Irene FISCHER, Mary SLUTSKY, F.R. SHIRLEY, P.Y. WYATT l i t Army Map Service, Washington, D.C.

NEW PIECES IN THE PICTURE PUZZLE OF AN

ASTROGEODETIC GEOID MAP OF THE WORLD

Abstract

The astrogeodetic geoid map presented to the IUGG at Helsinki in 1960 has been updated to reflect the accumulation of new data. The geoid map of North America has been recomputed, that of South and Central America enlarged. A geoid chart of Australia has been added. Various other improvements were made. Terrestrial gravity was used for interpolation, and satellite observations for inter- continental connection. World datum parameters were derived in various solutions. Satellite positioning of the major astrogeodetic datum blocks leads to an equatorial radius a = 6378142 m for a f la t teningf = 1/298.25. If surface gravity is included, the radius is larger. A reference figure a = 6378150 m and f = 1/298.3 is recom- mended for practical applications ?

1. Introduction

Quite a few geoid charts are to be found in the current technical literature. They were derived from satellite data and cover the world. Why should we still be interested in the laborious process of piecing an astrogeodetic chart together.

While such comments are heard occasionally, astrogeodetic charts are nonetheless used as standard fo r comparing the various satellite charts. One reason is the variety of these charts depicting presumably the same geoidal surface : the charts look similar at first sight, but less so i f specific geoidal heights at specific places are needed. The limitations of the classical astrogeodetic method to land areas permit a direct comparison only for stations within the same datum, after allowing for the difference in reference system. The land limitations are relaxed somewhat through the use of HIRAN, stellar triangulation, and SECOR, that is through geometric techniques by which the area of a specified geoidal datum can be extended. If such extension is carried far enough so as to overlap with the area of another geodetic datum, conversion formulae can be derived to put both areas on the same datum. Also dynamic satellite results in the form of geoid charts or tracking station coordinates can be used to connect the isolated astrogeodetic datum blocks. Furthermore, dynamic results provide the relation to the center of

199 6

Irene FISCHER, Mary SLUTSKY, F.R. SHIRLEY, P.Y. WYATT III

mass of the earth. The advice received from these various satellite results on how the astrogeodetic blocks should be put together in relation to each other and to the center of mass is, however, not exactly the same at this time.

The construction of astrogeodetic geoid charts for the purpose of detailed local information as well as for the derivation of world geodetic systems has been carried out at the Army Map Service within a series of studies of the Figure of the Earth over several years. The present study is but one of this series. For proper perspective some highlights of the series are briefly recapitulated here.

When the two long meridional arcs from Canada to Chile and from Finland to South Africa were completed in 1953 and 1954, a tentative size of the earth was derived from astrogeodetic deflections of the vertical, indica[ing that the equatorial radius of the International ellipsoid was much too large [ 1 ] . Then the method was changed from deflections to geoidal heights [ 2 ] . All available geoid charts were collected and new ones constructed, and the Molodenskiy correction was introduced to cope with scale distortions in poorly fitt ing extended nets. The result upheld that of the previous study, suggesting that also the International flattening was too large. For various reasons, however, the conventional flattening 1/297 paired with an equatorial radius of a = 6378270 m (the Hough ellipsoid) was adopted for the Vanguard Datum and the Tentative World Datum [ 3 ] , although the free-air solutions had shown smaller values. Soon after that the first satellites went up. The analysL of observations from the Vanguard satellite yielded the value 1/298.3 for the flatt~,ning [ 4 ] , which v~as adopted for the Figure of the Earth studies. The mathematical relationship between values of the flattening and of the radius for the same observational material would automatically yield a smaller radius. A recomputation, incorporating all astrogeodetic, gravimetric, and satellite data available at that time, was presented to the IUGG at Helsinki in 1960 [ 5 ] . The results were confirmed by also using elaborate statistical methods on the same data [ 6 ] . The Mercury Datum (a = 6378166 m for f = 1/298.3) is solution 4 in

reference 5b, using gravimetric orientation. The ellipsoid used in the South Asia Datum (a = 6378155 m for f = 1/298.3) is solution 3, derived purely astrogeodeti- cally. A third solution a = 6378160 m (p. 249, s was submitted to the Working Group on Astronomical Constants of the International Astronomical Union, Hamburg 1964. The present study updates the Helsinki paper. The astrogeodetic geoid chart has been extended to formerly blank areas, and satellite results were used for intercontinental connections.

2, Geoid Charts

The North American geoid chart presented at Toronto 1957 has been recomputed with some deflection values and a new method [ 7 ] . Geoidal profiles along meridians and parallels spaced 1 ~ apart were computed by formulas conform- ing to the projection method, and the geoidal height values at the intersection points were adjusted with weights reflecting the number of given deflection values pertinent to the adjacent l~ sections. The geoid profile along the 35 th

200

NEW PIECES IN THE PICTURE PUZZLE ...

parallel, observed and computed by the Coast and Geodetic Survey [ 8 ] , was held fixed after the reference was changed to conform with zero meters at the origin Meades Ranch. As a second backbone a dense meridional geoid profile in the center of the country was constructed by gravimetric interpolation of the astrogeodetic deflections. The southern part of this profile was computed by the Coast and Geodetic Survey, using the circular template method. For the northern part the Coast and Geodetic Survey had observed a 200-mile wide band of gravity values ; the profile was computed by the Army Map Service [ 9 ] with a new automated method of small uniform rectangular template compartments [ 10 ] . The North American Datum was tentatively extended along the North Atlantic HI RAN path through utilizing the six deflections on the icecap observed by the 1959 Interna- tional Glaciological Expedition to Greenland, together with the astronomic positions observed by the Army Map Service at some HI RAN stations. Also stellar triangulation provided some geoidal heights.

With more deflections in Mexico and Central America the geoid compu- tation was extended to South America. A geoid chart of South America on 1956 Provisional South American Datum and also on the more suitable Corrego Allegre Datum was prepared for the Pan American Consultation in Guatemala City, 1965, with all deflections available at that time [ 11 ] . Since then more values in eastern and southern Brazil became available and the chart could be extended through Uruguay and along the 25th parallel in Paraguay. It is hoped that in the near future an Argentinian geoid profile from the border with Uruguay to the border with Chile will become available and will strengthen the huge loop around the continent. Transformation formulae between 1956 SAD and 1927 NAD were derived at the junctions in Nutibara, Colombia, and in Trinidad. Figure 1 shows the geoid contours in the Western Hemisphere on 1927 North American Datum.

Figure 2 shows the geoid contours on European Datum for Europe, Asia, and Africa. The chart for Europe is very similar to G. Bomford's more detailed chart of 1963. Spain has been recomputed. New deflections in southern Italy and corrections to some Greek values have lowered the geoidal height at Athens by several meters. A loop around the Mediterranean Sea has been attempted by extending Dufour's geoid contours in Morocco, Algeria and Tunisia to Libya and Egypt, then using Tengstr6m's gravimetric profile from El Alamein to Athens [ 12 ] . There is still a misclosure of ten meters, the African route yielding a negative value of about - 6 m. For the time being the value of zero meters with an uncertainty of + 5 m was adopted at Athens. This lowers the profile through Turkey and Iran by ten meters and slightly changes the transformulation formulas at Koh-i-Malik between South Asia Datum and European Datum to the following :

A N ( E D--SA D ) = + 112 cos ~ cos ~ + 49 cos ~ sin ~k + 116 sin r + 94 sin2 ~ _ 233

South Asia Datum itself is, of course, not affected. Geoid contours on that datum were extended to Thailand and West Malaysia (Figure 3), and then

201

Irene FISCHER, Mary SLUTSKY, F.R. SHIRLEY, P.Y. WYATT III

converted to European Datum by the above formula.

The ED contours in the USSR were recomputed, replacing Dubovkiy's chart with Molodenskiy's. The Manchurian tie was strengthened considerably by establishing transformations between horizontal positions on Iman Datum, Svobodny Datum, Manchurian Principal System, and Pulkovo 1942 Datum [ 13 ] . A three-dimensional transformation between Pulkovo 1942 Datum and Manchurian Principal System could now be derived, based on position and height differences at Svobodny, and strengthened by height differences along the two arcs from Vladi- vostok to Khabarovsk and along the Amur, This changed somewhat the transfor- mation between Tokyo Datum and European Datum. New deflections permitted an extension of the geoid chart of Japan to the south, including the Ryukyu Islands.

In Africa, data outside the 30th meridian are beginning to come in slowly but surely. A geoidal profile along the 30th parallel south was observed and computed by B.M. Jones [ 14 ] . All deflections on South African Arc Datum were collected, including those along the sixth parallel south latitude [ 15 ] , and tentative geoid heights were computed with reference to the origin at Buffelsfon- tein. Similarly, a geoid chart referred to Adindan Datum was computed for the area between Muluk and Adindan including the 12th parallel from Chad to the Red Sea. Transformation formulae between these two datums were derived at their common point Mulu :, and between Adindan Datum and ED at Adindan. The new work in North Afric~ has been mentioned earlier.

Figure 4 shows a simplified version of the first astrogeodetic geoid chart of Australia [ 16 ] , referred to the New Australian Geodetic Datum with Johnston origin. The deflection values and the definition of the reference datum were provided by the Australian Department of National Development. The construction of the geoid was carried out in a combination of two methods. The computation of evenly spaced geoid profiles along meridians and paralles as described in reference 7, was combined with the computation of loops along the triangulation routes, using the geoidal increments between individual stations. Values obtained by the second method were held fixed in the over-all adjustment.

3. Intercontinental connections

The astrogeodetic geoid coverage, referred to three major geodetic datums, was represented by 644 hypothetical points in a roughly 4 ~ x 4 ~ distribution for computational purposes : 303 points in the Western Hemisphere on 1927 North American Datum ; 277 points in Europe, Asia, and Africa on European Datum ; and 64 points in Australia on Australian Geodetic Datum. Various techniques compete in suggesting how these three datum blocks should be connected to a uniform world datum.

SECOR and stellar triangulation which would extend and connect these datum blocks geometrically unfortunately did not yet have numerical results available at this time. A geometric solution was offered through Baker-Nunn

202

NEW PIECES IN THE PICTURE PUZZLE ...

camera observations, where station coordinates computed from these observations could be compared with their counterparts as computed from the latest astro- geodetic information. The differences at each station provide a datum shift ; they were averaged if more than one station was given on the same geodetic datum. After shifting the 644 points accordingly, a best fitting equatorial radius was determined for alternative values of the flattening.

Dynamic solutions are possible through the use of satellite or surface gravity geoid charts by the technique of a geoid match. The 644 astrogeodetic geoid heights were paired with the corresponding values read from a gravity chart. In a least squares solution in ten unknowns the flattening of each astrogeodetic datum block was changed to that of the gravity chart, and the result yielded the three datum shifts in three coordinates and a best-fitting equatorial radius. A second solution was then added for an alternative value of the flattening under the condition that the volume of the ellipsoid should be the same.

In deriving the Tentative World Datum [ 3 ] and the Mercury Datum [ 5 ] only surface gravity charts had been available. This time we had a variety of such charts derived from different types of data, by different techniques, and in various combinations. In order to see the influence of the data more clearly, we concen- trated on the independent geoid charts derived from Doppler satellite data [ 1 7 ] , Baker-Nunn satellite data [ 18 ] , and surface gravity (Uotila's material as used by Levallois [ 19 ] ). We did not use charts from combined data, because their purpose is to smooth over discrepancies rather than to point them out, Geoid charts from combined satellite and surface gravity show essentially the same general features as satellite geoids, since the lower harmonics are those of the satellite representation ; and surface gravity charts with data extrapolations based on correlations with satellite gravity blur the differences by this very assumption. In fact, the Levalloiso Uotila chart did lead to results which are rather different from those obtained by satellites. For confirmation, two other treatments of the same surface gravity material were investigated also, namely Kivioja's densification with topography and isostasy, as used by Levallois [ 19 ] , and a chart by the Aeronautical Chart and Information Center based on Uotila's own procedure [ 20 ] . Different results could be anticipated when comparing some features in these various charts. For example, the slope from Greece to Ceylon is about 110 m on satellite charts, but only around 50 m on surface gravity charts, The slope across Australia is respectively on the order of 100 m versus 20 m.

4, Results

The value of the flattening 1/298.3 obtained from the Vanguard satellite in 1958 [ 4 ] is essentially still valid today, except for the refinement in the second decimal of the denominator to around 1/298.25. The effect of this refinement on the length of the radius is insignificant.

Table 1 lists the various solutions obtained. If one expects a geodetic accuracy of a few meters, the variety of numbers is bewildering, On closer

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inspection we notice that the values for the equatorial radius form two clusters about 40 meters apart, depending on satellite or surface gravity advice. The difference due to the refinement in flattening is an immaterial 1.2 m as expected. The datum shifts are essentially equivalent to statements of where the center of the earth is. The differences in these shifts between the geometric and dynamic Baker- Nunn solutions (1, 2, 4) are mostly but not always within the 20-meter accuracy claimed. The differences between the dynamic Doppler and Baker-Nunn solutions (3, 4) are smaller. The three surface gravity solutions (5, 6, 7) seem roughly to support each other, although the differences reach 40 to 50 m in a few cases. The differences between satellite and surface solutions are particularly large in the y- axis. A simple explanation of such differences lies in the deficiency of surface gravity coverage compared with the overwhelming abundance of satellite data. Yet, in pre-satellite times, a so-called "absolute" orientation of astrogeodetic datum blocks was based on even less than today's holdings. Whether the discrepancies are solely due to this deficiency or to a real difference is still debated today.

The two geometric solutions 1 and 2 are based on the coordinates of the Baker-Nunn observation stations as given before and after the 'scale was changed from 6378165 m to 6378155 m due to a more recent GM value [ 18 ] . A mean datum shift was derived in either case from stations 1,9, 10 for NAD, stations 4, 15, 6, 8 for ED, and stations 3, 23 for Australian Geodetic Datum (AGD). The dynamic solution 4 is not affected by this change in scale.

Table 2 shows a comparison between the geoidal heights at the 1 2 Baker ~ Nunn stations as computed from their coordinates (ref, 18, vot. 1, p. 2) and as read from the gravitational geoid chart (ref. 18, vol. 2, po 134). These numb "rs provide an opportunity to test Kaula's I~ 21 ] contention that the mean di, ~rence of geometrical and gravitational geoid heights calculated from the same solution for the 12 Baker-Nunn stations may be taken as a correction to the initial semimajor axis, which leads to a new value of the size of the earth. Persuasive as his argument sounds, it does not seem to hold for various reasons. One obvious objection is the great discrepancy from station to station in Table 2 {and even more so in Kaula's Table 8), so that the inclusion of one station more or less would lead to a very different result, More fundamental is the aspect that these geoidal heights are rather insensitive to the nominal semimajor axis used init ial ly, in a similar way as a gravimetric geoid referred to the International flattening does not reflect the radius of the International ellipsoid. Table 2 stands whether we use the coordinates keyed to the scale 6378165 m or 6378155 m . and the gravitational heights are valid for both. Should we then subtract the 8 meters from the one or the other or from what ? In our procedure the astrogeodetic data impose a constraint leading to the dynamic solution 4. Should we subtract the 8 meters from this result to anticipate the corresponding geometric result ? Yet, of these geometric solutions 1 and 2, one is smaller and the other larger than the dynamic solution 4, Thus it seems that the mean difference between geometric and dynamic geoid heights has no bearing on the actual radius of the earth . it merely indicates the inner consistency of the two methods, which is wi thin the accuracy claimed.

205

Irene FISCHER, Mary SLUTSKY, F.R. SHIRLEY, P.Y. WYATT III

Table 2

Geometr ic and Dynamic Geoidal Heights on the

1966 Smithsonian Inst i tut ion Standard Earth

Stat ion No. Baker-Nunn

1

2

3

4

5

6

7

8

9

10

11

12

Mean

Geometric

--41 m

+ 1 7 m

- - 0 3 m

+ 3 0 m

+ 2 7 m

- - 7 2 m

+ 2 7 m

- - 3 5 m

--41 m

- - 5 2 m

+ 3 8 m

- - 8m

Dynamic

- - 2 4 m

+ 2 4 m

0

+ 3 7 m

+21 m

- - 4 5 m

+ 20m

- - 1 0 m

--21 m

- -29 m

+ 1 7 m

- - 0 2 m

Geometr ic -Dynamic

- - 1 7 m

- - 7m

- - 3 m

- - 7m

• 6m

- - 2 7 m

+ 7m

- - 2 5 m

- - 2 0 m

- - 2 3 m

+ 2 1 m

-- 6m

-- 8m

5. Conclusions

To crystallize the outcome of this study a mean of the three satellite solutions 2, 3, 4 was formed to represent the satellite result (sol. 8 in Table 3). Solution 1 was eliminated in favor of Solution 2, because the authors of the "Standard Earth" recommended the reduction of scale in their final coordinates. For their adopted value of the constant GM = 3.986 013 (102~ cm3/sec 2, derived by the Jet Propulsion Laboratory from Ranger 6, 7, 8, 9 and Mariner 4, the equatorial gravity value "re consistent with the solution would be 978.033 gal. Astrogeodetic geoid contours referred to this world datum are shown in Figure 5.

For contrast, Figure 6 shows the astrogeodetic geoid connected by surface gravity of Solution 5.

A representative solution should include both satellite and surface gravity data, giving the superior satellite techniques much more weight. In Solution 9 the

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NEW PIECES IN THE PICTURE PUZZLE ...

Table 3

Combined Solutions

Type of Solut ion

8. Satellite mean of Sol. 2,3,4

9. Weighted mear satellite and .surface

Datum shift to WD

f rom A.X A y AZ

NAD -- :11 m +161 m +181 m

ED -- 8 3 m - - 1 2 2 m - - 1 2 3 m AGD - - 1 1 6 m -- 5 5 m + 1 1 0 m

NAD -- 17m + 1 4 0 m + 1 8 2 m ED -- 7 7 m - 9 6 m - - 1 2 2 m AGD - - 1 0 5 m -- 3 7 m + 9 3 m

equatorial radius

f = 1/298.25

6378142.4 m

6378152.7 m

i f =___~t/.298.3

63781 ---G3

63781 51.8 m standardized : 6378150 m

mean satellite Solution 8 is given three times the weight of the mean from the surface solutions 5, 6, 7. The uncertainty of the resulting radius is estimated as

about + 10 m computationatly. A realistic uncertainty range of -+ 40 m rather than the generous + 80 m recommended by the International Astronomical Union in 1964 might be sufficient.

For practical applications a standardized reference figure a = 6378150 m and f = 1/298.3 is recommended. The rounding of the value for the radius affects the geoidal heights by an insignificant amount. By rounding the reciprocal of the

f lattening to one decimal a geodetically un impor tan t discussion of the correct second decimal is avoided. Furthermore, the rounded value provides the least disruption in routine work where tables are widely in use. The datum shifts of Solut ion 9 together with the ellipsoid parameters a = 6378150 m for f = 1/298.3 may be regarded as " A Modif icat ion to the Mercury Datum". Geoid contours on this datum are shown in Figure 7* . For GM as above the corresponding value of

"Ye = 9 7 8 . 0 3 0 gal.

Comparing the present result wi th that presented at Helsinki [ 5 ] we notice that the size of the earth has shrunk again by a few meters. Satell ite derivations seem to insist on a smaller size and the smaller GM value points in the same direction. To investigate the effect of the new astrogeodetic material, the best f i t t ing radius for the separate hemispheres was computed and compared with previous results. (See Table 4).

* -- This Figure7 is superseded by Figure 7a ; see Addendum.

207

Irene FISCHER, Mary SLUTSKY, F.R. SHIRLEY, P.Y. WYATT III

Table 4

Influence of new deflections (1960-1967)

Western Hemisphere

Eastern Hemisphere

f

1/298.3

1/298.3

a1960

6378153 m

6378248 m

al 967

6378193 m

6378199 m

number and spacing of stations

1990 196_Z 141,5 ~ x,5 ~ 303,4 ~ x 4 ~

160,5 ~ x 5~ 277.4 ~ x 4 ~ I

Both solutions are now almost the same, yet considerably larger than the global solutions. Maybe, the larger solutions derived from surface gravity could be explained by the fact that the great bulk of gravity observations are made on land. Only a few decades ago, lacking intercontinental connections, one would have concluded that the practically common value in Table 4 represented the size of the earth. This contradicts the idea that derivations from land areas tend to foreshorten the semimajor axis. It would seem, rather, that the continental areas on the whole are flatter, less curved, than the ocean areas.

Addendum

At the XlVth General Assembly of the IUGG at Lucerne, September 1967, a paper "The Geopotential to (14,14) from a Combination of Satellite and Gravimetric Data" was presented by R. H. Rapp, which contained a geoid chart computed from surface gravity only. The data used represent the current holdings

at Ohio State University and include the older data collections of Uotila and Kivioja, which we used in the form of Levallois' two geoid charts. We therefore replaced these two charts by Rapp's and proceeded as before.

The solution derived from a geoid match at the 644 astrogeodetic points with Rapp's values is given below, along with the final solution containing the replacement. The standardized rounded parameters a = 6378 150 m and f = 1/298.3 are recommended for practical applications where the fluctuations in the last digit are of no significance. Geoid contours on this datum, called "A Modi- fication to the Mercury Datum (1968)", are shown on the Figure below.

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NEW PIECES IN THE PICTURE PUZZLE ...

Type of Solution

10. Astrogeodetic and Rapp

t 1. Weighted mean satellite and surface, "A Modification tc the Mercury Datum 1968"

Datum shift to WD

from A x A y A z

NAD -- 43m + 1 0 6 m :F165m ED -- 7 2 m - 6 8 m - - 1 2 6 m A G D - - 1 0 5 m - 28m + 78m

NAD -- 18m + 1 4 5 m + 1 8 3 m ED -- 81m - -104m - -121m AGD - -105m -- 44m + 94m

Equatorial radius

f = 1/298.25 f = 1/298.3

6378159.8 m 6378158.6 m

6378149.0 m 6378147.8 m a = 7 m

standardized 6378150 m

REFERENCES

[1] B.H. CHOVITZ and I. FISCHER : A new determination of the figure of the earth from arcs, Trans. Amer, Geophys. Union, 37,534-555, 1956.

[2 ] I. FISCHER : The Hough ellipsoid or the figure of the earth from geoidal heights, Bull. Gdod., No. 54, 1959.

[3] I. FISCHER : A tentative world datum from geoidal heights based on the Hough ellipsoid and the Columbus geoid, J. Geophys. Res., 64, No. 1, 1959.

[ 4] J.A. O'KEEFE et al : Harvard Announcement Card No. 1408, June 24, 1958.

[5 ] I. FISCHER : (a) An astrogeodetic world datum from geoidal heights based on the flattening f = 1/298.3, j. Geophys. Res., 65, No. 7, 1960.

(b) The present extent of the astrogeodetic geoid and the geodetic world datum derived from it, Bull. G~od., No. 61, 1961.

[6] W.M. KAULA : A geoid and world geodetic system based on a combination of gravimetric, astrogeodetic, and satellite data, NASA, TN-D702.

[7 ] I. FISCHER : A revision of the geoid map of North America, J. Geophys. Res., 71, No. 20, 1966.

[8 ] D.A. RICE : A geoidal section in the United States, Bull. Gdod., No. 65, 1962.

[9 ] I. FISCHER, R. SHIRLEY and P. WYATT, III : A geoid profile in North America from a combination of astrogeodetic and gravimetric data, J. Geophys. Re&, 71, No. 20, 1966.

[10] I. FISCHER : (a) Gravimetric interpolation of deflections of the vertical by electronic computer, Bull. Gdod., No. 81, 1966.

(b) Slopes and curvatures of the geoid from gravity anomalies by electronic computer, J. Geophys. Res., 71, No. 20, 1966.

209

Irene FISCHER, Mary SLUTSKY, F.R. SHI R LEY, P.Y. WYATT III

[11] I. FISCHER and Mo SLUTSKY : Un Estudio Del Geoide en Sud America, Revista Cartogr. No. 14, 1965.

[12] E. TENGSTROM : Research on methods of determining level surfaces of the earth's gravity field, Part II I, Geodetic Institute, Uppsala University, 1964.

[13] A. GLUSIC : Inter-Datum Relationships in the Russian Far-East, Army Map Service Technical Report (in preparation).

[14] 8.M. JONES : (private communication), Dept. of Land Surveying, University of Natal, Durban, Republic of South Africa.

[15] P. MEEX : L'Arc Congolais du 68me Parall~lesud du lac Tanganika ~ I'Ocean Atlantique, Inst. Gdogr. Nat., Paris 1967.

[16] I. FISCHER and M. SLUTSKY : A preliminary geoid chart of Australia, The Australian Surveyor, 1967~

[17] R.J. ANDERLE : Geodetic parameter set NWL-5E-6 based on doppler satellite observations, NWL Report No. 1978.

[18] C. LUNDQUIST and G. VEIS : Geodetic parameters for a 1966 Smithsonian Institution Standard Earth, Smithsonian Astrophys, Obs., Special Report 200.

[19] J.J. LEVALLOIS and H.M. DUFOUR : Sur une Application de la formule de Stokes ~ la Recherche du G~oi'de Mondial et du Centre des Masses de la Terre, Ass. Int. de Gdoddsie, Lucerne 1967.

[20] U.A. UOTILA : Gravity Anomalies for a Mathematical Model of the Earth, IUGG, Berkeley 1963.

[21] W.M. KAULA : Tesseral Harmonics of the Earth's Gravitational Field from Camera Tracking of Satellites, J. of Geophys.Re&, Vol. 71, No 18, Sept. 1966, p. 4387~

210

Fisure I. Geold contours of Western Hemisphere on 1927 North American Datum, in meters.

211

Figure 2.

Geoid contour~ of Europe, Asia and Africa on European Datum, in meters.

212

213

44" 48" 52" 56" 60" 64" 68" 72" 76" 80" 84" 88" 92" 96" 100"104"108" 112" 116~ ~ 128"

ff -,~.

/ '50- / "~

SOUTH ASIA DATUM Geo]d cen~oers In n~;ers

/ \

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. \ "

~ / } /

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44" 48" 52" 56" 60" 64" 68" 72" 76" 80" 84" 88" 92" 96" 100"]04"108"112"116"120"124"128"

Fi2ure 3. Geoid contours of South As~ on South Asia llltum, in meters.

214

100 ~ 104" 108" 112" 116" 1 2 ~ 124" 128" 132" 1316" 140" 144" 14.8" 152" 156" 160" 164" 168" 172" 176 ~ 180" , j

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Figure 4. Geoid contours of Australia on Australian Geodetic Datum, in meters.

215 7

Figure 5.

Geoid contours of Astrogeodetlc World D~tu~ with Satellite Orientation, in meters.

2~6

I

217

Figure 6. Geoid contours of As t roseode t i r Geos connected

by.Surface Gravi ty ( L e v a l l o i s - U o t i l a ) in meters .

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