Possible geodetic applications of Eötvös´
torsion balance gravity gradient data were investigated in
the present study. As a practical example gravity data have
been predicted for two test areas in Hungary with the method of
least squares collocation. By evaluating the results with
gridded data interpolated from gravity measurements the standard
deviation of differences &plusmn; 1-1.7 mGal was found which proves the
usability  of such gradient data for geodetic applications

Tóth, Gyula

English

Possible geodetic applications of Eötvös ’ torsion balance gravity gradient data were investigated in the present study. As a practical example gravity data have been predicted for two test areas in Hungary with the method of least squares collocation. By evaluating the results with gridded data interpolated from gravity measurements the standard deviation of differences ±1 − 1.7 mGal was found which proves the usability of such gradient data for geodetic applications

Gyula Tóth

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PREDICTION BY EÖTVÖS ’ TORSION BALANCE DATA IN HUNGARY

Periodica Polytechnica (Budapest University of Technology and Economics)

PERIODICA POLYTECHNICA SER. CIV. ENG. VOL. 46, NO. 2, PP. 221–229(2002)PREDICTION BY EÖTVÖS’ TORSION BALANCE DATA INHUNGARYGyula TÓTHDepartment of Geodesy and SurveyingBudapest University Technology and EconomicsH–1521 Budapest, Hungarye-mail: gtoth@geo.fgt.bme.huReceived: Jan. 25, 2002AbstractPossible geodetic applications of Eötvös’ torsion balance gravity gradient data were investigated inthe present study. As a practical example gravity data have been predicted for two test areas inHungary with the method of least squares collocation. By evaluating the results with gridded datainterpolated from gravity measurements the standard deviation of differences±1 − 1.7 mGal wasfound which proves the usability of such gradient data for geodetic applications.Keywords: Eötvös’ torsion balance, least squares collocation.1. IntroductionNowadays there is an increasing interest in measuring gravity gradients of theearth’s gravity field. Mainly these measurements are intended to be made on mo-bile (air, marine and space mounted) platforms (BELL et al., 1997; PAWLOWSKI,1998). Undoubtedly an advantage of these gradient measurements is their relativeinsensitivity to small platform accelerations which constitutes a principal problemfor aerial gravimetry, for example. Moreover, for space gradiometric devices (e.g.GOCE, VISSER, 1999) the exponential gravity signal decrease for high altitudesis counteracted by the signal amplification at shorter wavelengths due to the dou-ble differentiation for second-order gradients of the gravity field (RUMMEL et al.,1993).Beside the planned satellite mission it is a historical fact that gravity gradio-metric measurements for the first time were successfully taken at the surface ofthe earth due to Loránd Eötvös’ terrestrial gradiometric device, the torsion balance.This means an advantageous position for Hungary since a great deal of gradiometricmeasurement data has been collected here. Even in time of Eötvös these data havebeen used for geodetic tasks as well (see HOMORÓDI, 1966). In our present paperwe would like to highlight some ideas about the application of this kind of data togeodesy by taking into account not only modern computing facilities but also recenttheoretical developments in the solution of the Geodetic Boundary Value Problem(GBVP).222 GY. TÓTHAt first we introduce shortly the principle and history of measurements takenby Eötvös’ torsion balance device. Then we continue with Hungarian torsion bal-ance measurements highlighting the data collected at ELGI (Eötvös Loránd Geo-physical Institute). Then a short discussion will present how it is possible to usesuch data in solving the GBVP. Finally a prediction experiment by the method ofleast squares collocation will show how to get gravity values from torsion balancegradient measurements.2. Measurement Principle of Eötvös’ Torsion Balance and a Short History ofMeasurementsFig. 1. Principle of Eötvös’ torsion balanceThe principle of measurement with the Eötvös’ torsion balance is shown inFig. 1. This device was called by Eötvös horizontal variometer, and through gravitychange between the two massesm a force arises which exerts a small moment onthe horizontal swinging arm of length 2l. This moment is counteracted by themoment exerted by the threadw, and the corresponding angle position can be reador photographically registered on film for certain kind of torsion balance devices.As of the original torsion balance, at each site the readings should be taken in5 different azimuths in order to determine the corresponding 5 unknowns (readingfor the torsion-free position, two gradient and two curvature parameters). Laterin a more advanced design two such balances were mounted at opposite positionto each other and hence readings had to be taken at only 3 azimuths to computethe 6 unknowns. (twin/double balances). We would like to mention that recentlyPREDICTION BY EÖTVÖS’ TORSION BALANCE DATA IN HUNGARY 223DOROBANTU (1999) has done research at the University of Munich to equip thedevice with automatic electronic reading.The first measurements with torsion balance were led by Eötvös himself, firstat the foot of Gellért Hill (1889), then at Sághegy near Celldömölk (1891), thenon the ice sheet of Lake Balaton (winters of 1901–1903). The measured gravitygradients were in agreement with lake bathymetric data. After the first successfultests for resource exploration at Morvamező near Egbell (Gbely) in 1916, the torsionbalance device gained an increasing field of application for gravity exploration inGermany, Hungary and Bohemia. The first hydrocarbon reservoir in the USA wasdetected by the device in 1924 (Nash Dome).The introduction of astatized spring gravimeters by the end of the thirties madean end to the torsion balance measurements in the United States. The reason behindthis was not only the large field work required by the torsion balance (the terrainaround the field point had to be carefully levelled along eight directions for terraincorrection computations), but also the interpretation of the gradient and curvaturedata was not so straightforward as for gravimetricg data. An assertion to thisis the fact that the curvature data measured were simply discarded for resourceexploration due to its interpretation difficulties. Torsion balance measurements,however, are continued in Europe; for example in Hungary even in the sixties it wasused for field work.3. Torsion Balance Measurements and Data in HungaryIn the external gravity field three elements of the gradient tensorE (Eötvös orMarussi tensor), and the difference of two other elements, altogether 4 data fromthe 5 independent elements can be determined from torsion balance gradiometry.IfE = grad g =[Wx x Wxy WxzWxy Wyy WyzWxz Wyz Wzz]denotes the elements of the Eötvös tensor, thenWxz, Wyz, Wxy andW = Wyy−Wx xcan be measured by the balance. It is customary to call the first two of them,(Wxz, Wyz) components of gradients, and the other two, (Wxy andW) componentsof curvatures sinceWxz = gx , Wyz = gy. are components of the horizontal gravitygradient vector andWxy andW are in connection with the curvature of the levelsurface.Conventionally the gradient values are illustrated by vectors and curvaturesby line segments, respectively, as illustrated inFig.2. The following equationstanα = WyzWxz, Wsz =√W 2xz + W 2yz224 GY. TÓTHFig. 2. Gradient and curvature componentscan be used to compute the azimuth and magnitude of gradient vectors, andκ =Wsz/Wszg connects the magnitude with curvatureκ of the plumb line. The differ-ence of main curvature parametersR = g(1rmin− 1rmax)=√W 2 +(2Wxy)2can be expressed, as well as the azimuth of maximal curvature from the expressiontan 2α = −2WxyW.This may also be plotted on a map as a line segment of azimuthα and lengthR.Original files of measurements with the torsion balance are stored at theLoránd Eötvös Geophysical Institute (ELGI). In the past this material has beentreated somewhat carelessly henceforth some original files have been lost and theycannot be recovered. During the years 1995-2000 in the framework of a researchproject 11795 point gradient and curvature data have been saved to computer filesand given to Technical University of Budapest by ELGI. These data mainly coverthe central part of Hungary (data distribution is shown inFig.3). This figure alsoshows two test areas chosen (the larger one, ‘A’ is mostly plain area while the otherone, ‘B’ is a hilly area with moderate topography). These areas were chosen forextensive tests and these will be described in much more detail later on in the presentstudy.A statistical analysis was performed on the above test data including gradientand curvature parameters and topographic effects, also contained in the originalfiles. These statistics are inTable 1 below.PREDICTION BY EÖTVÖS’ TORSION BALANCE DATA IN HUNGARY 225Table 1. Statistics of gradients and curvatures for test areas A and B (number of data: 752and 691 resp.; Eötvös Unit, 1E.U.= 10−9 s−2), t – topographic effect, and thelinear correlation coefficients of a specific gradient/curvature parameter and itstopographic correctionArea A Wxz Wyz 2Wxy W txz tyz 2txy tMin. –16.3 –22.9 –29.1 –43.2 –10.3 –11.4 –36.7 –28.2Max. 42.5 27.6 32.3 33.7 8.5 13.3 22.4 16.2Mean 8.5 0.6 4.3 0.5 0.0 –0.3 0.2 –0.1Std.dev. ±6.9 ±6.4 ±8.2 ±10.0 ±2.3 ±2.6 ±5.0 ±4.8Corr.coeff. 0.300 0.419 0.488 0.455Area B Wxz Wyz 2Wxy W txz tyz 2txy TMin. –38.4 –65.8 –175.3 –324.3 –42.3 –38.4 –167.9 –292.6Max. 75.9 67.4 299.7 216.0 35.1 38.9 257.7 163.1Mean 13.1 –8.6 2.1 –7.8 1.5 –2.4 –1.4 –5.5Std.dev ±15.1 ±17.3 ±51.6 ±65.9 ±7.9 ±8.4 ±44.6 ±56.4Corr.coeff. 0.649 0.708 0.927 0.918The high correlation is very pronounced with topography for area B of gradi-ents and especially of curvatures. This confirms the well-known fact that the torsionbalance measurements are very sensitive to nearby topographic masses, somethingmust be noted for geodetic applications as well.4. Possible Utilization of Torsion Balance Measurements in GeodesyTorsion balance measurements have already been used for geodetic purposes. Themathematical tool used was line integration of gradients and curvatures, whichallowed to express differences of gravity and deflection of the plumbline betweenany two points. Following this technique it became possible to interpolate gravityand deflections of the vertical in a whole measurement network when some pointshave known gravity or deflection values (VÖLGYESI, 1998).A great theoretical achievement in the solution of GBVPs can be thanked toGELDEREN and RUMMEL (2001). They for the first time derived certain combina-tions of the Eötvös tensor that can be used to solve the GBVP and the correspondingkernel functions that may be used in surface integrals for the solution, just as theStokes’ integral for gravity anomalies. By using these integrals, combinations ofWxz, Wyz and 2Wxy , W of torsion balance measurements will be boundary valuesfor the determination of the outer gravity field (parametrized e.g. by the disturbingpotential). The practical application of these integrals is an open field for research.226 GY. TÓTHp Fig. 3. Distribution of torsion balance measurement sites. The upper rectangle shows testarea ‘B’, the other shows ‘A’. The coordinate system is GSR80,geodetic coordinatesA more traditional approach in physical geodesy is the least squares collo-cation (TSCHERNING, 1994), the advantage of which lies in the combination ofdifferent kinds of gravity measurements in the gravity field determination process,and this of course includes torsion balance data as well. Another advantage is thatany other parameter of the gravity field can be obtained in one computational step.An example can be found in literature for its application for the GGSS gravity gradi-ent data (JEKELI, 1993) processing (ARABELOS and TZIAVOS, 1992; ARABELOSand TSCHERNING, 1999). A known limitation of the method is that the numberof measurement points cannot go beyond several thousand due to the size of theequations to be solved during collocation. This method was chosen for the presentstudy to predict gravity anomalies for the two test areas by using only (Wxz, Wyz)gradients measured by the torsion balance.5. Prediction of Gravity Anomalies by Least Squares CollocationGravity anomalies have been used for prediction on the one hand since it can servea good starting point for numerous geodetic and geophysical applications (geoiddetermination, gravity field interpretation, inverse problems), and on the other handa grid of interpolated Faye anomalies for the two test areas was available and thesedata served well for evaluation purposes. We have concentrated on using gravityPREDICTION BY EÖTVÖS’ TORSION BALANCE DATA IN HUNGARY 227gradients although usage of curvature components in collocation poses no problemor even the combination of the two is easily manageable.The first step in gravity field modelling by collocation is to remove all possibleshort and long wavelength effects (trends) from our data in order to make theresiduals more smooth, easily interpolatable, and with zero mean. The so callednormal effect (VÖLGYESI, 1982) has been removed for the purpose as well as long-wavelength gradient components of the EGM96 geopotential model and the abovementioned topographic effect.The normal effect and the geopotential model are only parts of the trendremoval. Since there is no way to remove a localized trend effect by applying thesecorrections only, we decided to take into account a local trend in reduced (Wxz, Wyz)gradients before determining empirical covariance functions. This process wassimilar to the one used by HEIN and JOCHEMCZYK (1979) for the torsion balancedata in Germany when modelling local covariance functions from torsion balancegradients. Two kinds of such local trend models – as functions of the position –were applied to reduced gradient components:• Linear trend function• Quadratic trend functionIn the next step an analytic covariance model has to be fitted to the empiricalcovariance function through an essentially iterative process. Then parameters ofthe analytic covariance function of Tscherning–Rapp Model 2 withB = 4 in thedenominator (see TSCHERNING, 1994, Eq. 19) were yielded.The last step was to compile the job file for GEOCOL program written byTscherning in Fortran. The collocation step predicted residual gravity anomalieswhich do not contain the gravity effects of the geopotential model in addition to thenormal effect and topography.Finally these gravity predictions were compared with interpolated 1′ × 1.5′residual Faye anomalies corrected also to the effect of the EGM96 geopotentialmodel. This comparison is valid since both values are topographically reducedones. Statisics of the differences for the two test areas are summarized inTable2for the cases of linear and quadratic trend models.The standard deviation of differences is seemingly better for linear trendmodel and for the plain area, what of course, was something to be expected. The±1 mGal standard deviation of differences for the plain area can be said to be verylow especially if we take into account the fact that according to PAPP (1993) thesame prediction error of gravity anomalies is to be expected by using the pointgravity anomalies themselves!Our results are in harmony with the study of VÖLGYESI (1998) as well, whonoted differences around±0.60′′ − 0.65′′ in deflections at evaluation points by in-terpolating these data from curvatures measured by the torsion balance. At the sametime according to TÓTH et al. (2000) the standard deviation of differences betweenastronomically measured and gravimetrically predicted deflection components isabout the same value,±0.5′′ − 0.6′′. These results confirm the conclusion that228 GY. TÓTHTable 2. Statistics of the differences of residual gravimetric and (W xz, Wyz)-predicted grav-ity anomalies for test areas A and B. (units are mGal).T 2 denotes linear,T 3quadratic trend, respectively, andF indicates that a least squares fitting plane hasbeen removed from the differencesA–T2 A–T2–F A–T3 A–T3–F B–T2 B–T2–F B–T3 B–T3–FMin. –2.52 –4.00 –2.14 –4.54 –11.95 –3.37 –12.99 –3.61Max. 3.66 2.21 4.77 2.09 3.31 4.08 1.50 4.95Mean 1.30 0.00 1.38 0.00 –5.31 0.00 –6.07 0.00Std.dev. ±1.10 ±1.02 ±1.36 ±1.05 ±3.96 ±1.67 ±3.82 ±1.77torsion balance measurements may be used for the determination of certain gravityfield parameters as well as gravity anomalies.6. Conclusions and OutlookOur studies for two selected test areas have demonstrated the fact that the Hungar-ian torsion balance measurements can be advantageously applied for gravity fielddetermination. This means that these data can be combined with gravity anomaliesfor more reliable gravity field determination. However, they in themselves maybe appropriate to solve the GBVP. Because of the great number of such data thecollocation method meets numerical difficulties, hence we propose to apply othernumerical methods as well (e.g. FFT-based numerical integration methods).Besides the gradient components, the application of curvature parametersneeds more studies, since according to the work of ARABELOSand TZIAVOS (1992)this may further improve the prediction. Also a detailed comparison with the resultsof line integration methods is planned.It can be told that Hungary has a very advantageous position as regards terres-trial gradiometry and it would be pity to let these valuable gradient measurementsgo unused that was collected (although for resources exploration) through laboriousfield work during decades. The present study wants to point out that practical workwith this exceptional kind of data may yield fruitful results in the future for geodesy.AcknowledgementThe author acknowledges the financial support kindly provided by the National ScientificResearch Fund (T030177), as well as of the Physical Geodesy and Geodynamics ResearchGroup of the Hungarian Academy of Sciences for the research involved in the present paper.PREDICTION BY EÖTVÖS’ TORSION BALANCE DATA IN HUNGARY 229References[1] A RABELOS, D. – TSCHERNING, C. C., Gravity Field Recovery from Airborne GradiometerData Using Collocation and Taking into Account Correlated Errors,Phys. Chem. Earth (A), 24No. 1 (1999), pp. 19–25.[2] A RABELOS, D. – TZIAVOS, I. N., Gravity Field Approximation Using Airborne Gravity Gra-diometer Data,Journal of Geophys. Res., 97 No. B5 (1992), pp. 7097–7108.[3] BELL, R. E. – ANDERSON, R. – PRATSON, L. F., Gradiometer Spinning onto Gulf Trends,The American Oil & Gas Reporter, 1997.[4] DOROBANTU, R., Gravitationsdrehwaage.IAPG/FESG, No. 4 (1999), Technische UniversitätMünchen.[5] van GELDEREN, M. – RUMMEL , R., The Solution of the General Geodetic Boundary ValueProblem by Least Squares,Journal of Geodesy, 75 (2001), pp. 1–11.[6] HEIN, G. – JOCHEMCZYK, H., Reflections on Covariances and Collocation with Special Em-phasis on Gravity Gradient Covariances,Manuscripta Geodaetica, 4 (1979), pp. 45–86.[7] HOMORÓDI, L., Felsőgeodézia (Geodesy), Tankönyvkiadó, Budapest, 1966, (in Hungarian).[8] JEKELI , C., A Review of Gravity Gradiometer Survey System Data Analyses,G ophysics, 58No. 4 (1993), pp. 508–514.[9] PAPP, G., Trend Models in the Least-Squares Prediction of Free-Air Gravity Anomalies,Peri-odica Polytechnica, 37 No. 2 (1993), pp. 109–130.[10] PAWLOWSKI, B., Gravity Gradiometry in Resource Exploration, The Leading Edge, January1998.[11] RUMMEL , R. – SANSÓ, F. – van GELDEREN, M. – BROVELLI , M. – KOOP, R. – MIGLI -ACCIO, F. – SCHRAMA, E. J. O. – SACERDOTE, F., Spherical Harmonic Analysis of SatelliteGradiometry.Netherlands Geodetic Commission, New Series,39 (1993), Delft.[12] TÓTH, GY. – RÓZSA, SZ. – ANDRITSANOS, V. D. – ÁDÁM , J. – TZIAVOS, I. N., Towardsa cm-Geoid for Hungary: Recent Efforts and Results,Phys. Chem. Earth(A), 25 No. 1 (2000),pp. 47–52.[13] TSCHERNING, C. C., Geoid Determination by Least-Squares Collocation Using GRAVSOFT.in: International School for the Determination and Use of the Geoid, Milan, October 10–15,1994, pp. 135–164.[14] VÖLGYESI, L., Geofizika, (Geophysics) Tankönyvkiadó, Budapest, 1982, (in Hungarian).[15] VÖLGYESI, L. Geoid Computations Based on Torsion Balance Measurements,Second Conti-nental Workshop on the Geoid in Europe, Reports of the Finnish Geodetic Institute, 98 No. 4(1998), Masala, Finland, pp. 145–151.[16] VISSER, P. N. A. M., Gravity Field Determination with GOCE and GRACE,Advances inSpace Research, 23, No. 4 (1999), pp. 771–776.

PREDICTION BY EÖTVÖS´ TORSION BALANCE DATA IN HUNGARY

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PREDICTION BY EÖTVÖS´ TORSION BALANCE DATA IN HUNGARY

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