ANNO XLIII - BOLLETTINO DI GEODESIA E SCIENZE AFFINI - N. 3, 1984

Geoid Computations in the ltalian Area

B. BENCIOLINI - L. MUSSIO - F. SANSÙ

Istituto di Topografia, Fotogrammetria e Geofisica, Politecnico di Milano

P. GASPERINI - S. ZERBINI

Istituto di Geofisica, Università di Bologna

Summary. - A gravimetric geoid has been determined in the italian area: this paper re­ports procedures and results. The data which have been used are the gravity anomalies suppliedby BGI.

A collocation procedure has been applied locally using the Rapp ('79) 180 x 180 model asreference field. The geoidal height m.s.e. is almost everywhere less than 0.5 m on the land. Verticaldeflections are estimated too and the discrepancies with the measured values show significantlarge values in the tectonically active areas.

A comparison is made with the Cruz and Rapp geoid showing significant systematic dif­ferences.

CALCOLO DEL GEOIDE NELL'AREA ITALIANA.

Sommario. - Del calcolo di un geoide gravimetrico nell'area italiana si riportano proce­dure e risultati. Si è fatto uso delle anomalie della gravità fornite dal BGI.

La procedura è basata sulla collocazione applicata localmente con riferimento al modello diRapp ('79), 180 x 180. Lo e.q.m. delle altezze sul geoide è quasi ovunque inferiore a 0.5 m sullaterraferma. Si stimano anche le deviazioni della verticale; le discrepanze con i valori misurati sonomaggiori nelle zone tettonicamente attive.

Un confronto con il geoide di Cruz e Rapp mostra un significativo sistematismo nelledifferenze.

214 B. BENCIOLINI- P. GASPERINI- L. MUSSIO- F. SANSÙ- S. ZERBINI

INTRODUCTlON

The present paper reports the status of the research developed by theMilan group with the cooperation of dr. S. Zerbini and dr. P. Gasperini, of theUniversity of Bologna, in the field of computational physical geodesy. Thefinal tasks of the research are the comparisons of several methods for deter­mining the anomalous potential (1), represented as usual by the geoid orbetter by the quasi-geoid (1), and the actual computation of the geoid in theItalian region with the best possible accuracy (target ±20 cm).

The first step of the research was the computation of an astrogeodeticgeoid as reported in the 2nd Intemational Symposium on the European andMediterranean Geoid (Rome, September 1982).

Wath we present here is the 1983 step of the research, Le. mainly the,computation of a first gravimetric geoid.

1. - GENERAL INFORMATION ABOUT THE METHOD

1.1. - TRE COLLOCATIONPROCEDURE

The collocation procedure has been used at this step of the research, for itis particularly simple to use (no a-priori smoothing and interpolation of data isrequired) flexible (any kind of data can be used) and powerful (any kind ofgeodetic quantity can be predicted).

The program we have used is the C.C. Tscheming program with a fewmodifications (see C.C. Tscheming, 1974).

the program allows a three-steps collocation, but only twosteps havebeen used till now, the first of which is the computation of the requestedquantities using a global mode! of the earth anomalous potential representedby spherical harmonic coefficient.

1.2. - PARTITIONINGOFITALYINTOZONES

The Italian region has been partitioned into 13 rectangular zones whichhave been trated independently; this partitioning was necessary in order toavoid too long, and expensive, computations.

It is not a completely satisfactory procedure, but at this stage it seems tobe the best compromise between a completely rigorous (or at least morerigorous) procedure, and the need to reduce the computations to a mana­geable amount.

Justifications and problems of this procedure will be pointed out later onin the papero Table l and figure l illustrate the zone boundaries.

(1) For the sake of brevity we shall use throughout the paper the word «geoid» instead of«quasi-geoid».

GEOID COMPUTATIONS IN THE ITALIAN AREA

2. - TRE DATA

2.1. - GRAVITY DATA

215

The gravity data we used are the data collected and distributed by theB.G.I., which we like to thank for the kind cooperation.

10116 data have been supplied, 6551 of which are land data and 3564 aremarine data. Only land data have been used, mainly because of the difficultyof finding reliable weights for the two kinds of measurements.

TABLE 1ZONES BOUNDARIES

Zone cpÀ

1

44.00 -;.-45.50 8.00 -;.-11.00

2

44.00 -;.-46.00 10.00 -;.-13.00

3

43.00 -;.-44.50 10.00 -;.-14.00

4

41.00 -;.-43.50 11.00 -;.-15.00

5

40.00 -;.-42.00 14.00 -;.-17.00

6

40.00 -;.-41.50 16.00 -;.-19.00

7

37.50 -;.-40.50 15.50 -;.-17.50

8

36.50 -;.-38.50 12.00 -;.-16.00

9

38.50 -;.-41.50 8.00 -;.-10.00

lO

44.00 -;.-46.00 6.50 -;.- 8.50

11

45.00 -;.-46.50 7.50 -;.-10.00

12

45.50 -;.-47.00 9.50 -;.-12.00

13

45.50 -;.-47.00 11.50 -;.-14.00

2.2. - POTENTIAL COEFFICIENTS

The Rapp 180 model, complete up to degree' and order 180, has been usedin alI computations.

2.3. - SEASAT RADAR ALTIMETRY DATA

The Seasat satellite was launched in 1978 and it has been collecting datafor three months. The Seasat contained a radar altimeter which continuouslymeasured the distance from the satellite to the sea surface; the accuracy of thealtimeter was approximately ±1O cm. From these data the geoid can be ob­tained directly if the orbit of the satellite and the oceanographic corrections

(3.1)

216 B. BENCIOLINI- P. GASPERINI- L. MUSSIO- F. SANSÙ- S. ZERBINI

are known. Oceanographic corrections consist of tides and both time-depen­dent and steady-state variations produced by currents.

The data from Seasat experiment have been processed by Rapp andRowlands and one global network and four regional network adjustmentswere carried out (Rapp, 1981; Rowlands, 1981). One of these regions, the NorthAtlantic, included data from the Mediterranean Basin; in this area, after ad­justment of the orbital parameters, the cross-over errors were ±26 cm (Cruzand Rapp, 1982). Cruz and Rapp compared the adjusted data from the Seasatand Geos-3 satellites in the Mediterranean and the comparison indicated thepresence of a systematic difference and a slope between the two surfaces. Alocal adjustment of the Seasat data in this region was then carried out. AIso, inthis new adjustment, aH tide values were set to zero because the Schwiderskicorrection is not valid in this area. The cross-over discrepancy of this adjust­ment is ± 15 cm. The data set used in this work is this latest one (Cruz andRapp, 1982).

It is worth mentioning that, in generaI, in the Mediterranean basin tidecorrections are smaH except for the Adriatic Sea where they can be up to 1 mand more.

The Seasat geoid has been used, here, only for comparison purposes.

2.4. - VERTICALDEFLECTIONS

Vertical deflection measurements have been supplied by IAG SSG 5.50,which we heartily thank. A total amount of 323 vertical deflection points whereavailable, which have been used for comparison only.

3. - TRE COVARIANCE FUNCTION

3.1. - COVARIANCEFUNCTIONMODELS

The choice of a priori covariance function or, switching to analitycal de­scription of coHocation, the choice of a proper approxitnation space is thecentraI point when coHocation is applied.

The construction of the maybe most famous covariance function modelssuitable for coHocation has been treated by Tscherning and Rapp (1974) andby Tscherning (1976): the recommended model for the anomalous potentialcovariance function is:

C(P,Q) = I~O (1- 2~1+ 24) (RBj'Yp 'YQ)Hl Pl(COS ~P'Q)

while aH the other needed auto and cross-covariance functions are derivedfrom this basic one.

GEOID COMPUTATIONS IN THE ITALIAN AREA 217

When working in smaH areas the mentioned global covariance functionmust be replaced by a local one in order to avoid numerical instabilities. It hasbeen found (Tscherning, 1976) that a reasonable model for a local covariancefunction can be expressed again by (3.1) where a certain number of coeffi­cients of the first degrees is set to zero.

3.2. - COVARIANCE FUNCTIONS IN PRACTICE

The actual use of model (3.1) as local covariance function requires theestimation of three parameters, Le.A, which is a scale parameter, the radius R,of the Bjerhammar sphere, and the order, n, of the local function, Le. thenumber of terms to be deleted.

AHthe quantities must be set to proper values in order to obtain a goodagreement between the model-covariance function and the «true» local-cova­riance function according to the main features.

The available gravity anomaly data have been used to estimate zone byzone 13 empirical covariance functions.

The great variety of geophysical situations we have in Italy is immediatelyevident by inspecting the shape and the power of the different functions.

The empirical covariance functions of zone 4 (CentraI Italy) and of zonelO(Alpine Region) are represented in fig. 2 as an example of these differences.In table 2 are shown the values of the functions at the origin; the values of thenoise variances and the positions of the first zero (\);0)'

The great variety of values must be again remarked.

TABLE 2MAIN PARAMETERS OF COVARIANCE FUNCTIONS

Zone ~g signal variance~g noise variance\);0n22

(mgal) (mgal)

1

3662 20230'1602

2930 1540'1203

2214 19150'904

1234 2750'905

922 (-2)32'1506

1052 12130'1607

1781 20422'2208

3142 2537'1409

696 222'220lO

3264 1828'16011

3800 (-2)37'13012

3209 525'20013

1758 1131'160

218 B. BENCIOLINI - P. GASPERINI - L. MUSSIO - F. SANSÒ - S. ZERBINI

As it is expected from the many researches in this field (see for instanceSiinkel, 1981),the local covariance functions perform a very typical behaviorwith high value at the origin and short correlation length in rough tectonicallyactive areas (e.g. Alps and Sicily) and vice-versa in plain or settled areas.

The A parameter of expression (3.1) is internally computed by the pro­gram using the gravity anomaly signal variance. The local function degree isselected by fitting the first zero-point of the empirical covariance function; thiscan be done following Tscherning and Rapp (1974),because the higher is theorder, the nearer to the origin is the first zero. The radius R has been heldfixed to the value 0.9998RE because it is not clear thoretically how couldreasonably be glued solutions estimated by covariance functions referring todifferent domains of harmonicity.

The noise variance, which is estimated in this way together with the co­variance function can happen to be negative: in this case we have always resetan arbitrary, small positive value (e.g. 1 mgal\

4. - PRACTICALCOMPUTATIONMANAGEMENT

4.1. - SINGLE ZONES TREATMENT

As already explained, the first step of data treatment is the zone parti­tioning and then the estimation of a covariance function for each zone.

Mereover in some areas of particularly high data density, some of themhave been neglected. This resulted in a decrease of the data budget to about3000values, with a much more regular density distribution.

The second step is just the estimation of geoidal heights in a regular gridwith a 101 mesh-side; moreover, the solution of normal equation is saved andcan be used for further computations. In fact, if the normal equation solutionis available, any potential-related quantity can be computed in a separate runtaking advantage of the restarting capability of the program.

Seasat altimeter geoid and vertical deflections have actually been com­pared in this manner.

4.2. - THE ZONES PATCHING

The 13 regions into which we have partitioned Italy are as mentionedstrongly overlapping; there are therefore a lot of grid point where the geoidalheights have been computed using two, or sometimes more, partially differentsets of data. How to combine these different estimates in order to obtain thebest unique estimate is not a practically solved problem, till now.

The main difficulty is to obtain a good estimate of the covariances bet­ween the heights of the common points, estimated from the different zones.

GEOID COMPUTATIONS IN THE ITALIAN AREA 219

In the present case, the simple weighted average has been used as finalestimate; this is by no means a rigorous solution but it becomes anywaymeaningful if the discrepancies are small.

The mean square values of residuals are computed, too, and are used as atool to decide whether the weighted averages are acceptable or some reasonsfor the discrepancies must be sought for.

4.3. - A REMARK ON LOCAL APPLICATIONS OF COLLOCATION

The principle on which the collocation is used in a local mode relies on thefolIowing heuristic reasoning: the information on the law frequency compo­nents of the gravity field comes from large averages of measured quantities(f.i. gravity anomalies) and cannot be influenced significantly from local ob­servations. Whence a reliable high order reference field must be used in orderto introduce this information. If the corresponding coefficients are fixed andconsidered as known up to a certain degree only higher frequency componentsmust be estimated; correspondingly an «optimal» estimate can be obtained byusing a covariance function where alI the low frequencies have been cutoff (l),what is in fact done as previously described to model the local features of theempirical covariance function. It folIowsfrom this reasoning that the degree ofthe reference fiald should always be larger than the low frequency componentcut off in the local covariance function: this we remark to prevent otherbeginners, as we consider ourselves, from making the same errors that wemade in a first step of computations of the Italian geoid.

5. - PRESENT RESULTS

5.1. - THE GRAVIMETRIC GEOID «ITALGEO 83»

As already mentioned, thegeoidal heights have been computed on a re­gular grid with a 10' mesh-side. The covered area is the whole Italian land andsome smal marine regions, reteined only to simplify the data management.

The final result is tabulated in the appendix.The geoidal height m.s.e.s estimated from collocation formulae are almost

everywhere less than 0.5 m, with the exception of sea zones, Le.zones far awayfrom the data area.

In the overlapping zones we have also computed the m.s.e. of theweighted averages: their distribution is shown in table 3.

(2) This corresponds to a stepwis estimation of the components of Ton two subspace of theHilbert space admitting C(P,Q) as a reproducing kernel: the two projectors on these subspacescoincide as· a matter of fact with the low frequency and high frequency components of kernelC(P,Q).

220 B. BENCIOLINI - P. GASPERINI - L. MUSSIO - F. SANSÙ - S. ZERBINI

TABLE 3DISTRIBUTION OF M.S.E. OF POINTS IN OVERLAPPING AREAS

rj - intervai (m) Number of points

0.00 + 0.10

238

0.10 + 0.20

1410.20 + 0.30

69

0.30 + 0.4066

0.40 + 0.5036

0.50 +0.60

20

0.60 +0.70

14

0.70 +0.808

0.80 +0.90

4

0.90 + 1.001

> 1.00

7

This should not be taken as an overall measure of the accuracy of theestimates, but more as an internaI consistency of the method.

In any way it is not by chance that 47 of the 54 m.s. weighted residuaisgreater than 0.50 m, were Iocated at a Iatitude higher than 45°30', Le. in theimmediate sorroundings of the AIps.

5.2. - A FIRST COMPARISON WITH AN ASTROGEODETIC GEOID

The Italgeo 83 has been first of alI compared with a geoid derived, aiso bycollocation, from astrogeodetic data.

This has been computed by C.C.Tscherning in '82 and communicated tothe authors in private formo It consists of about a hundred of estimated on­duiations spaced half degree by half degree from Iatitude 43° to 47°; 74 of themare in common with our predicted values.

The average of the discrepancies between the two and their mean squareerror are respectiveIy

x = 0.82 m

5,1 = 1.36 m

This seems for the moment acceptabie considering that the two geoidshave been computed from completely independent data Syts and no Iocaioptimization of the datum shift has been used to transform the verticaideflections from the ED50 to the GRS80 system.

GEOID COMPUTATIONS IN THE ITALIAN AREA 221

5.3. - COMPARISON BETWEEN GRAVIMETRIC ANO ASTROGEOOETIC OEFLECTIONS

From Italgeo 83, deflections of the vertical have been predicted at thesame points where we had astrogeodetic measurements, for comparison pur­poses.

The results are summarized in table 4 in form of histograms of the di­screpancies

Os = Sobserved - Spredicted

OT) = T)observed - T)predicted

The gravimetric deflections are firstly predicted in the GRS80 system andthen transformed to the ED50 system.

The columns of table 4 refer to dasses of amplitude of one second and arecentered at the heading value. The column «outside» refers to discrepancieswhich are larger than 10".5in absolute value.

Though these results are not exiting, one point is absolutely dear: thelargest discrepancies between gravimetric and observed deflections happen inareas where the topography is extremely rough, like lO, 11, 12, 13,which referto the alpine arc, or to tectonically active areas like zone 5, which indudes thevery strongly seismic region Irpinia or zone 7 with the seismic area of theStraits of Messina.

A reason for the relatively high values of the discrepancies, should besought for in the lack of any correction for the topography, which on the otherhand is known to have a large influence on the deflections 1;, T). Moreover, thegravimetric deflections have been for the moment predicted only at the meansea level, since we were not aware of the height of the measuring points: thismight give a significant difference, specially in montaneous areas.

Furthermore the mean density of gravimetric points is not very high, thusit makes no wonder that only poor predictions can be made, where the gravityfield shows a rough pattern.

5.4. - COMPARISON BETWEEN ITALGEO 83 ANO ALTIMETRY OERIVEO-GEOID

As we said previously a set of Seasat altimetry data adjustment bt Cruzand Rapp (cfr. Cruz and Rapp 1982)has been compared to the Italgeo 83ondulations.

To this aim only points dose to the Italian shores have used so that thecollocation solution could give a really informative prediction there (Le.,witha prediction error of about 0.50-0.70m). It is worth nbting that being bothgeoids referred to the GRS80 system, no datum shift has been applied beforecomparing one to the other.

TABLE 4

Zone N. -10-9 -8 -7 -6 -5 -4 -3 -2 -1 O123456789lOOutsideO~oliSrPTI

1

O~ O1O1OO1253421OO1OOOOO 2-2.5 4.3

OT}

2OO11O11337121OOOOOOO O-1.7 3.4

2 o~

O1O131534442211OOOOOO 4-2.6 5.7

OT}

OO1O1112OO4936212O21O O 1.53.7

3 o~

OO1O142O3322OOOOOOOOO 1-3.2 3.1

OT}

OOOO1OO3233O3211OOOOO O 0.02.8

4 O~

OO1OOO2132513O11OO1OO 1 0.54.5

OT}

O1O1OO2124123O212OOOO O 0.03.9

5 o~

2OOOOOOOO21421O11O121 11 1.810.6

OT}

OO122OOO2O1OOO2125134 3 4.76.4

6 O~

OOOO1OOOO231O8OOOO1OO O 1.62.9

OT}

OOOOOOOO11O331322OOOO O 2.62.4

7 O~

OOOOOOOOO2O3O331O111O lO 0.79.8

OT}

11O1O1OOO21OOOO11OO21 13 6.18.9

8 o~

OOOOOOOO2186735313212 1 3.13.3

~T}OOO1OO1246246512332OO 3 1.45.3

9 o~

11OO1O23222O142422OOO 2-0.2 5.6

OT}

O33O2221231OOO1121O22 3-0.4 7.1

lO o~

21O1O2OOO13113OOO22OO 8-1.4 9.7

OT}

OOOOOOOOOOOOO1O321O11 18 1.020.7

11O~1O23112OO1OO1OOOOOOOO 5-8.7 6.2

OT}

OO1O1OOO15O42OO1O1OOO 1 0.64.5

12 o~

OOOOOOOOOOOOOOOOOOOOO 15-6.052.7

OT}

OOO1OOOOOOOOOOOOOOOOO 14 19.222.8

13 o~

O31213OO1O1OOO11OOOOO 18-8.67.0

OT}

OOOOOOO31O11242523 ' 223 1 4.64.2

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GEOID COMPUTATIONS IN THE ITALIAN AREA 223

A total of 175 points has been processed and the differences Naltimetry ­

Ngravimetry have been computed: the resulting overall average and the meansquare value are

x = -1.06 m

SJ1 = 1.56 m

As we expected, also according to the remarks in Cruz and Rapp (1982),the mean square error of the differences is significantly too high to be ex­plained as the sum of the prediction error from the land gravity determinedgeoid and the estimation error of the adjustment altimetric ondulations.

For this reason we have tried a comparison in a regional mode partitio­ning the Italian coast as shown in fig. 3.

On the figure for each zone are indicated the number N of points pro­cessed, the zone average difference X, and the zone mean square error SJ1.

Apart from minor oxillations, some generaI remarks can be drawn:

the mean square errors are compatible with the expected theoreticalvalues;

the mean zone values show a clear systematic pattem with an in­creasing difference from north to south seas;

- the - 3.08m difference (NaIt. - Ngrav) in the Southem Adriatic Sea isin quite reasonable agreement with the analogous - 3.70 m found in com­paring the Seasat geoid with the Greek gravimetric geoid DGGG80(cfr. Cruzand Rapp, 1982).

Q. - FINAL COMMENTSAND INDICATIONS

Considering the work done till now we can say that the first steps in thedirection of the production of a high accuracy geoid in Italy and sorroundingareas have been done: in particular a team of researchers has begun to betrained in such geodetic computations. A first gravimetric geoid to which wecan assign a prediction error smaller than 0.50 m in not too rough areas hasbeen completed. Future work will be done to enlargethe data base used andto improve the collocation method by implementing estimates of the effects oftopographic masses (cfr. for instance, H. Siinkel, 1981):to this aim we mentionthat a ·large data set of gravity values as well as of mean topographic heightsis becoming available (M.T. Carrozzo et al., 1981a-1981b).

A further deeper analysis of the biases of altimetric computed geoid isalso deserved, with particular care to the .J.ocaltidal models.

224 B, BENCIOLINI - p, GASPERINI - L. MUSSIO - F, SANSÙ - S. ZERBINI

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GEOID COMPUTATIONS IN THE ITALIAN AREA 225

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226 B. BENCIOLINI - P. GASPERINI - L. MUSSIO - F. SANSÙ - S. ZERBINI

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GEOID COMPUTATIONS IN THE ITALIAN AREA

ACKNOWLEDGMENT

227

This work has been realized with the finantial support of the «GruppoNazionale per l'Informatica Matematica (CNR)>>,by a computer use contract.

REFERENCES

B. BENCIOLINI, L. MUSSIO, M.C. ROUFOSSE, F. SANSÙ, S. ZERBINI (1982), Astrogeodeticand Altimetric Geoid Computations in the Italian Area. Presented at the 2nd ln­temational Symposium on the European and Mediterranean Geoid, Roma.

M.T. CARROZza et Al. (1981a), Realizzazione di un archivio delle quote medie. Atti delPrimo Convegno Gruppo Nazionale Geofisica della Terra Solida, Roma.

M.T. CARROZza et Al. (1981b), Carta gravimetrica d'Italia: tecniche automatiche per lasua realizzazione. Atti del Primo Convegno Gruppo Nazionale Geofisica della TerraSolida, Roma.

RH. CHOVITZ(1981), Modern Geodetic Earth Reference Models. EOS 62, 65-67.S.Y. CRUZ, RH. RApp (1982), Sea Surface Heights in the Mediterranean Area {rom Seasat

Altimeter Data (to appear in Bollettino di Geofisica Teorica e Applicata).W.A. HEISKANEN, H. MORITZ (1967), Physical Geodesy. W.H. Freeman & Co., San

Francisco and London.

F.J. LERCH, S.M. KLOSKO, RE. LAuBSCHER, C.A. WAGNER (1980), Gravity Model Im­provement Using Geos-3. GEM 9 and lO.

J.J. LEVALLOIS,H. MONGE (1978), Le geoide européen version 1978. Proceedings of thelntemational Symposium on the Geoid in Europe and the Mediterranean Area,Ancona.

H. MORITZ (1980), Advanced Physical Geodesy. H. Wichman Verlag, Karlsruhe.RH. RApp (1979), Geos-3 Data Processing for the Recovery of Geoid Undulation and

Gravity Anomalies. Joum. Geophys. Res. 84, 3784-3785.R.H. RApp (1981), Gravity Field Determinations with Sea~at Altimeter Data. Paper pre­

sented at the fall meeting of the American Geophysical Union, Abstract in EOS 62,844.

M.C. ROUFOSSE (1978), Interpretation of Altimeter Data. Proceedings of the 9th GEOPConference «An· lntemational Symposium on the Applications of Geodesy toGeodynamics»; Dept. of Geodetic Science, Report n. 280, The Ohio State Univer­sity, Columbus, 261-266.

H. SONKEL (1981), The Estimation of Free-Air Anomalies. OSU Report N. 315 of De­partment of Geodetic Science ..

C.c. TSCHERNING(1974), A Fortan IV Program for the Determination of the AnomalousPotential Using Stepwise Least Squares Collocation. OSU Report N. 212 of theDepartment of Geodetic Science.

C.c. TSCHERNING(1976), Covariance Expression for Second and Lower Order Derivativesof the Anomalous Potential. OSU Report N. 225 of the Department of GeodeticScience.

228 B. BENCIOLINI - P. GASPERINI - L. MUSSIO - F. SANSÙ - S. ZERBINI

C.C. TSCHERNING (1978), A User Guide Geopotential Approximation by Stepwise Col­location on the RC-4000 Computer. Kebenhavn.

C.C. TSCHERNING(1979), Comparison 01 Some Methods for the Detailed Representationof the Earth 's Gravity Pield. Presenté~ at the GeneraI AssembIy of IAG, Camberra.

C.c. TSCHERNING(1982a), Geoid Determil1,ation for the Nordic Countries Using Colloca­tion. Presented at the Symposium on Geoid Definition and Determination, Tokyo.

C.C. TSCHERNING (1982b), Determination of a (quasi) Geoid for the Nordic Countriesfrom Heterogeneous Data Using Collocation. Meeting of the Nordic GeodeticCommission, GiivIe, Sweden, Sept. 1982.

C.C. TSCHERNING(1983), A Brief lntroduction to Geoid Modelling Using Collocation withsome Results from Scandinavia and Greenland.

C.C. TSCHERNING, R.H. RApp (1974), Closed Covariance Expressions for GravityAnomalies, Geoid Undulations, and Deflections of the Vertical lmplied byAnomaly Degree Variance Models. osu Report N. 208 of the Department ofGeodetic Science.

\

APPENDIX

TRE NUMERICAL REPRESENTATION OF ITALGEO 83

GEOlD COMPUTATIONS IN THE ITALIAN AREA231

LAT.

LON.GEOIOLAT•LON.GEOIOLAT.L.ONeGEOIOD

'" D'" ••• DMO'" '" OM DM M

47

9 3049.674794050.60479 SO51.3447

1051.8447lO lO52.1047lO 2052.1347

lO 3051.9947lO 4051.744110 5051.17,.7

1150.354711<10"9.984711 20't9.4947

11 3047.944111 4047.734111 50-H.83,47

Il48.104112 lO48.414712 2049.1347

12 3049.694112 4050.054712 SO50.2141

1350.194713 lO50.044713 2049.1841

13 3049.454713 4049.074713 SO48.6447

14/te. l'CI46 SO9 3051.0946 SO9 4051.1746 SO

9 SO51.9846 50lO'51.6746 SO10 lO5l·l2

\46 50

10 2050.5846 SOlO 3049.7046 50lO 4049. 746 SO

lO SO49.1846 501148.5446 5011 lO47.8046 50

11 2047.8646 5011 3046.8646 SO11 4046.5841, 50

11 5046.9646 SO1247.2546 5012 lO47.6146 50

12 2048.3546 5012 3048.9546 5012 4049.3546 SO

12 SO49.5146 SO1349.5146 SO13 lO49.4146 SI)

13 2049.2546 5013 3049.0546 SO13 4048.8046 ~O

13 SO48.5146 501448.1546 409 3051.704()40

9 4052.0546 409 5051.7246 40lO50.4646 40

lO lO48.8346 40lO 2048.8046 40lO 3047.994" 40

lO 4047.2146 4010 5046.7646 401146.3546 40

11 lO46.1146 4011 2046.5946 4011 3046.52

46 4011 4046.7446 4011 SO47.346 40124.1.•63

46 4012 lO47.4346 4012 2047.5246 4012 3047.78

46 4012 4048.0546 4012 5048.0546 401347.99

46 4013 1047.9646 4013 2047.9646 4013 3041.97

46 4013 4047.9846 4013 5047.9.46 40h47.78

46 307 3052.4146 30.,4051.9646 307 SO5h35

46 30850.63~ 30a lO49.9546 308 2049.49

46 308 3049.2646 308 4049.1046 308 SO48.89

46 30948.5746 309 lO48.2146 309 20,47.92

46 309 3049.5846 309 4049.9746 309 5050.00

46 30lO49.5446 30lO lO48.584~ 30lO 2047.72

46 30lO 3047.6446 30lO 4047.1446 30lO 5046.67

46 301146.2446 3011 lO·45.8946 30Il 2045.98

46 3011 3046.6446 3011 4047.4846 3011 SO48.05

46 301247.~l346 3012 lO46.9946 3012 2046.67

46 3012 304f.6046 3012 4046.5446 3012 So.~.21

46 301346.0246 3013 lO46.0546 3013 2046.24

46 3013 3046.5546 3011 4046.8946 3013 SO47.18

46 301447.2346 207 3052.4046 2074051.7.

46 f!O

7 5050.8246 20849.7146 208 lO48.6146 20

8 2047.8746 20A3047.9746 208 4048.0646 20

8 SO41.A246 20947.3446 209 lO46.6046 20

9 2045.7246 209 3046.8446 209 4047.6846 20

q 5047.8546 20lO47.7646 20lO 1047.0846 20

lO 2046.5646 20lO 3046.9246 2010 404~.7146 20

lO 5046.2846 201145.8446 2011 1045.6246 20

11 2046.0346 2011 3047.3446 2011 4047.8246 20

Il 5047.9846 201246.8646 20, 12 lO.6.254" 20

12 2045.8546 20l? 3045.7446 2012 4045.4146 20

12 5045.0346 201344.8046 2013 lO44.8446 20

l3 2045.2446 2013 3045.6946 2013 4046.1(146 20

13 5046.5046 201446.6446 lO7 3052.26·46 lO

7 4051.4546 lO.,SO50.3546 lO849.0246 lO

A lO47.1346 108 2047.3746 108 3047.6946·lO

8. 4047.3646 lOA 5046.8046 io946.30

232 B. BENCIOLINI ....,...P."GASPERINI- L MUSSIO -.:. F: SANSÒ- S. ZERBINI

4f1l0

91045.6246 lO<) ZO44.8046 lO<) 3045.4346 lO·

9 4045.864.l> lO<) 5045.9246 lOlO45.9946'1;0

lO lO46.2946 lOlO 2046.2746 lOlO 3046.7946 lO

lO 4&46.184·6lOlO 5046.5046 lO1145.9146 rO"

111045.7946 lO11 2046.1046 lO11 3047.6146 1.0

11 4047.9746 lO11 0;047.3746 lO124~.5846 lO

12·1045.4546 lO12 2045.0746 lO12 3044.66

46 {O1~·4044.2346 lO

12 SO43.92·46 lOB 3043.8~46()l. lO43.C)~46 O13 2044.394ftlO 44.8 .

46 lO13 4045.3146 lO13 5045.8046 lO1446.04

4663052.q446f) 4053.29466 5053.36

46753.io467 lO53.12467 2052.10

467 ~O52.l~461 4051.61467 5050.68

46849.64468 lO48.78468 2048.25

468 3041.664684046.48468 5045.79

46945.4246<) lO44.99469 2044.63

46<) 3045.53469 41)45.89469 5046.02

46lO45.8546lO lO46.3546lO 2046.47

4(,lO 3047.0546lO 4046.6846lO 50.".12

.61145.874611 lO46.314611 20.6.97

4611 3047.594611 4047.484611 50.".60

461245.844612 lO.4.964612 20.4.30

.612 3043.794612 4043.494612 5043.37

.61343.384613 lO43.494613 2043.78

4613 3044.144613 4044.634613 50.45.16

461445.4945 50(,3053.7l45 SO6 4053.69

45 506 5053.0A45 507"52.1045 507 lO51.5.

45 507 2050.7045 SO7 3050.4445 507 4050.25

45 507 5050.0345 50A49.5145 508 lO49.05

45 50A 2048.1245 50A 3.046.6845 508 4045.57

45 50A 5045.03455qQ44.5045 509 lO44.19

45 509 204.4.().845 509 3044.7845 509 4045.07

45 509 5045.2745 50lO45.1845 50lO lO45.43

45 50102046.1045 50lO 3046.2445 SOlO 4046.13

45 50lO 5045.904S 501146.1845 50111041.01

45 5011 2047.4'445 5011 3047.4545 5011 40"6.84

45 50115045.Rl45 501244.9545 SO12 lO44.24

455012 204J.7~45 5012 3043.3845 5012 4043.25

45 5012 5043.2245 501343.2.845 5013 lO41.40

45 50132043.6645 SO13 3043.9345 SO13 4044.31

45 5013 SO44.1545 501445.0845 406 3054.54

45406 4054.3645 40'.l 5053.5645 40752.05

45 40.,lO50.7345 407 2050.0445 407 3049.85

45 407 4049.3445 407 5049.1645 40Il48.89

.5 408 lO48.1145 .0e 2046.8245 408 3045.66

45 40

8 4044.4145.40A 5043.8245 40Cl--.3.4645 40

9 lO43.2745 409 2043.2445 409 .3043.Sl.5 40

9 .043.8445 409 5044.2145 40lO44••945 40

lO lO.4.784$ 40lO la45.2145 .0lO 3045.40"5 40

lO 4045.434540lO 5045.6945 4011.6.5140;~O

11 lO46.5945 4011 2046.2745 4011 3045.9645 40

11 4045.3645 4011 5044.7245 401244.2745 40

12 lO43.8145 4012 2043.5145 4012 3043.3745 40

12 40.43.]445 4012 5043.4445 401343.6145 41)

13 lO43.8045 4011 2043.994r;4013 3044.16

4~ 40

13 404~·~345 4613 5044.5545 401444·n4 306 305.: 345 30~ 4054.9645 306 5054.

45 30752.9{'45 307 lO5h5845 307 2050.36

45 )07 3050.11~5 307 4049.8245 307 50.9.1>6

45 3084e.0745 3081046.624S 308 2045.43

45 308 3044.5145 30A 4043.4545 3085042.90

GEOID COMPUTA:TIONS IN TRE rtALIAN AREA233

45 30

942.5145 309 1042.2245 309 2041.9945 .30

9 3042.0245 309 4042.2245 309 5042.6145 3.0

lO42.92•.5.30101043.3045 30lO 2043.7245 30

lO 3043.9745 30lO 4044.4745 30ltl5044e7145 30

1145.1245 3011 lO.45.3045 3011 2045.2745 30

11 3045.0145 3011 4044.5245 3011 5044.1245 30

1243.7a45 3012 lO43.514S 3012 2043.40

45,30

12 3043.4445 3012 4043.584S 3012 5043.1945 30

1344.0645 3013 lO44.4145'3013 2044.5645 30

13 3044.664S 3013 4044.7145 3013 5044.8045 30

1444.9445 20l'. 3055.4545 206 4054.9445 20

6.5054.0545 20752.944S 207 lO51.7545 20

7 2050.6345 207 3050.1845 207 4049.52

n ~gè i84a,.~~:~~8846.69:~ ~8B 18:~:~~44.

8 3043.444520

a SO42.0745 20941.664S 209 lO41.2945 20

9 2040.9245 209 3040.64'45 209 4040.5345 20

Cl 5040.5645 20lO40-.6645 20lO lO40.804520

102041.2645 20lO 3041.7645 20lO 4042.~045 20

lO 5042.7545 201143.1145 2011 lO43.4545 20

11 2043.7145 2011 3043.8845 2011 4043.8945 20

11 5043.7045 201243.4145 2012 lOU.2345 20

12 2043.1845 2012 3043.2245 2012 4043.34452.0

12 5043.5245 201343.7545 lO6 3055.26'45 lO

6 4054.1945 lO6 SO52.9445 lO751.7445 lO

7 lO50.9.645 lO7 2050.5145 lO7 3050.0145 lO

7 4041!.6845 lO7 5047.214510845.8945 lO

8 lO44.6145 lO8 2043.5445 lO8 3042.6445 lO

A 4041.8345 lO8 5041.3845 lO940.9645 lO

9 lO40.5645 lO9 2040.1845 lO9 3039.8245 lO

9 4039.55'4S lO9 SO39.4145 lOlO39.3445 lO

lO lO39.3645 lOlO 2039.6845 lOlO 3040.1145 lO

lO 4040.5645 lOlO 5041.0345 lO1141.5545 10 . 11 lO

42.0545 lOn 2042.5145 lO11 3042.9045 lO

J.l 4043.1645 lO11 SO43.1745 lO1243.0.045 lO

12 lO42.sa45 lO12 2042.8945 lO12 3042.9745 lO

12 4.0403.1245 1012 5043.3445 lO1343.6045

6 3055.40456 4054.09456 5052.9245

752.00457 lO50.96457 2050.4445

7 3044.6845.7 4048.29457 5046.8245

845.12458 lO44.42458 2043.4.3~5

8 3042.48458 4041.52458 5041.0445

940.67459 lO40.40459 2040•.0745

9 3039.5445Q 4039.12459 5038.8645

lO38.754516 lO38.4645lO 2038.5645

lO 3038.8345lO 4039.214510 5039.7245

1140.294511 lO40.884511 2041.4545

11 3041.914511 4042.224511 5042.3345

1242.304512 lO42.294512 2042.3645

12 3042.494512 4042.714512 5042.9945

1343.3144 506 3056.1444 SO6 4055.3044 50

6 5054.2244 50752.8444 507 1051.4444 50

7 2050.6044 501 3049.3244 SO7 4047.904450

7 5046.6444 50845.5844 508 lO44.6244 SO

8 2043.1144 SO8 3042.8044508 4042.0144 50

A 5041.65.44 50941.5344 5091041.3344 50

9 2040.9644 50Cl 3040.3544 509 4039.8744 50

9 SO39.3144 50lO38.7944 50lO lO3B.4044 50

10 2038.1944 SOlO 3038.2044 SOlO 4038.4244 50

lO 5038.85'44 501139.3344 SO111039.8844 50

11 2040.4444 5011 3040.9644 SO11 4041.31

234 B. BENCIOLINI .:....P. OASPER:lNI .....•.L. MUSSIO --F. SANSO .......S.ZERBINI

44 50

11 SO41.464450li.'41.4944 SO12 lO41.5244 50

12 204l.b']4~$0l? 3041.8444 5012 4042.1544 50

12 SO42.5144 SO1342.9144 406 3056.6044 40

6 4055.9844406 5054.8944 40753.5344 40

7 1052.3544 40·1 2051.1244 407 3049.8144 40

1 4048.2clJ44 407 SO47.09.44 40846.3644 40

8 1045.7544 408 2045.0044 408 3044.3444 40

8 4043.8244 40R 5043.4444 40943.1544 40

9 10.43.1344 409.2042.6344 409 3042.174440

9 4041.5144 409 5040.1544 401039.8444 40

10 lO39.2344 4010 2038.7644 4010 3039.5144 40

10 4038.4244 4010 5038.4844 40113".6844 40

11 lO39.0044 4011 2039.4544 4011 3039.9344 40

11 4040.2944 4011 5040.5544 401240.6844 40

12 lO40.6944 4012 2040.8444 4012 304lei544 40

17.4041.5544 4012 5042.0044 401342.4744 30

6 3056.41.44 306 4055.9244 306 SO~4.1244 30

153.1444 301 lO52.194.4307 2051.4844 30

7 3050.4144 307 4049.1244 307 5048.0844 30

R41.6544 308 1047·U

44 308 20"'6~5644 30

(3 3046.2744 30A 4045.44 308 SO45.1644 30

944.8944 309 1044.6444 309 2044.4544 30

9 3044.0244 309 4043.2744 309 SO42.4144 30

1041.1)544 30lO lO41.1344 30lO 2040.4344 30

lO 3040.0544 30lO 4039.5944 3010 5039•.4144 30

1139.2644 3011 1039.1244 3011 2039.1744 30

11 3039.3144 3011 4039.4544 3011 5039.5544 30

1239.6944 3012 1039.1344 3012 2039.8144 30

12 3040.1444 3012 4040.S844 3012 504l.lì44 -30

1341.b844 30n lO41.8544 3013 2042.S044 30

13 304~.ì.344 3013 4043.1044 3013 SO44.2044 30

1444.6344 204'; 3055.9344 206 4055.-5444 20

6 5054.5844 20153.2644 207 lO52.1544 lO

7 2051.5544 207 3050.8844 207 4050.1044 2·0

1 5049.4044 20R48.7644 208 lO48.3544 20

8 2041.8344 20A 3047.2144 208 4046.6344 20

8 5046.2144 20945.9244 209 1045.6544 20

q 2045.3344 209 3045.1144 209 4044.6944 20

9 SO44.2044 201043.6344 20lO lO4~.3044 20

lO 2042.8,144 2010 3042.2944 2010 4041.:7944 20

lO 5041.3344 20n40.88-44 2011 lO40.2544 20

11 2040.1244 20Il 3039.89442011 4039.6644 20

11 SO39.4844 201239.4144 2012 lO39.4544 20

12 2039.6344 2012 3039.9444 2012 4040.4044 20 12 50

40.9344 20l~41.4944 2013 lO41.704420

3 2042.33442013 3042.•9344 2013 4043.4944 20

13 SO43.9844 201444.4144 lO6 3055.n44 lO

6 4054.9'744 lO6 SO54.4344 lO753.5944 10

1 lO52.7444 lO7 2052.0744 lO7 3051.624410

7 4051.04.44 lO1 5050.2944 10849.4544 lO

8 lO48.#J944 108 2048.2244 108 3047.1444 lo

F.l4.041.2744 10~ SO46.9844 10946.1544 lO

9 lO46•.5444 lOq 2046.3444 lo9 3046.1244 lO

9 404'5.8144 lO9 5045.4844 lOlO45.0144 lO

lO lO44.7144 1.0lO 2044.4644 lO10 3044.2844 lO

10 4043.874410lO 5043.3944 lO1142.71'44 10

11 lO42.2944 lO11 2041.9944 1011 3041.4944 10

n 4040.9744 1011 SO-40.5144 10124.0.1444 lO

12 1040.0144 1012 2040.1044 1012 3040.2644 10

12 404Ò~5944 lO

12 5041.0244 lO1341.5144 lO

13 lO41.1044013 2042.2644 1013 3042.80

GEOID'COMPUTATIONS IN rtm ITALIAN AREA 235

44 lO 13 4043.3144 lO13 5043.7144 lO1444.1644

6 30~~:3~44~ 40~4.0944'

• 19U:3l44'7 44.lO3.0344

447 30S2•.Ò2447 4051.38447 5050.53

44849.50448 lO48.90448 20••8.40

448 3047.97448 4047.62448 SO47.45

44947.31H9 lO47.21449 2047.09

449 JO46.94449 4046.7~449 5046.51

441046.0044lO lO45.8444lO 2045.56

44lO 3045.12440lO 40404.8844lO 5044.14

441144.234411 lO43.854411 2043.48

44011 3043.074411 4042.634411 5042.06

441241.514412 lO41.204412 2041.10

4412 30'41.004412 4041.134412 SO41.38

441341.134413 lO41.844413 2042.30

4413 3042.764413 4043.204413 5043.59

4401443.9243 SOlO46.54-43 50lO lO46.37

43 50lO 2046.26435010 3046.1643 SOlO 4045.89

USOHl 5045.6143501145.4143 5011 lO45.12

4;)50

11 2044.8943 5011 3044.5543 5011 4044.5343 50

11 5043.9843501243.3243 SO12 lO42.7743 50

122042.4543 5012 3042.1343 5012 4042.01.4350

12 5041.9743 501341.9543 SO13 lO42.1'343 SO

13.2042.4643 5013 3042.8443 5013 4043.19-43 50

135043.4843 501443.7343 40lO47.33

4~ 41)18 1847·~843 4819 ~g47·n4l 40n 30:l:U4 40

4f.i.43 446.4 40

43 4011 lO46.4943.40Il 2046.2843 4011 3045.86

43 4011 4045.674,3~4011 SO45.3343 401244.99

43 4012 lO44.4443 4012 2043.9343 4012 3043.41

43 4012 4043.2143 4012 5043.0043 4.01342.10

43 4.013 lO42.6343 4013 2042.8243 4013 3043.11

43 4013 4043.32434013 SO.43.4843 4{)1443.62

43 30lO/te.0643 '30lO lO48.0143 30lO 2041.91

43 30lO 3047.964330lO 4047.9943 30lO 504.7.90

43 301147.(f0433'()li lO47.7943 3011 20~1.63

43 3011 3047.1843 lan 4'046.7643 3011 5046.49

43 3012.,.6•.17·433012 lO45.8043 3012 2045.48

43 3G12 3045.04-43 3012 40'44.6443 3012 1504••.29

43 301343.9543 3013 lO43.1543 3013 2043.73

43 3013 3043.1943 3013 4043.8543 3013 5043.88

43 301443.9343 3014 lO44.1543 30'14 2044,,25

43 3014 3044.4143 la14 4044.6143 3014 SO44.86

43 301545.1543 20lO48.6243 20lO lO48.63

43 20lO 204A.5943 20lO 3048.4743 20lO 4048.52

43 20lO 5048.634.3201148.7343 20Il lO48.63

43 2011 2lt4R.3143 2011 3048.0643 2011 4047.74

43 2011 5047.4743 201247.2843 2012 lO46.99

43 2012 2046.6243 2012 3046.2643 2012 4045.83

43 ?O12 5045.3943 20P44.9743 2013 lO44.71

43 2013 20.,.4.3543 2013 lO4'h0643 2013 4043.93

43 2013 SO43.8843 20·1443.8543 2014 1043.99

.,.32014 204_,,1043 2014 3044.2843 20H 4044.52

43 2014 5044.8143 20}C;45.1543 lOlO49.00

43 lOlO lO4Q.0443 lOlO 2049.0443 lOlO 3049.04

43 lOlO 4049.2243 lOlO 5049.4043 lO1149.31

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43 lO

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GEOIDCOMPUTATlONS IN TRE ITALIAN AREA237

42

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41 2012 3049.7841 :2012 4049.7141 2012 SO49.61

41 201349.5541 2013 lO49.5641 2013 2049.54

41 2013 3049.5041 2013 4049.39412013'·5049.34

41 201449.4841 2014 lO49.4541 2014 2049.46

41 2014 3049.6441 2:014 4049.8241 2014 5050.00

41 201550.1041 20l~ lO50.2141 2015 2050.26

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238 a. BENCIOUNI- P. GASPERINI-L. MUSSIO- F. SANSO- S. ZERBINI

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GEOID COMPUTATlONS IN·TRE ITALIAN AREA 239

40 4014 4050.4940 4014 50 '50.3740 4'01550.434'040 l~ 18~g:~~40 40l~~8~g:~t48 4'0l~ 30~0.444'04'0

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16 304.7.1940 2016 4041.0240 2016 5047.054'020

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240 B.BENCIOLlNI .......•P. GASPERlNL ••...L. MUSSIO •.•..F. SANSO~ S. ZERBINI

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9 304e.06399 4041.743~9 5047.23

GEOID COMPUTATIONS IN THEITALIAN AREA241

39

lO46.693915 3'045.623915 4045.5139

15 5045.57391645.753916 lO45.9639.

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13 4045.1538 2013 5044.9538 201444.873820

14 lO44.8738 2014 20·44.9636 ~O14 3045.1638 ·2·0

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15 lO4!i.1.738 2015 2045.7338 2015 3045.0538 20

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16 lO42.1738 2016.2040.6938 lOi6 3039.203820

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17 3032.97381246.163812 lO46.19

242 B. BENCIOLINI - P. GASPERINI - L. MUSSIO - F. SANSO - S. ZERBINI

38

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37 lO

15 3040.0937 lO15 4038.0537 lO155036.18

GEOID COMPUTATIONS IN THEITALIAN AREA 243

37 lO37373737373737373736 5036 5036 5036 5036 5036 403~ 4036 4036 4036 4036 4036 3036 3036 3036 3036 30

1612 20

12 50J 2013 5014 2014 SO15 2015 5016 2014 lO14 4015 lO15 4016 lO1414 3015}I; 301616 3014 2014 5015 2015 5016 20

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3737373737373737373736 5036 5036 5036 5036 5036 4036 4036 4036 4036 4(136 3036 3036 3036 3036 30

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. 36 3036 3036 30

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