Calibration/validation of GOCE data by terrestrial torsion balance observations

Transcription

1 Calibration/validation of GOCE data by terrestrial torsion balance observations Gy. Tóth 1, J. Ádám 1, L. Földváry 1,4, I.N. Tziavos 2, H. Denker 3 1 Department of Geodesy and Surveying, Budapest University of Technology and Economics, P.O.B. 61, H-1521 Budapest, Hungary. and Physical Geodesy and Geodynamics Research Group of BUTE- HAS 2 Department of Geodesy and Surveying, Aristotle University of Thessaloniki, Univ. Box 440, Thessaloniki, Greece 3 Institute für Erdmessung, Universität Hannover, Schneiderberg 50, D Hannover, Germany 4 Institute for Astronomy and Physical Geodesy, Technical University of Munich, D Germany Correspondence to: Gy. Tóth ( gtoth@sci.fgt.bme.hu) Abstract One promising method for the external validation and calibration of the upcoming GOCE satellite mission data is the use of ground gravity field data continued upward to satellite altitude. There is a unique situation for Hungary in this respect since surface gravity gradients are available at points over an approximately km 2 area, measured by the classical Eötvös torsion balance. The concept of this contribution is to test the usability of these point gravity gradient observations for upward continuation to the GOCE satellite orbit in combination with different geopotential models and other gravity field information. The computations are based on the least squares collocation method and the direct numerical integration of the torsion balance data. For the latter method, the spectral combination technique and the classical integration kernels are considered. Furthermore, various other data sources, such as the T gravity gradients based on the gravity and terrain data collected within the frame of the European Geoid Project, are utilized for comparisons. Besides the comparisons between the different satellite gravity gradient computations, an error analysis of the results is presented. Keywords. GOCE, vertical gravity gradients, upward continuation, Eötvös torsion balance 1 Introduction Ground gravity field data will be useful for calibration/validation of the gravity gradients that are to be measured by the upcoming GOCE satellite gravity field mission (Arabelos and Tscherning, 1998; Denker, 2002). In this paper the possibility of using ground gravity field gradients for calibration of Level 1b gravity gradients of GOCE is investigated by using gravity field information provided by the Eötvös torsion balance. The intent is instrument calibration and/or data validation. As it is well known, torsion balance data provide short wavelength gravity field information, and therefore it is interesting whether these data can be used in the context of space gravity field data calibration. It is because short wavelength information is mainly lost when we upward continue our ground gravity field data. Another question is whether the spatial extension of our data is enough for upward continuation to the GOCE altitude (approximately 250 km). In this respect the truncation error is expected to be quite critical. The issue of the combination of gravity anomalies and gravity gradient data is also important to be investigated, and the question is whether this technique can yield accurate enough gravity field predictions for the GOCE orbit. The algorithms used in this study were the least squares collocation and spectral combination method. We have used collocation because of its flexibility, particularly its capability to use and to predict various gravity field quantities and since it provides statistical quantities, estimates of covariances and cross-covariances 2 Torsion balance data The original design of the Eötvös torsion balance goes back to Since then a great number of such measurements were carried out in Hungary, mainly for oil field exploration. The approximate number of the measurement sites is in Hungary, but only about of them are stored in a database. The processing of the old field books is still going on at the Loránd Eötvös Geophysical Institute. The data distribution can be seen in Fig. 1. In the plot also those areas are marked where there are still unprocessed field books of measurements exist. At each station the topographic effect of the nearby and remote terrain was also calculated and stored in the database. The standard error of the W xz W yz data is estimated to be about 1.2 E (Eötvös) and this figure is slightly greater for the topographic

2 effect. The W yy-xx W xy gravity gradients have a greater error of about 2 E. Throughout this study 2 E error standard deviation was used for the spectral combination computations using the W xz W yz data and 1 and 2 E for the collocation computations using both W xz W yz and W yy-xx W xy gravity gradients, respectively. The statistics of all the torsion balance gravity gradients and their topographic effects are shown below in Table 1. In all further computations the topographic effects were removed from the gradients and also the normal effect on the W xz and W yy-xx observations were taken into account (see for example Hein,1981, p 21). Table 1. Statistics of Hungarian torsion balance data and their topographic effects. All figures shown are in E (Eötvös). Data set no. min max mean std W xz t xz W yz t yz W yy-xx t yy-xx W xy t xy Fig. 1. Location of torsion balance measurements in Hungary (status: January 2003.). Hatched areas indicate places where torsion balance measurements exist but not yet stored in the database 3 Methodology Least squares collocation. The standard formulation of the well-known collocation method is not repeated here. The reader is referred to, for example, Tscherning (1994). Spectral combination technique The least squares spectral combination method (Wenzel, 1982) in our case, yields an optimal combination in the spectral domain of two or more gravity field quantities to obtain second vertical derivative at satellite altitude. The gravity field quantities at a point with spherical polar coordinates ( r,, which are used in the combination are the T vertical gravity gradients, computed from torsion E balance observations T, gravity anomalies T G, and from a geopotential model T M. These are combined in a least squares sense to yield the estimate ˆ M, + T G E T = T φ + T. (1) Here the torsion balance part with spectral weights w l can be written by introducing the mean Earth radius R in the form of T E = l= 2 R r l+3 w δt l l ( λ ), (2)

3 where 1 2l * * δtl ( = Pl ψ ( Txz α Tyz α dσ π l (cos ) cos + sin ) 4 σ (3) is the degree l surface spherical harmonic of T coming from torsion balance measurements T,, which are reduced by the effect of the xz T yz geopotential model and topography. In the above 1 equation (cosψ ) denotes associated Legendre P l functions of order 1, ψ is the spherical distance between evaluation and data point, and α* is the azimuth of the evaluation point measured at the data point. For further explanation of eq. (3) the reader is referred to van Gelderen and Rummel (2001) and Tóth et al (2002). We have followed a two-step process of first converting T xz, T yz data to T at the ground level (zero height) using the appropriate kernel function E ψ ) in T 1 + = E ψ )( Txz cosα * Tyz sinα dσ, 4 *) π σ (4) and then the formulas of spectral combination with spectral weights were used. The kernel function E ψ ) is plotted on Fig 2. We can see that it facilitates the computation of vertical gravity gradients on the ground level well since it is only necessary to integrate the torsion balance data up to a reasonably small integration radius of about as input. The prediction was taken at 595 points of the simulated GOCE orbit. We have simulated GOCE gradient observations for 3 months of 2006, between and Our noise-free GOCE orbit simulation was counting for the gravitational effect of the Earth (using EGM96 model), and the tidal effects of the Sun and the Moon, plus oceanic tides (Schwiderski model was applied). The initial conditions of the satellite orbit were providing similarities to a reliable GOCE scenario, with a mean altitude of 255 km, a mean inclination of 96.5º, and a mean eccentricity of Then, the gravity gradients were computed/simulated along this orbit. No noise simulation has been perfomed. The estimated covariance function of W xz, W yz that was used for the predictions is illustrated in Fig 3. Fig 3. Covariance function of W xz, W yz gravity gradients between zero height of data and the mean prediction level of 250 km. The simulated gravity gradients were transformed from the local orbital frame to the local North, East, Up frame and compared with the gravity gradients of the EGM96 gravity field model up to degree and order 120. The residuals were very small, the standard deviation was only me. Since we have used the same geopotential model for the collocation prediction, the residuals with respect to the simulated GOCE gravity gradients would be the same. Hence no further comparison was meaningful in this respect. Fig 2. Normalized kernel function E (r,ψ) to transfer (T xz, T yz ) torsion balance data into T vertical gravity gradients at zero height h=0 (cf. eq. 4). 4 Numerical experiments In the collocation experiment we used all the four gravity gradients (combinations) measured by the torsion balance. To reduce the number of observations we have selected 936 stations of W xz, W yz and 938 stations of W yy-xx, W xy closest to the nodes of a regular geographical grid. Hence altogether 3748 gradients, reduced by the EGM96 gravity field up to maximum degree 360, were used Table 2. Statistics of predictions at 595 simulated measurement sites. All units are me. data set Min Max mean Std W xz W yz W yy-xx W xy The statistics of the predicted residual quantities can be found in Table 2. The prediction error of these quantities varies between 4-6 me and on average is about 5.5 me. We can see that the predicted gravity gradients are very small. This may be attributed partly to the smoothing property of the

4 collocation but most significantly to the relatively small data collection area of the gradients. The correlation length of the covariance function of residual gravity gradients at altitude (Fig 3.) is considerably larger than the corresponding correlation length (about 2.5 km) at ground level. This means we would need an extended dataset to reduce the truncation error. The truncation error will be investigated numerically below for the spectral combination method. The next numerical experiment was made by the spectral combination technique. First, the torsion balance T xz, T yz data were gridded on a grid. The extent of this grid was in latitude and longitude, respectively. Second, as we mentioned, T vertical gravity gradients were computed by integration over the grid at the ground level by eq. (4). For the statistics of these input grids the reader is referred to Table 3. Third, spectral weights were computed using an exponential error covariance function for both gravity anomalies and vertical gravity gradients. For gravity anomalies, which were available over a grid, the Bψ C( ψ ) = Ae model was used with parameters A=2 [mgal 2 ], B=4 [1/ ]. This value of B means a 19 km error correlation length of gravity anomalies, the same value that was used for computing the EGG97 European Gravimetric Geoid (Denker and Torge, 1998). The T vertical gravity gradients were assigned the same error model but with A=2 [E 2 ], B=7.7 [1/ ]. This value of B means 10 km correlation length, a slightly smaller value than for the gravity anomalies. Since we have no real information on the error covariance function of torsion balance data, the correlation length chosen is only a guessing. Table 3. Statistics of input grids used for the spectral combination solution. The reference field was the EGM96 geopotential model up to degree and order 360. data set Min Max Mean Std T xz [E] T yz [E] T [E] g [mgal] Fig 4. Spectral weights assigned to gravity anomalies and vertical gravity gradients computed from exponential error covariance models (Case B). The dashed lines show spectral weights for the case of equal correlation length of about (19 km) of g and T. We have computed two different solutions by spectral combination. The first one (Case A) was made using the error degree variances of the EGM96 geopotential model and the ground T data only (up to spherical harmonic degree and order 360). The second solution (Case B) was computed by combining gravity information from the EGM96 geopotential model, vertical gravity gradients and gravity anomalies (triple combination). The spectral weights for Case B combination as function of spherical harmonic wavelength can be seen on Fig 4. The dashed lines show the (slightly different) weights for equal correlation length of gravity anomalies and gravity gradients. It is remarkable that gravity anomalies dominate the solution between the km wavelengths. This means that a very little contribution is to be expected from surface gravity gradients at the altitude of GOCE. This is confirmed by the results in Table 4. where a 2.3 me Rms signal was computed in the Case B solution from T data. The spectrally modified kernel functions with the spectral weights assigned for the EGM96 error degree variances and the above two error models can be seen in Fig 5. Fig. 5. Integral kernels for the spectral combination solution. The kernels are different for T when these data are used alone (Case A) or in combination with g (Case B). To carry out an omission error analysis the truncation error was estimated using the spectral transfer function technique (Wenzel, 1982), adapted to our case. Due to space restrictions we do not give the formulas here, instead we present some plots. The truncation error as function of the truncation cap size for the Case A (T -only) and Case B (T + g) spectral combination solutions is plotted in Figs. 6. and 7. Looking at these plots it can be seen that we have a large truncation error when the torsion balance data are integrated up to about 1, as is the case with our data. The truncation error is obviously smaller (about 1.5 me Rms) for Case B solution, but in this case also the signal coming from T data is small. This serious truncation error may explain the large differences, discussed below, between this computation and the one computed at

5 the Insitute für Erdmessung (IfE solution, see Denker, 2002). Fig. 6. Truncation error for different cap sizes for the spectral weights assigned to the T -only solution (Case A). Fig. 8. Differences for T computed using Case B (T + g) spectral combination solution and that of based on the European gravity data. Fig. 7. Truncation error for T at different cap sizes for the spectral weights assigned to the T + g solution (Case B). Finally, a comparison of T gravity gradients over Hungary took place at 5 5 grid points with that of computed at the IfE from EGG97 gravity data by spectral combination technique (Denker, 2002). The differences of the vertical gravity gradients for the T + g solution can be seen on Fig 8. The same differences are shown on Fig 9. for the the T -only spectral combination solution. The statistics of the results and differences of Case A and Case B solutions with respect to the IfE solution are shown in Table 4. It can be seen that the residual values computed in this experiment are smaller (especially for Case A) and go zero outside Hungary. This also indicates significant truncation errors in the results. The two spectral combination solutions (Case A and B) show big differences. The solution with gravity anomalies (Case B) agrees better with the IfE solution at least they show a similar low over Hungary in the residual vertical gravity gradients. But even in this case we expect large truncation errors in the integration of gravity anomalies since our gravity data are extended over a area only. Table 4. Statistics of T T (EGM96) residuals computed at H=250 km by spectral combination (Cases A and B) and differences with respect to the solution based on the European gravity data (IfE). All units are me. data set min max mean std Case A Case A IfE Case B Case B, contribution from T Case B IfE Fig. 9. Differences for T computed using the Case A (T -only) spectral combination solution and that of based on the European gravity data

6 5 Conclusions, recommendations The most important conclusion that can be drawn from the above study is that the torsion balance data in their present data coverage fail to perform a meaningful calibration/validation for the (relatively) high altitude of the GOCE satellite. This is mainly because we found the ommission error at this altitude very serious for torsion balance gravity gradients. The comparisons with the IfE solution supports our conclusion in this respect. Moreover, if one considers the anticipated accuracy of the GOCE gravity gradients of about 4 me/ Hz for the mhz measurement bandwidth, 1 me accuracy of the calibration/validation gradients is required. As we have seen, the truncation errors exceeds this accuracy by at least a factor of 10. Another factor in explaining the differences is the supplementing character of ground and space gravity gradients. Space gravity gradient data correlate best with ground gravity anomalies and/or geoid heights, not with ground gravity gradients. This is due to the smoothing effect of upward continuation. Therefore we conclude one may use torsion balance data only as a supplementary information giving a few me signal, in addition to a gravity anomaly dataset of good data extent as it was the case of computations by Denker (2002). This combination would be a meaningful one. On the other hand, torsion balance data are still valuable in checking ground gravity field information. As it can be seen from the spectral weights in Fig 4, below of about a 30 km wavelength torsion balance data may provide better gravity field modeling than gravity anomalies. Therefore it would be interesting to use these data in local geoid determination using the spectral combination technique as discussed in this paper above. In the context of GOCE, torsion balance data can support gravity field determination in the short wavelengths, due to their above mentioned supplementary character. Acknowledgements The GOCE simulation has been performed by the orbit integrator at the IAPG, Technical University of Munich, developed by M. Wermuth, B. Frommknecht and P. Steigenberger. The support of the Greek-Hungarian Interngovernmental Scientific and Technology Fund is also acknowledged. References Arabelos, D., and C.C. Tscherning (1998). Calibration of satellite gradiometer data aided by ground gravity data. Journal of Geodesy, 72: Denker, H. and Torge, W. (1998). The European gravimetric quasigeoid EGG97 An IAG supported continental enterprise. In: R. Forsberg et al. (eds.), Geodesy on the Move Gravity, Geoid, Geodynamics, and Antarctica, IAG Symp. Proceed., Springer Verlag, Berlin, Heidelberg, 119: Denker, H. (2002). Computation of Gravity Gradients for Europe for Calibration/Validation of GOCE Data. Proceedings of the Gravity and Geoid rd Meeting of the Internat. Gravity and Geoid Comm., Aug , Thessaloniki, Hein, G.W. (1981). Untersuchungen zur terrestrischen Schweregradiometrie. Dissertationen Reihe C 264, Deutsche Geodätische Kommission, München. Tóth Gy., Rózsa Sz., Ádám J. and Tziavos, I.N. (2002). Gravity field modeling by torsion balance data a case study in Hungary. In Ádám J. and Schwarz, K-P., editors, Vistas for Geodesy in the New Millennium, Vol 125. IAG Symposia, pp Tscherning, C.C. (1994). Geoid determination by Leastsquares Collocation using GRAVSOFT. In: International School for the Determination and Use of the Geoid, Milan, October 10-15, 1994, pp van Gelderen, M. and Rummel, R. (2001). The solution of the general boundary value problem by least squares. Journal of Geodesy, 75:1-11. Wenzel, H.G. (1982). Geoid computation by least squares spectral combination using integral formulas. Proceed. Of the General Meeting of the IAG, Tokyo, May 7-15, pp