distance, between the grid node, termed the computation point P and the ith discrete crossover point within the cell, termed the running point Q, is computed by :

Image for - Determination of Gravity Anomalies over the Arabian Sea
(8)

where R is the mean radius of the Earth, ψ, is the spherical distance between P and Q and

Image for - Determination of Gravity Anomalies over the Arabian Sea
(9)

the weight function can be calculated as (Jiang, 2001) :

Image for - Determination of Gravity Anomalies over the Arabian Sea
(10)

where r is the search radius and S is the spherical cap, which was set at 2°. The interpolator was finally given by:

Image for - Determination of Gravity Anomalies over the Arabian Sea
(11)

where N is the number of data points used in fitting the interpolated value at the grid node and μ is the smoothing factor, which was set at 2 for this study.

Computation gravity anomaly using inverse vening-meinesz formula: The concepts of determining marine gravity anomalies from satellite radar altimetry are as follows. The altimeter essentially measures the distance between the satellite and the instantaneous sea surface along the nadir using pulse-limited radar at a series of footprints along the sub-satellite tracks (Fu and Cazenave, 2001).

The Inverse Vening Meinesz formula used in this study to compute gravity anomalies using altimeter-derived components of deflection of vertical (Eqs. 1-5) as input data and is given as (Cheng et al., 2001; Hwang, 1998):

Image for - Determination of Gravity Anomalies over the Arabian Sea
(12)

where the component of deflection of vertical along the azimuth α is related to the geoid slope ( Image for - Determination of Gravity Anomalies over the Arabian Sea)

Image for - Determination of Gravity Anomalies over the Arabian Sea
(13)

where R is the mean radius of the earth ξ and are η the altimeter-derived north-south and east-west components of the deflection of the vertical, respectively.

Image for - Determination of Gravity Anomalies over the Arabian Sea
(14)

Image for - Determination of Gravity Anomalies over the Arabian Sea
(15)

substituting Eq. (13) into Eq. (12) yields

Image for - Determination of Gravity Anomalies over the Arabian Sea
(16)

Image for - Determination of Gravity Anomalies over the Arabian Sea
(17)

considering the Eq. (14) and (15) Eq. (17) is written as:

Image for - Determination of Gravity Anomalies over the Arabian Sea
(18)

where

Image for - Determination of Gravity Anomalies over the Arabian Sea

Image for - Determination of Gravity Anomalies over the Arabian Sea
(19)

Image for - Determination of Gravity Anomalies over the Arabian Sea
(20)

and

Image for - Determination of Gravity Anomalies over the Arabian Sea
(21)

so

the 1D convolution of Eq. (17) is expressed as:

Image for - Determination of Gravity Anomalies over the Arabian Sea

Image for - Determination of Gravity Anomalies over the Arabian Sea
(22)

Image for - Determination of Gravity Anomalies over the Arabian Sea
(23)

Image for - Determination of Gravity Anomalies over the Arabian Sea
(24)

The corresponding spectral expressions for 2D spherical convolution and 2D-FFT of Eq. (17) given below:

Image for - Determination of Gravity Anomalies over the Arabian Sea
(25)

where

Image for - Determination of Gravity Anomalies over the Arabian Sea
(26)

Image for - Determination of Gravity Anomalies over the Arabian Sea
(27)

Image for - Determination of Gravity Anomalies over the Arabian Sea
(28)

where and represent the 2D-FFT operator and its inverse. With Eq. (28) the gravity anomalies at all gridded points are computed simultaneously.

RESULTS AND ANALYSES

The differences between the altimeter-derived gravity anomalies (Fig. 1 and 2) and the EGM96-derived gravity anomalies are equal to the residual gravity anomalies computed utilizing the inverse Vening Meinisz formula and they represent the short-wavelength component of the gravity anomaly signal.

Table 1: Statistics of the differences between EGM96-generated and altimeter-derived gravity anomalies via 1D-FFT over the Arabian Sea (unit, mgal)
Image for - Determination of Gravity Anomalies over the Arabian Sea

Table 2: Statistics of the differences between EGM96-generated and altimeter-derived gravity anomalies via 2D-FFT over the Arabian Sea (unit, mgal)
Image for - Determination of Gravity Anomalies over the Arabian Sea

Image for - Determination of Gravity Anomalies over the Arabian Sea
Fig. 1: 2.5'x2.5'Map of altimeter-derived gravity anomalies

Image for - Determination of Gravity Anomalies over the Arabian Sea
Fig. 2: 2.5'x2.5' 3D map of altimeter-derived gravity anomalies

Table 3: Statistics of the deflection of vertical over the Arabian Sea via 1D-FFT(unit, mgal)
Image for - Determination of Gravity Anomalies over the Arabian Sea

Table 4: Statistics of the deflection of vertical over the Arabian Sea via 2D-FFT(unit, mgal)
Image for - Determination of Gravity Anomalies over the Arabian Sea

Table 1 and 2 show the descriptive statistics of the differences between the ltimeter-derived gravity anomalies and the EGM96- derived gravity anomalies for 1D and 2D-spherical FFT, respectively, Table 3 and 4 show the statistics of the gravity anomalies over the Arabian Sea computed from deflections of vertical via 1D and 2D-spherical FFT, respectively.

CONCLUSIONS

This study describes the procedure and accuracy for marine gravity anomalies over Arabian Sea from multi-satellite altimetry. for the present research, it could concluded that the influence of the Dynamic Ocean Topography on the deflection of vertical should be taken into account in order to improve the accuracy of the determination of gravity anomalies. The gridded residual vertical deflections over unobserved area can be supplied by the corresponding model value when the inverse Vening-Meinesz formula is used to derive the marine gravity anomalies.

REFERENCES
1:  Cheng, L.J., N.J. Sheng, C.J. Yong and C.D. Bo, 2001. Determination of gravity anomalies over the South China sea by combination of Topex/Poseidon, ERS2 and geosat altimeter data. Acta Geodaetica Catographica Sinica, 30: 197-201.

2:  Dadzie, I., 2005. Derivation of gravity anomalies and geoid over the South China sea from multi-satellite altimetry data. Ph.D.Thesis.

3:  Emadi, S.R., M. Najafi-Alamdari, K.N. Toosi and M. Sedighi, 2003. Determination of the earth gravity field parameters in persian gulf and Oman sea with the satellite altimetry data. Islamic Azad University, Tehran, South Unit.

4:  Fu, L.L. and A. Cazenave, 2001. Satellite Altimetry and Earth Sciences: A Handbook of Techniques and Applications, International Geophysics Sciences. Vol. 69, Academic Press, San Diego, USA., pp: 463.

5:  Hwang, C., 1998. Inverse vening meinesz formula and deflection-geoid formula: Applications to the predictions of gravity and geiod over the South China Sea. J. Geodesy, 72: 304-312.
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6:  Jiang, W.P., 2001. The application of satellite altimetry in geodesy. Ph.D Thesis. Chinese Academy of Sciences.

7:  Smith, W.H.F. and D.T. Sandwell, 1994. Bathymetric prediction from dense satellite altimetry and sparse shipboard bathymetr. J. Geophys. Res., 99: 21803-21824.
Direct Link  |  

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