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Gravity Anomalies and Geoid Heights derived from ERS-1, ERS-2, and TOPEX/POSEIDON Altimetry in the Antarctic Peninsula Area

A. Marchenko, Z. Tartachynska, A. Yakimovich, F. Zablotskyj
National University “Lviv Polytechnic”, S.Bandera St., 12, Lviv, Ukraine (email: march@polynet.lviv.ua)

Abstract.

Gravity anomalies in the area between the Antarctic Peninsula and South America were determined using satellite altimetry observations of the Sea Surface Heights over this region. The satellite data applied in the analysis include ERS-1, ERS-2, and TOPEX/POSEIDON altimetry from 1992 to 2001. The solutions for gravity anomalies and geoid heights at (2' x 4') and (3' x 3') grid points, respectively, are evaluated by the Tikhonov regularization method. The estimation is based on the kernel functions described by singular point harmonic functions. Comparison with the independent KMS99 and KMS01 solutions of the Geodetic Division of the Danish National Survey and Cadastre was performed.

1. Introduction

This paper represents further continuation of the recent study by (Marchenko and Tartachynska, 2003) on the inversion of the Sea Surface Heights (SSH) into the gravity anomalies in the closed region of Black sea and will focus here on the recovery of the gravity anomalies and geoid heights N from the ERS-1, ERS-2, and TOPEX/POSEIDON altimetry (SSH) in the marine part of the region between the Antarctic Peninsula and South America. Computations of the gravity anomalies and geoid heights from the combination of ERS1, ERS2, and TOPEX/POSEIDON Sea Surface Heights (SSH) in the area at longitude from 60°W to 70°W and latitude from 60°S to 70°S are discussed, where the Ukrainian Antarctic station Vernadsky is located in the central part of the chosen region.

The following data sets corrected by CSL AVISO for different geophysical and instrumental effects are used:

The first 3x3E1E2TP solution for gravity anomalies and geoid heights is evaluated at the coordinates of the KMS2001 (3' x 3') grid points over the marine part of studying area by means of the Tikhonov regularization method using kernel functions (analytical covariance functions) described by singular point harmonic functions. The dependence of the regularization parameter on the variance of the studying field and the variance of the noise is considered. An optimal kernel function was adopted as the modified Poisson kernel or the so-called dipole kernel. The second 2x4E1E2TP solution at the coordinates of KMS1999 (2' x 4') grid points was constructed especially for further comparison with the KMS1999 gravity anomalies inverted by FFT method in the Geodetic Division of the Danish National Survey and Cadastre from multimission satellite altimetry data (see, Andersen and Knudsen 1998; Knudsen and Andersen, 1998). The comparison of 3x3E1E2TP geoid heights with KMS2001 3' x 3' solution is analyzed.

2. Method

As before (Marchenko, and Tartachynska, 2003) the traditional “remove-restore” procedure was used to get the initial information for further determination of the gravity anomalies ,

(1)

where SSH are the corrected Sea Surface Heights, assumed to be coincided with the geoid height N; NEGM96 is the long wavelength part of N adopted according to the EGM96 gravity field model (360, 360).

Then the prediction of the residual gravity anomalies and the residual geoid heights was estimated at some point P (preferably inside the studying area) applying the regularization method ,

(2) ,

(3)

where l is the q-vector consisting in this case of the components ; q is the number of the observations ; C is the (q x q) - covariance matrix of the residual geoid height is the (1 x q) - cross-covariance matrix between and ; is the (1 x q) ­ auto-covariance matrix of ; is the (q x q) ­ covariance matrix of the measurements noise n; is the Tikhonov regularization parameter (Neyman, 1979; Moritz, 1980; Marchenko and Tartachynska, 2003). Having the values (1) at some set of scattered points and the above covariance matrixes, the residual gravity anomalies and the residual geoid heights can be estimated straightforward at chosen grid points by the regularization method. After solving this basic problem the predicted gravity anomalies and geoid undulations N can be restored at the same grid by means of the EGM96 gravity field model ,

(4) .

(5)

For further use of the relationships (2), (3) the following problems have to be solved:

  1. The construction of the analytical covariance function K(P,Q) of the anomalous potential T.
  2. The choice of a suitable method for the computation of the regularization parameter .
  3. Preprocessing or prediction of 19959 regular distributed SSH values by the collocation method at 3' x 3' grid points, because of a large total number (=557954) of observations.

The analytical covariance function or reproducing kernel K(P,Q), described only by singular point harmonic functions (Marchenko and Lelgemann, 1998; Marchenko, 1998), is chosen in the following way

(6)

where R is the Earth`s mean radius; RB is the Bjerhammar’s sphere radius; rP and rQ are the geocentric distances to the external points P and Q; GM is the product of the gravitational constant G and the planet’s mass is the dimensionless potential of eccentric radial multipole of the degree n; ßn represents some dimensionless coefficient. Expressions for the analytical auto-covariance function of geoid heights and cross-covariance function between gravity anomalies and geoid undulations (based on the covariance propagation) can be found in (Marchenko, 1998).

Note now that the traditional determination of the regularization parameter in (2) or (3) according to (Tikhonov and Arsenin, 1974; Neyman, 1979) requires in the frame of a special iterative process the inversion of matrixes with a dimension equal to the number q of observations. So, when a number of observations are large we come to a time consuming procedure. As before (Marchenko, and Tartachynska, 2003) to avoid this difficulty another possible value of is used ,

(7)

leading to the estimation of prior to the matrix inversion in (2) and (3).

Simplest illustration of possible values of the regularization parameter a given by (7) can be made under several assumptions. First one, geodetic measurements of one kind only are considered. Second one, the matrix can be represented as =dI, where d is the variance of a noise and I is the unite matrix. Third one, the matrix C can be described by the Dirac delta function and can be written as C=C0I, where C0 is the variance of a studying field. With these assumptions the expressions for the regularization parameter corresponded to (7) are found as

= 1, (8) .

(9)

In fact, the first root (8) corresponds in (2) and (3) to the least-squares collocation solution. The second root (9) corresponds to the relationship (7) under the adopted assumptions and can serve for the illustration of a possible dependence of on the given C0 and d. Note again that the formulae (7) and (9) represent only possible upper limit of a (Marchenko, and Tartachynska, 2003), which requires a further improvement of the considered estimation of .

4. Results and Conclusions

Removing the contribution of the geopotential model EGM96 (360,360) from altimetry data (SSH) the residual geoid heights were adopted as initial information. Then the empirical covariance function (ECF) of the residual geoid heights was constructed and approximated by the analytical covariance function (ACF) based on the radial multipoles potentials or ACF of the so-called point singularities (Marchenko and Lelgemann, 1998). As a result, the optimal degree n = 1 in the formula (6) was chosen from ECF approximation that corresponds to the dipole kernel function (Poisson kernel without zero degree harmonics). The optimal ACF has the following essential parameters: (a) the variance of the field var()=0.1214 m_; (b) the correlation length ; (c) the curvature parameter =4.074.

Then to avoid a large total number of observations (=557954) the computation of the regular distributed SSH values at 19959 (3' x 3') grid points by the collocation method was made before the application of the relationships (2) and (3), using the nearest scattered SSH values around every grid point within the radius search = 5' (mean value of applied SSH for the prediction is equal 47) and Gaussian covariance function on this step. Fig. 1 illustrates such regular SSH data distribution at 3' x 3' grid points where predicted SSH values are known. In the following this regular (3' x 3') grid was adopted as initial information for the recovery of the gravity anomalies and geoid heights by the Tikhonov regularization method to obtain the solution 2x4E1E2TP at 27342 (2' x 4') grid points and the solution 3x3E1E2TP at 24356 (3' x 3') grid points filled all studying area.

 

Fig.1. Distribution of the predicted at 19959 points SSH values of the (3' x 3') regular grid

Table 1. Statistics of the predicted residual geoid heights and gravity anomalies at (2' x4') and (3' x 3') grid points

Statistics , m, (3' x 3') grid , mGal, (2'x 4") grid
Minimum -2.81 -90.48
Maximum 0.30 44.22
Mean -1.13 -15.81
Standard deviation 0.38 14.39

Fig. 2. Accuracy of the geoid prediction from ERS-1, ERS-2, and TOPEX/POSEIDON altimetry. Contour interval: 0.01 m

Fig.3. Accuracy of the gravity anomalies inversion from ERS-1, ERS-2, and TOPEX/POSEIDON altimetry. Contour interval: 1 mGal

Table 2. Statistics of the 2x4E1E2TP gravity anomalies and 3x3E1E2TP geoid heights restored at (2' x4') and (3' x 3') grids, respectively, and their accuracy estimations

Statistics 3x3E1E2TP solution 2x4E1E2TP solution
  N, m N, m , mGal mGal
Minimum 3.22 0.04 -112.06 3.91
Maximum 20.39 0.34 100.64 16.44
Mean 12.45 0.05 5.18 4.63

According to the expressions (2), (3) and (7), the prediction of the residual gravity anomalies and the residual geoid heights was done by the regularization method at the adopted grids points with the resolution (2' x4') and (3' x 3'), completely filled all marine part of the studying area. Note that the regularization parameter consists the value 3.55 computed according to (7). Statistics of the estimated and and their accuracy are shown in the Table 1. Accuracy distributions are shown in Fig. 2 and Fig. 3. Fig. 4 and Fig. 5 illustrate the gravity anomalies and geoid heights computed by the regularization method and based on the adopted 19959 (3' x 3') grid values SSH, obtained preliminary from ERS-1, ERS-2, and TOPEX/POSEIDON altimetry (see, Fig. 1).

Fig. 4. Gravity anomalies inverted from ERS-1, ERS-2, and TOPEX/POSEIDON altimetry. Contour interval: 10 mGal

Fig. 5. Geoid heights from ERS-1, ERS-2, and TOPEX/POSEIDON altimetry. Contour interval: 0.5 m.

Table 3. Comparison of the predicted (3' x 3') geoid heights and (2' x4') gravity anomalies with the KMS2001 and KMS1999 solutions, respectively

Statistic
Minimum -0.98 -89.46
Maximum 1.26 40.77
Mean 0.03 -14.95
Standard Deviation 0.16 8.72

Fig. 6. Differences between (2' x4') inverted gravity anomalies and KMS2001 solution. Contour interval: 10 mGal

 

Table 3 illustrates the comparison of the constructed above 3x3E1E2TP geoid solution and 2x4E1E2TP gravity anomalies, obtained from ERS-1, ERS-2, and TOPEX/POSEIDON altimetry, with the (3' x 3') KMS2001 SSH and (2' x4') KMS1999 gravity anomalies derived also from multimission satellite altimetry data. Note here a good accordance of the 3x3E1E2TP and KMS2001 solutions in terms of the mean and standard deviations of the predicted geoid heights. Nevertheless, we get large differences between 2x4E1E2TP and KMS1999 gravity anomalies demonstrated by the Table 3 and Fig. 6. These discrepancies have rather a systematic character, which possibly connects with the initial data distribution (see Fig. 1). On the one hand, larger differences have located typically around islands and near the seashore where initial data may be absent and inverted gravity anomalies reflect mostly the results of the extrapolation. This conclusion has confirmed by the accuracy distribution of the geoid heights and gravity anomalies shown in the Fig. 2 and Fig. 3, respectively. On the other hand, such deviations may be caused by difference in the adopted methods of data processing. The Tikhonov regularization method was applied in this paper for the whole studying geographical region without any separation to cells and based on the ERS-1, ERS-2, and TOPEX/POSEIDON data. In the case of (2' x4') KMS1999 solution the inversion of gravity anomalies was done by “piecewise processing” of multimission satellite altimetry within every (1° x5°) chosen rectangular cell using FFT method (Andersen and Knudsen 1998; Knudsen and Andersen, 1998). As a result, further improvement of the considered above solutions in the frame of the regularization method is expected after including gravimetry, GEOSAT altimetry, etc. data.

Acknowledgments.

We are very much indebted to AVISO for their support in receiving the corrected SSH of ERS-1, ERS-2, and TOPEX/POSEIDON altimetry used in this paper.

References

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