A Figure of the Earth for Australia
by B.P. Lambert, F.I.S.Aust.
(Originally presented at the 1962 Conference of the Australian and New Zealand Association for the Advancement of Science and published in The Australian Surveyor, Vol.19, No.3, pp.178-185.)
[This version of Lambert’s paper has added sections, defined by the use of square [] brackets, to present information not included by Lambert and thus simplify its reading today.]
Introduction
The prime object of a national geodetic survey is the determination of the coordinates of a fairly close network of inter-connected survey stations distributed over the whole area of the survey.
The desired coordinates consist of a radius vector from the point of intersection of the earth's polar axis and its equatorial plane combined with angular values expressed as latitudes and longitudes.
For convenience in computation, it is customary to adopt a geometrical Figure of the Earth to which all survey measurements can be reduced. Then, instead of using radius vectors an elevation can be given with reference to the adopted geometrical figure.
The geometrical figure usually used is an ellipsoid of revolution developed by rotating an oblate ellipse around the earth's north/south axis. Theoretically the centre of the ellipse should be located at the intersection of this axis with the plane of the equator and the surface of the ellipsoid should coincide with world mean sea level.
The semi major (equatorial) axis is normally denoted by
the letter a, the semi minor (polar) axis by the letter b and the
flattening (= a-b/a) by the letter f.
In 1924, on the basis of the data then available, the International Union of Geodesy and Geophysics recommended use of a Figure in which a = 6 378 388 metres and f = 1/297. However, more recent investigations indicate that the approximate values of a and f are 6 378 200 metres and 1/298.3.
The astro/geodetic approach to determination of the parameters of a reference ellipsoid
If the earth surface conformed to a mathematical figure, it would be possible to measure meridian arcs or sections of parallels of latitude and from these measurements, in conjunction with carefully observed astronomical values, directly calculate the parameters of the ellipsoid.
However, due to the irregular attractive force of the earth's topography and the heterogeneous distribution of masses within the earth's crust, it is impossible to directly determine a direction either parallel, or normal, to the surface of the Figure of the Earth.
This means that in fact there is available no true reference system for astronomical observations and that the bubbles of survey instruments do not normally coincide with a truly level surface.
The latter disturbance affects the determination of the difference of elevation between the surface of the terrain and the geometrical reference surface that has been assumed as coinciding with mean sea level.
In geodesy, reference is therefore made to another surface called the geoid, which for the purposes of this paper may be accepted as the surface established in relation to the terrain surface by the reductions to sea level that have been computed at ground stations from survey operations. It deviates slightly, and for the most part irregularly, from the mathematical Figure of the Earth.
The separations between the geoid surface and the reference ellipsoid are called undulations of the geoid and these undulations seem to consist of a combination of minor undulations, somewhat analogous to small, irregularly shaped, coastal type, sand dunes of fairly even height, which in turn appear to be superimposed on irregular major undulations some of which conform to a regular ellipsoidal surface and others which have gentle slopes and/or block shifts usually extending over fairly large areas but sometimes concentrated into small belts.
In the making of geodetic surveys, it is customary to start with an arbitrarily assumed Figure of the Earth and, either by direct analysis of the comparisons between computed geodetic coordinates and astronomically observed coordinates, or by analysis of the undulations of the geoid as deduced from these differences, arrive at the most appropriate figure for the area under survey.
In either case, particular coordinates and a particular orientation are assumed for the origin of a survey and an arbitrary reference ellipsoid adopted as a basis for survey computations.
When the first approach is used, an analysis is made of the discrepancies between the observed astronomical values and the computed coordinates of survey stations situated either along arcs of latitude, or of meridian, or over areas, and an appropriate least square fit is made to give the most probable corrections for the coordinates of the origin, the orientation of the survey and the parameters of the ellipsoid.
The resultant differences between observed coordinates and survey coordinates as computed on the new figure, are accepted as measures of the deviation of the vertical at the survey stations.
When the second approach is used the initial differences between observed and computed coordinates are accepted as measures of the deviation of the vertical with respect to the adopted arbitrary coordinates and reference ellipsoid.
These deviations are treated as slopes of the surface of the geoid and the geoid contours with reference to the arbitrary ellipsoid are calculated from them.
A least squares calculation is then made to give the most probable changes required in the origin and the reference ellipsoid in order to give a best fitting new ellipsoid.
Returning to the analogy of sand dunes it is necessary to measure a dense enough pattern of slopes from which to determine the contours of the dunes themselves and show up the underlying basic irregularities of the major undulations.
If, for example, a grid pattern of slope measurements is made of insufficient density the result will be a set of random values. In which case, if slopes are interpolated along the grid lines and contours deduced therefrom, the result will most likely be a set of contours that look like those of a sand dune pattern but in which the ridges, knolls, valleys and depressions will be strongly correlated with the grid pattern itself and more widely spaced than those which fit the actual minor undulations.
If lines of survey are developed as a vertical section of the geoid care has to be taken in order to ensure that the profile does not tangle up with another set of false undulations which normally develop with the consecutive addition of successive random quantities.
Because of these difficulties when deducing and analysing geoidal undulations, the most practical first approach appears to be that of developing a grid pattern of geodetic surveys with closely spaced astro/ geodetic stations and making a statistical analysis of the angular deflections.
As a knowledge of geoidal undulations is an essential scientific requirement it is desirable to eventually cover a country with a dense network of combined astro/geodetic stations, and to deduce the geoid undulations therefrom.
The desirable intervals between consecutive stations would have to be determined after analysis of results obtained from experimental spacings; Brig. G. Bomford has suggested an interval of 15 miles(1)[10 kilometres].
Many authorities contend that the values for deviation of the vertical can be largely ameliorated by applying corrections based on an assumption of isostatically compensated topography.
The method of calculating corrections most commonly used in geodetic survey operations is that developed by Hayford of the United States Coast and Geodetic Survey. The depth of compensation adopted is usually in the vicinity of 100 kilometres(2).
Some other authorities contend that the use of these corrections introduces systematic errors into the investigations.
In practice it seems that the isostatic approach improves the results obtained over short arcs and small areas, but over very extensive surveys the uncompensated effects tend to balance and much the same result is obtained from either approach.
In the Division of National Mapping an initial figure has been adopted as a basis for investigation of geodetic surveys in which a = 6 378 148 metres and f = 1/298.3.
[The source of these values may be found in Volume 1 of the 1971 National Aeronautics and Space Administration, Directory of Observation Station Locations, where it was stated that ”for a short period in 1962 computations were performed on the so-called “NASA” spheroid (a = 6 378 148 m; f= 1/298.3) with the origin at Maurice”. Trigonometrical station Maurice was on Maurice Hill some 16 kilometres south west of Orroroo, South Australia. Please refer to map below.
Map showing the progress of the Australian geodetic survey to 1960.
Red and blue shaded areas are first and second order triangulation networks respectively; the red lines first order traverse routes and red dots Laplace stations. The location of the first order station Maurice is shown in green and the position of the line along latitude 32S added for clarity. ]
Analyses have been made of deduced deflections to arrive at the best arc radius that will fit observations at 54 astro/geodetic stations on and adjacent to the 32 degree (South) parallel of latitude between Perth and Sydney.
In one analysis the observations were corrected on the assumption (Hayford's method) of isostatic equilibrium at a depth of 96 kilometres. Computation was stopped at the outer limit of Hayford's Zone 7 as Fischer(3) has demonstrated that the total effect of succeeding zones is negligible in respect of Australia.
The flattening of 1/298.3 was then applied to this isostatically determined radius to give a = 6 378 150 ± 100 metres.
In the second analysis no isostatic correction was made and the best fitting arc radius when combined with the same flattening gave a = 6 377 999 ± 110 metres.
In 1960, Milan Bursa of the Geodetic Research Institute, Prague, analysed observations at about 1 000 astro/geodetic stations in Europe and arrived at a = 6 378 112 metres and f = 1/298.4.
Analyses of geoidal undulations have been undertaken in recent years :
at Oxford University under the direction of Brig. G. Bomford utilising data available over Europe and parts of Asia and Africa; derived values were a = 6 378 201 metres and f = 1/297.65; these results were published in 1959.
Fischer also published in 1960 the results of a similar analysis of much the same materials, and also of surveys in the Americas, to arrive at a = 6 378 280 metres; f = 1/296.8; and a = 6 378 192 metres; f =1/297.7 respectively; when the flattening was held at 1/298.3 the a values became 6 378 248 and 6 378 153 metres, respectively. (10)
The utilisation of geodetic satellites for determination of the figure of the Earth
The development of the geodetic satellite equipped with a flashing light and telemetric devices should permit of worldwide geodetic surveys in which the angles from ground to satellite are deduced from photographic measurements of the flashing light against a star background.
These angular measurements will be referred to the celestial coordinates of the stars and therefore be free of any deviation of the vertical effect.
The resultant coordinates deduced from these surveys will consist of true geocentric angular coordinates and deduced radius vectors from the centre of the earth.
From a series of well spaced stations fixed by the satellite survey it should be possible to determine a very accurate Figure of the Earth and the geoidal elevation at each station.
If these stations can be located at the grid intersections of the geoidal profile survey, they could serve as control bench marks between which the sections of the astro/geodetic profile could be adjusted.
This procedure is largely a matter for the future, but possibly the not too distant future. (The USA currently plans such a geodetic satellite survey.)
A moon photography programme of observations(4) was undertaken at about 20 stations throughout the world during and immediately after the International Geophysical Year (IGY); Stromlo and Perth Observatories participated in this programme. The observations, which are still being processed by the US Naval Observatory, should produce somewhat similar but possibly less accurate results than those expected from the geodetic satellite surveys.
O'Keefe and others(5) in 1959 deduced, from a combination of radar and angular measurements to the moon, a flattening of 1/298.32 ± 0.05 and a value of a = 6 378 255 ± 30 metres.
Kaula of the USA National Aeronautics and Space Administration stated at the Symposium on Geodesy in the Space Age (Ohio State University, 1961) that the value of a derived from lunar distances is 6 378 251 ± 80 metres.
The gravity approach
Theoretically, it should be possible to determine the flattening of the earth by making numerous gravity measurements and then analysing the variation of these measurements in relation to the latitudes of the points of observation. It should also be possible to compute deflections of the vertical and undulations of the geoid.
However, in the formation of the earth the distribution of masses near to the surface has been most complicated, with the result that an enormous number of gravity determinations spread over the earth's land and sea surfaces will be necessary before accurate results can be obtained from these analyses.
On the other hand, recent observations of the performance of satellites in their orbits, where the effects of gravity forces are more uniform, have permitted a very accurate deduction of the flattening and the present best value appears to be f = 1/298.24 ± 0.01.(6)
Considerable statistical analysis, research and speculation is at present being made in respect of available gravity data with the object of deducing quantitative, systematic patterns.(7)
Combined approaches
In 1948 Sir Harold Jeffreys(8) published details of a general adjustment of the then known data concerning the dimensions of the Earth in which he took into account the results of astro/geodetic figure determinations, gravity anomalies and astronomical constants.
[Harold Jeffreys (1891-1989) was a British mathematician and geophysicist who determined a flattening of 1/297.10 ± 0.36 and a value of a = 6 378 099 ± 116 metres.]
Sir Harold's technique has recently been partially applied within the División of National Mapping to more recent arc measurements. Four Meridian arcs and two parallel arcs (previously analysed by Chovitz and Fischer(9) ) were used together with the best fitting arc derived from the United States Coast Geodetic Survey's measurements along the 35 degree (North) parallel(11) and the best fitting arc along the 32 degree (South) parallel as derived from Australian measurements.
Following Sir Harold's recommendation no corrections were made for isostatic compensation.
The investigation was limited firstly to an analysis based on the uncertainties disclosed by the measurements themselves and secondly to an analysis using these uncertainties corrected for variations between 10° and 30° squares (proposed by Sir Harold as a result of his gravity investigations).
The results of this investigation are as follows :
|
with original uncertainties : |
|||||
|
a and f variable |
|||||
|
= |
6 377 981 ± 61 metres |
f |
= |
1/299.0 ± 1.4 |
|
|
f held at 1/298.3 |
|
|
|
||
|
a |
= |
6 377 971 ± 54 metres |
f |
= |
1/298.3 ± 0.6 |
|
|
|
|
|
|
|
|
with uncertainties corrected for gravity variations : |
|||||
|
a and f variable |
|||||
|
a |
= |
6 378 138 ± 71 metres |
f |
= |
1/298.3 ± 0.7 |
|
f held at 1/298.3 |
|||||
|
a |
= |
6 378 138 ± 64 metres |
f |
= |
1/298.3 ± 0.6 |
Within the Division of National Mapping another analysis was made of arc values that had been computed on the basis of isostatic compensation. Chovitz and Fischer's(9) (isostatic) meridian arc solutions 17, 18, 21, and 22 were used and their parallel arc solutions P3 and P4 together with the Division's own isostatic analysis of the 32° (South) parallel.
[The details of the Chovitz and Fischer work mentioned above, are tabled below :
|
Solution |
Location (USA) |
A |
1/f |
|
17 |
Pacific meridian |
6 378 408 ± 170 |
295.6 ± 1.5 |
|
18 |
98th meridian |
6 378 508 ± 302 |
294.5 ± 2.5 |
|
21 |
24th meridian |
6 377 665 ± 141 |
303.8 ± 1.4 |
|
22 |
32nd meridian |
6 377 825 ± 109 |
302.3 ± 1.1 |
|
P3 |
44th parallel |
6 378 309 ± 116 |
297 |
|
P4 |
52nd parallel |
6 378 108 ± 105 |
297 |
]
The data was weighed on the basis of the probable error of the original computations.
The results were as follows :
|
a and f variable |
|||||
|
a |
= |
6 378 225 ± 56 metres |
f |
= |
1/298.5 ± 0.6 |
|
f held at 1/298.3 |
|
|
|
||
|
a |
= |
6 378 231 ± 52 metres |
f |
= |
1/298.3 ± 0.6 |
In 1960 Fischer(10) published a paper giving the results of analyses of geoid undulations over the Americas and over Europe, Africa and Asia. In these analyses one approach was based on the most probable fitting of a geoid as computed from astro gravimetric deflections and another on the matching of the astro/geodetic undulations with a series of undulations computed separately from gravimetric data.
With the flattening held at 1/298.3 the first approach gave a = 6 378 155(ii) metres and the second gave a = 6 378 160(iii) metres.(*)
Kaula in 1961(6) derived a figure from a most intensive analysis of gravimetric, astro/geodetic and satellite data and derived a Figure with f = 1/298.24 ± 0.01 and a = 6 378 163(iv) ± 21 metres.(*)
(*) These investigations appear to be based on much the same geodetic and gravity data.
Finally, within the Division of National Mapping all figure determinations known to have been made since 1900 were accepted as giving in each case separate dimensions for the a and b radii and it was assumed that the varieties of observations and treatments used over that period could reasonably be expected to give a random series of values.
It was further assumed that if each group exhibited a good statistical pattern and the mean b value, corrected for the flattening of 1/298.3, was close to the value directly determined for the mean a value, then a reasonable figure determination would be obtained.
In the first attempt, 144 determinations were considered but the comparison between the a and b mean values indicated that the results had been strongly influenced by the large number of figure determinations in which f had been arbitrarily held at 1/297.0.
In the second attempt, all determinations in which the flattening had been arbitrarily held at a value different to 1/298.3 were eliminated and all remaining determinations with flattenings between 1/294.3 and 1/302.3 were accepted.
The number of determinations then considered was 76 and good statistical patterns were obtained for each radius. The a value directly determined was 6 378 183 ± 17 metres, while the a value derived by applying a correction for the 1/298.3 flattening to the b radius, was 6 378 163 ± 13 metres.
The mean of these two values (a = 6 378 173 metres(i)) has been accepted as the best value obtainable from this approach.
Conclusion
It is considered that the best value for the a radius will be the mean of the following values :
6 378 173 from (i) above
6 378 155 from (ii) above
6 378 160 from (iii) above
6 378 163 from (iv) above
6 378 163 metres, the mean.
Rounding off to the nearest 5 metres, it is concluded that the best Figure of the Earth that can be determined from currently available data is an ellipsoid with the following dimensions :
major equatorial axis (a) 6 378 165 ± 15 metres.
flattening (f) 1/298.3.
However, within the next few years there will almost certainly be an improvement in the accuracy of figure determination as a result of both the moon photography programme and the geodetic satellite programme.
The final answer may differ somewhat from the above, but it is to be hoped that, following on these determinations, International agreement will be reached on a Figure of the Earth for worldwide acceptance and use.
Australian Geodetic Surveys are scheduled for their first overall computation and adjustment by the end of 1965. However, it is unlikely that a new International Figure will have been adopted at that time. In the meantime, this first computation should be made on the basis of the dimensions set out above (a = 6 378 165 metres and f = 1/298.3), or on a Figure very close to this, and the results converted to the New International Figure as soon as it is adopted.
[The Figure defined by the values a = 6 378 165 metres and f = 1/298.3, became known as the “165” spheroid and was used for later Australian determinations. Later, however, at their 1964 meeting in Hamburg, the International Astronomical Union (IAU) adopted the values a = 6 378 160 metres and f = 1/298.3. The 1967 Lucerne meeting of the International Union of Geodesy and Geophysics (IUGG) adopted the values a = 6 378 160 metres and f = 298.247167427 for use when a greater degree of accuracy was required. Noting these values in the light of existing Australian work, the Australian National Mapping Council agreed that the parameters, a = 6 378 160 metres and flattening f = 1/298.25 (exactly), were an appropriate fit for Australia and adopted these values for the Australian National Spheroid (ANS) in 1966, and the basis of the Australian Geodetic Datum (AGD) of 1966. The parameters of the various abovementioned spheroids were :
|
Spheroid name |
Semi-major (equatorial) axis (a) |
Flattening 1/f |
|
NASA |
6,378,148 metres |
298.3 |
|
165 |
6,378,165 metres |
298.3 |
|
Clarke 1858 |
6,378,293.645 metres |
294.26 |
|
ANS |
6,378,160 metres |
298.25 |
]
References
(1) Geodesy, by Brigadier G. Bomford.
(2) The Figure of the Earth and isostasy from measurements in the United States (U.S. Coast and Geodetic Survey Publication No. 82,1909).
Supplementary investigation in 1909 to the Figure of the Earth and Isostasy, by Hayford, 1910.
(3) The influence of the distant topography on the deflection of the vertical, by Bernard Chovitz and Irene Fischer, Bulletin Geodesique No. 54, December, 1959.
(4) Photographic Determination of the moon's position, and applications to the measure of time, rotation of the earth, and geodesy, by W. Markowitz, Astronomical Journal, March, 1954.
(5) Ellipsoid Parameter from Satellite Data, by John O'Keefe, Nancy Roman, Benjamin Applee and Ann Eckels (Geophysical Monograph No. 4 published by the National Academy of Sciences, National Research Council Publication No. 708, 1959).
(6) A Geoid and World Geodetic System based on a combination of gravimetric, astro-geodetic and satellite data, by William M. Kaula (National Aeronautics and Space Administration Publication NASA TND-702).
(7) Bulletin Geodesique No. 63 of March, 1962.
(8) "The Figures of the Earth and the Moon", by Sir Harold Jeffreys (Monthly Notices, Royal Astronomical Society, Geophysical Supplements Volume 5, pp. 219-247).
(9) A New Determination of the Figure of the Earth from arcs, by Bernard Chovitz and Irene Fischer (Transactions American Geophysical Union, October, 1956, Vol. 37, No. 5).
(10) An astro/ geodetic world datum from geoidal heights based on the flattening f = 1/298.3, by Irene Fischer (Journal of Geophysical Research, July, 1960, Vol. 65, No. 7).
(11) A geoidal section in the United States, by Donald A. Rice (paper presented at XIIth General Assembly of the IUGG, Helsinki, 1960).