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Processing gravity data in the territory of AndalusiaAntonio J. Gil a & Gracia Rodríguez‐Caderot b

a Dept. Ing. Cartográfica, Geodésica y Fotogrametría, Escuela Politecnica Superior ,Universidad de Jaén , Jaén, 23071, Spainb Dept. Astronomía y Geodesia, Facultad Ciencias Matemáticas , Universidad Complutense deMadrid , Madrid, SpainPublished online: 10 Jan 2009.

To cite this article: Antonio J. Gil & Gracia Rodríguez‐Caderot (1998) Processing gravity data in the territory of Andalusia,Marine Geodesy, 21:1, 81-89, DOI: 10.1080/01490419809388123

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Processing Gravity Data in the Territory ofAndalusia

ANTONIO J. GIL

Dept. Ing. Cartográfica, Geodésica y FotogrametríaEscuela Politecnica SuperiorUniversidad de JaénJaen, Spain

GRACIA RODRíGUEZ-CADEROT

Dept. Astronomía y GeodesiaFacultad Ciencias MatemáticasUniversidad Complutense de MadridMadrid, Spain

This article focuses on the study of gravity data in Andalusia to analyze its behaviorin this territory. Four zones that cover the greatest part of the region are selected, toobtain information as general as possible. Initially, an iterative method using least-squares prediction to detect gross errors in gravity data is considered. A comparisonof the difference between the observed and the predicted values with a critical limit isused to find outliers. A statistical analysis is carried out to check the effect of differentreductions on the observed gravity anomalies. Also, empirical covariance functionsare computed from original free-air gravity anomalies, from those reduced with theOSU9/A geopotential model, and from the trend-reduced free-air anomalies. Theresults show that there is different behavior of gravity anomalies corresponding todifferent topographic structures.

Keywords Earth's gravity field, empirical covariance function, geopotential model,gravity anomaly, least-square prediction, linear regression

This article reports on preliminary work to study the behavior of the observed gravityanomalies in the Andalusia region in southern Spain before computing a very high-resolution gravimetric geoid to be compared to a GPS/leveling-derived geoid in that region.

The accuracy of a gravity field approximation depends mainly on two factors: distri-bution of the available data and variation of the gravity field, therefore a statistical studyof data is necessary.

In the estimation of geoid undulations by fast Fourier transform (FFT) (Sideris, 1994)or fast Hartley transform (Tziavos, 1993), a regular grid of gravity data is used, so it isnecessary to know the behavior of the free-air anomalies to select the optimum procedurefor interpolation of observed data. On the other hand, modeling the gravity field by least-squares collocation (Tscherning, 1984; Gil et a!., 1993a) requires that a system of equa-tions be solved with as many unknowns as the number of data used. Practical computations

Received 15 December 1996; accepted 4 September 1997.Address correspondence to Dr. A. J. Gil, Escuela Politecnica Superior, Universidad de Jaen,

23071 Jaen, Spain.

81Marine Geodesy, 21:81-89. 1998

Copyright © 1998 Taylor & Francis0149-0419/98 $12.00 + .00

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82 A. J. Gil and G. Rodriguez-Caderot

show that the maximum number of points treated together is nearly 3,000 (Broveili &Sanso, 1992), so knowledge of the gravity field variation is necessary to select data withthe optimum grid size.

Description of the Selected Areas

We have selected four areas in Andalusia, from flat to high topographic relief, whichshow the features of different zones, in order to study the behavior of gravity anomalies.

Most of the Iberian Peninsula is occupied by the Meseta, a large plateau that is almostcompletely surrounded by mountain ranges. The region of Andalusia is located in thesouth. The valley of the Guadalquivir River is separated from the plateau by the SierraMorena and from the Mediterranean by the Sistema Penibetico (Baetic Cordillera) in-cluding the Sierra Nevada in the southeast, but the river has unimpeded passage to theAtlantic in the southwest.

The first area is in the Sierra Morena in northern Andalusia; its limits are 38° s 4>^ 38.5° and —5.4° < X < —4° in latitude and longitude, respectively. The second areais in the valley of the Guadalquivir River, and its limits are 36.8° < <)) < 37.6° and-6 .3° < X < -5 .7° . The third area, with limits 36.9° < (}> < 37.3° and -4 .5° < X <— 3.25°, is located along the Sistema Penibetico. The fourth area is in eastern Andalusia,and its limits are 37.2° < 4> < 38° and - 3 ° < X < - 2 ° .

Data Processing

Computation of Gravity Anomalies

Instituto Geografico Nacional supplied Andalusian free-air gravity anomalies which hadbeen calculated with International Gravity Formula 1967. Figure 1 illustrates the distri-

36.00-7.50 -6.50 -5.50 -4.50 -3.50 -2.50

Figure 1. Distribution of available gravity anomalies in Andalusia.

-1.50

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Processing Gravity Data 83

bution of the available gravity anomalies. Initially, we computed gravity anomalies in theGeodetic Reference System 1980 by (NGS, 1986)

Agiwo = Agl967 - (0.8316 + 0.0782 sin2 $ - 0.0007 sin4 c|>) (1)

where 4> is the geodetic latitude of the point.Figure 2 shows a contour map of the free-air anomalies in Andalusia.

Gross Errors Detection

When one is interested in gravity field approximation, it is necessary to be sure that thedata do not contain gross errors. Therefore, the first task, before using data in computa-tions, is to search for outliers. Several methods have been proposed in the geodeticliterature. For large numerical databases, semiautomatic methods which identify grosserrors are used, followed by a detailed analysis of each value. Some methods predict avalue at each point from the closest points stored in the data file. A comparison of thedifference between the observed and the predicted values with the error estimate can thenbe used to identify.a possible gross error (Tscherning, 1991).

A gravity anomaly in a point P, Agl?, is predicted from a set of values Ag(, / = 1,. . . , n, in the neighborhood, spaced as regularly as possible in all directions by

AgP = Z, a,- Agi (2)

where

a, = \ClP)T{Cu + Do)~< (3)

with CIP = the covariance between the observation i and the predicted value at P

C,, = the covariance of the observationsDu = the covariance of the observation errors (associated to the points / and j)

The error estimate is then computed as

- AgP) = Co - (CIF)T(CU + D, )" ' (Cp (4)

A measurement is rejected or considered suspect if

\AgP - AgP\ > K[<jP{AgP - A'gP) + cr~,(AgP)Yn- (5)

where k is a constant generally having the value 3, and u2{AgP) is the error variance ofthe observation AgP.

From Eqs. (4) and (5) it is observed that gross errors are easily found when Co is assmall as possible, and therefore Eq. (4) should be applied when data have been smoothed,removing the long wavelengths by taking into account a geopotential model and the localgravity field irregularities caused by topographic relief.

The method that we have used (Gil et al., 1993b) works with two disjoined sets ofobservations in each area. The goal of this method is to detect gross errors by comparingobserved and predicted values in an iterative way in order to avoid doubtful points that

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84 A. J. Gil and G. Rodriguez-Caderot

39

36-7.50 -6.50 -5.50 -4.50 -3.50

Figure 2. Contour map of free-air anomalies.

-2.50 -1.50

could contaminate the predicted values. At each step, the predictions are made withouttaking into account the doubtful points detected in the previous step. To compute predictedvalues we have used the remove-restore technique. Table 1 shows some statistical resultsof validated gravity data in the four areas.

Free-air anomalies are positively correlated with the point height, due to the effectof the topographic masses in the short-wave range. First, the degree of correlation isestimated using the simple linear regression model.

Ag = a + bH (6)

where a depends on the mass distribution in the Earth's crust, b is a function of the meandensity of the topographic masses (Torge, 1989), and H is the height of the station. Table2 illustrates these parameters in the four zones. One can observe that the highest degreeof correlation between Ag and H appears in the area of Sierra Morena, and the lowest inthe valley of the Guadalquivir River.

Gravity Anomalies from a Geopotential Model

Gravity anomalies can be computed in spherical approximation from a geopotential coef-ficient set by

Ag = (C,,,,, cos mk + S.m sin m\)P,Jcos (7)

where -d and A. are the geocentric colatitude and longitude, respectively, of the point atwhich Ag is to be determined; C,im and 5,,,,, are the fully normalized spherical geopotential

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Table 1Statistics of the gravity and height data in the selected areas

Area

1234

Numberof points

346212421529

Max

1,042412

1,4742,168

Heights (m)

Min

1772

323224

Mean

56255

709833

Mean

34.01-15.56-28.18

0.17

Free-air

SD

23.3827.0230.6934.74

anomalies (mgal)

Max

93.2052.8486.09

168.07

Min

-20.81-54.48-80.45-73.70

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86 A. J. Gil and G. Rodriguez-Caderot

Table 2Parameters of the linear regression between free-air anomalies and height

Area a b

1234

-16.77-23.30

-107.26-40.54

0.090.140.110.05

0.730.280.570.38

Table 3Comparison of free-air and OSU91 A-model implied anomalies

Areas

MeanSDMaxMin

1

-5.7115.0840.62

-51.00

2

-12.2613.2342.70

-35.84

3

-38.6129.2657.95

-94.24

4

-21.8627.58

145.54-81.56

Units: mgal.

•-7.50 -6.50 -5.50 -4.50 -3.50 -2.50 -1.50

Figure 3. Contour map of OSU9lA-reduced anomalies.

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Processing Gravity Data 87

Table 4

Parameters of the linear regression between reduced gravity anomalies and height

Area a b r

1 -38.50 0.06 0.73

2 -19.87 0.14 0.573 -97.35 ' 0.08 0.444 -38.50 0.06 0.73

coefficients of the anomalous potential; the P,,,,, are the fully normalized associated Le-gendre functions; nmax is the maximum degree of the geopotential model; and G is themean gravity.

In Gil et al. (1994), a comparison of different geopotential models in Spain ispresented. In this work the geopotential model-implied anomalies have been computedfrom the OSU91A model (Rapp et al., 1991) at the same points where observations wereavailable. A contour map of the reduced anomalies Ag — Ag,m)J is presented in Figure3. Table 3 shows the mean, the standard deviation, and the maximum and minimumvalues of the reduced gravity data.

The standard deviations of the reduced anomalies in nearly all areas are much lowerthan the corresponding standard deviations of the original free-air gravity anomalies,except in zone 3, they are quite similar, showing that this geopotential model does notreduce satisfactorily the gravity field in this area.

To study whether the OSUI9A-reduced data are correlated with altitude, the regres-sion model (6) has been applied, giving the results shown in Table 4. Although thespherical harmonics subtracted from the free-air gravity anomalies contain topographiceffects, there are some cases (see Table 4) where the reduced free-air gravity values arestill correlated with altitude. Table 5 illustrates the effects of various reductions on thepoint gravity anomalies.

Table 5Effect of two reductions on the point gravity anomalies

Free-air anomalies minusFree-air anomalies minus height OSU91A minus height

correlated part correlated part

Area

1234

SD

15.9029.5125.2832.18

Max

43.3268.2343.34

109.40

Min

-29.17-39.76-43.58-69.15

SD

10.2410.8626.2519.43

Max

35.3532.9794.83

128.90

Min

-28.72-18.41-48.26-51.63

Units: mgal.

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88 A.J. Gil and G. Rodriguez-Caderot

Empirical Covariance Functions

One can gain more statistical information on the behavior of the gravity anomalies throughthe empirical covariance function.

For the estimation of C(tJ/) from anomaly data given on the surface of a sphere, wecan write (Heiskanen & Moritz, 1967)

I f 2 " f2lT T= — A#(fl,\) Ag($'X) sin ft d-d da d\ (8)

where -9 is a polar angle, \ is the longitude, and a is an azimuth. In practice the sphereis not completely covered by anomalies; an expression that may be used to compute thecovariance is

2 g; gjC(«|>) = (9)

n

where n is the number of points taken at a given spherical distance \\>. In practice thedistance i|i to a special product at distance i|/,v is determined by the equation

A\\>

where AiJ; is a suitable chosen range.The accuracy of the empirical covariance values depends on the data distribution.

For instance, the determination will be optimum if a regularly distributed set of anomaliesis available. Therefore, before computing the empirical local covariance functions, apartof the data sets was selected holding one value in each square of 3' covering the areas.Since the observations represent the areas of equal size, no values are estimated in theempty cells. The program EMPCOV (Tscheming et al., 1992) is used to compute theempirical covariance values.

Tables 6 and 7 show the variance Co. the correlation length £, and the ratio C ^ forthe free-air and OSU91A-reduced anomaly covariance functions in the selected areas.One can observe that the variance Co for the reduced-anomaly covariance function de-creases in all areas compared to the variance Co of the free-air anomaly covariancefunction.

Table 6Parameters of the free-air anomaly covariance functions

Area . Q k CJ%

1 531 9 592 642 12 53.53 601 6 100.17-4 819 11 74.45

Units: C,, (mgal-), 4 (arcmin), C,/£ (mgalVarcmin).

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Processing Gravity Data .

Table 7

Parameters of the OSU91 A-reduced anomaly covariance functions

Area C,, ' i

1 192 5 38.4

2 155 7 22.143 588 7 84.04 358 5 71.6Units: C,, (mgaF), | (arcmin), C,/£ (mgalVarcmin).

Conclusions

In this article we studied the gravity field in the territory of Andalusia by seeking corre-lations between free-air anomalies and altitudes and checking the effect of differentreductions on the point gravity anomalies.

Initially, a method to detect gross errors in observed data was applied in an iterativeprocess to avoid doubtful points that could contaminate the predicted values.

Empirical covariance functions were computed in order to give more statistical in-formation, since these functions show the internal consistence of data. The results indicatedifferences among the selected areas corresponding to different topographic structures.

References

Brovelli, M., and F., Sansó. 1992. The Geomed project: The state of the art. Mare Nostrum 2 :1 -33.

Gil, A. J., M. J. Sevilla, and G. Rodriguez-Caderot. 1993a. A gravimetric geoid in central Spainfrom gravity and height data. Bull. Géod. 67:41-50.

Gil, A. J.. M. J. Sevilla, and G. Rodrfguez-Caderot. 1993b. A method for gross-error detection ingravity data banks. Int. Geoid Serv. Bull. 2:25-31.

Gil, A. J., G. Rodrfguez-Caderot, and C. de Lacy. 1994. Comparing different geopotential modelsin Spain. Mar. Geod. 17(3):18i-19l.

Heiskanen, W. A., and H. Moritz. 1967. Physical geodesy. San Francisco: W. H. Freeman.NGS (National Geodetic Survey). 1886. Geodetic glossary. Rockville, MD: U.S. Government

Printing Office.Rapp. R. H., Y. M. Wang, and N. K. Pavlis. 1991. The Ohio State 1991 geopotential and sea

surface topography harmonic coefficient models. Rep. 386. Columbus. OH: Dept. of GeodeticScience and Surveying. The Ohio State University.

Sideris, M. G. 1994. Geoid determination by FFT techniques. Lectures notes of the InternationalSchool for the Determination and Use of the Geoid. pp. 165-229. Milan: International GeoidService.

Tscheming, C. C. 1984. Local approximation of the gravity potential by least squares collocation.Proceedings of Beijing International Summer School on Local Gravity Field Approximation,Beijing. China.

Tscheming, C. C. 1991. A strategy for gross-error detection in satellite altimeter data applied inthe Baltic-Sea area for enhanced geoid and gravity determination. In Determination of thegeoid. Present and future, IAG Symposium 106, pp. 468-475, Berlin: Springer-Verlag.

Tscheming, C. C., R. Forsberg, and P. Knudsen. 1992. The Gravsoft package for geoid determi-nation. First Continental Workshop on the Geoid in Europe, Prague.

Torge. W. 1989. Gravimetry. New York: Walter de Gruyter.Tziavos, I. N. 1993. Gravity field modelling using the fast Hartley transform. Proceedings of XVII

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