Specific reference figures are recommended for consistent use in geophysics and geodesy. The selection of appropriate reference figure for geophysical studies suggests a relationship between the Antarctic negative gravity anomaly and the great shrinkage of the Antarctic ice cap about 4-5 million years ago. The depression of the south polar regions relative to the north polar regions makes the Southern Hemisphere flatter than the Northern Hemisphere, thus producing the third harmonic (pear-shaped) contribution to the earth's figure

Khan, M. A.

Okeefe, J. A.

English

NASA Technical Reports Server

RECOMAENDED REFERENCE FIGURES 
FOR GEOPHYSICS AND GEODESY 
M. A. KHAN 
JOHN A. O'KEEF€ 
(NASA-TM-X-7CSe ;) E E C 3 H E E N D E D  R E F E E E N C E  N74-17118 
FIGURES FOE G Z G - ' H Y S I C S  AND GEODESY (FASA) 
23 p HC $4.25 C S C L  C 8 E  
Ucclas 
G3/13 30794 
https://ntrs.nasa.gov/search.jsp?R=19740009005 2020-03-23T11:06:02+00:00Z
RECOMMENDED REFERENCE FIGURES 
FOR GEOPHYSICS AND GEODESY 
M. A. Khan 
Jchn A. O'Keefe 
October 1973 
GODDAitD SPACE FLIGHT CENTER 
G-eenbelt, Maryland 

RECOMMEND ED REFERENCE FIGURES 
FOR GEOPHYSICS AND GEODESY 
M. A. Khan 
John A. O'Keefe 
ABSTRACT 
Specific reference figures are recommended for consist- 
ent use in geophysics and geodeey. The selection of appro- 
priate reference figure for geophysical studies suggests a 
relationship between the Antarctic negative gravity anomaly 
and the great shrinkage of the Aitarctic ice cap about 4-5 
miilion years ago. The depression of the south polar regions 
relative to the north polar regions makes the eouthern hemi- 
sphere flatter than the northern hemisphere, thus producing 
the third harmonic (pear-shaped) contribution to the earth's 
figure. 
RECOMMENDED REFERENCE FIGURES 
FOR GEOPHYSICS AND GEODESY 
The advent of artificial earth satellites made possible the determination of 
independent values for the geometric flattening of the earth on the one hand and 
the flattening of the corresponding equilibrium figure on the other hand. Since 
that time, there have been frequent debates over the question which flattening 
should be used for each particular purpose. Even the question of ?-doptirg a 
common reference figure both in geodesy and geophysics has been raised (see, 
for  example, Moritz, 1973). The desirability of having a common reference 
figure for both of these disciplines stems from the overlap of dynamic geodesy 
with geophysiw, psrticularly in relation to the geogravity field. The determina- 
tion and description of the geopotential lie in the domdn of dynamic geodesy, 
but its interpretation in terms of subsurface mass distribution lies properly 
within the domain of geophysics. But the possibility of using a common figure 
has to be examined rather cardully. A s  a first step, we define the purposes 
requiring a reference figure in each of these disciplines. 
In geodesy, we study (a) departures of the geoic? from a reference figure; 
(b) deflections of vertical defined by this reference figure; (c) the locations of 
various points on the actual surfa?e of the earth (station locations); (d) the dis- 
tances between two selected points and similar problems. All these quantities 
1 
are essentiallv geometric in  character, though they may hc +finable snci deter- 
minable by dynamic methods. Thus, in geodesy we deal ivith quantities \vhicii, 
although they are  often determined dpnamically a re  cssentiaillv geometric i n  
nature. Consequently, in geodesy we need a reference f igkre  which best fits thc 
actual earth in the geometric sense. It is the shape one would find if one fitted 
a long hire along a meridian of the earth, at the level of the geoid. 
The best way to determine this geometric shape would indeed be just that 
i.e., to fit long pieces of wire along the various meridians at  the level of the 
geoid and -4ie their average shape. A slightly easier alternative in practice 
vyould be the geodetic survey of the entire earth. But a complete geodetic survey 
cowrage is not possible for obvious reasons 01 inaccessibility I 70q of earth's 
siirface is oceans), expense, and even political reasons. It was,  hawever, pre- 
ciselv these scattered survey data which were used tc determine the pre-satellite 
ellipsoidal flattening of 1/297.0 - a figure which was used in the now outdated 
Intemzitional Gravity Formula. 
The artificial earth satellites provided an excellent and much more accurate 
alternative. The second harmonic coefficient J, in the geopotential is  directly 
related to the earth's flattening f by 
2 
where: m = o' a: (1-f)/GM. In this relation, w is earth's rate of rotation, 
a,, its equatorial radius, GM, the product of its mass and gravitational constant. 
The quantity 0 (f 3, denotes quantities of the order cf f 3. The coefficient J, can 
wst be determined from rate of regression of the nodes of a close earth satel- 
lite orbit. The flattening f in the above equatio. is in fact, what we can call dy- 
namic flattening. The regression of the node is produced by the second harmonic 
of the earth's gravitational field. This takes account of mass at any depth within 
the earth, as is clear from the well-known relation between the moments of in- 
ertia of a body and the second degree harmonics of the gravitational field. 
To a rather coarse approxir,iation, we could say that since the geoid 
is an equipotential surface of the geopotential, its shape is also a manifes- 
tation of the same field and hence the geometric definition of the second ha:- 
monic is the same from the geometric standpoint as from the dynamical. This, 
however, does nut allow for the effects of mass above sea level. If we actually 
should bore holes through the earth's crust everywhere and determine the actual 
level physica!ly corresponding to mean sea level, this level would clearly be 
affected by the hundreds, o r  thcusands of feet of rock above it and we could not 
relate the two withcut taking account of the continental masses. 
To a much better approximation, we can say that we don't care what the 
water would really do down there, a mile below Denver, Colorado, for instance. 
What we are  really interested in is the co-geoid; this is a surface which is so 
3 
ar raged  that it - does correspond to the external gravity field. 
that depth of the co-geoid, below the land surface is just about equal to 
the height as  determined by the ordina7 processes of surveying on the 
earth's surface, especially when account is taken of the variations in gravity. 
It is the co-geoid again that is determined by the processes of astronomical 
levelling. 
It turns out 
But even her2 we are  not out of the woods. It turns out that there is no un- 
ambigious way to produce an imaginary continuation of the external gravitational 
field down into the earth. To see this, imagine a gigantic ball of r w k  suspended 
above the earth's surface. Clearly some lines of gravitational force will radiate 
from it. It follows that no distribution of mass which is confined to the interior 
of the earth can represent the effect of this ball on the outside. Thus we see 
that the concept of the co-geoid must necessarily be ambiguous in reality. 
Fortunately, these problems a re  well below the precision to which we are  
working here; the same is true of the effects of the atmosphere, and hence f can 
be treated as the geometric flattening of the earth. The term geometrical flat- 
tening is preferable also because the term dynamic flattening has been used by 
some in another sense (see, for example, Jeffreye, 1962). The most recent 
number for f is f = 1/298.255, believed to be accurate to 5 parta In lo8.  
The geometrical flattening is the one preferred by geodesists because it gives 
the smallest corrections to reduce baseline lengths from the co-geoid to the ellipsoid. 
4 
In geophysid &dies, except probably in regional and local studies such as  
geophysical prospecting the purpose is to use the gravity anomalies to study the 
state of stress distribution in the earth's interior arising from an anomalous 
111888 distribution. Hence a logical reference figure is a figure of zero stress. 
Such a figure is called the equilibrium or hydrostatic figure d the earth because it 
is the shape the earth would have assumed at its present rotational velocity if it; 
were in a fluid state. The flattening f h of such a figure depends principally on 
its mean or  polar moment of inertia and its rotational velocity a s  illustrated by 
the first order relation: 
5 15 C 
where C is the polar moment of inertia. The rotational velocity enters Equation 
2 via m. The mathematical and com&ational details of the theory have been 
given by O'Keefe (1959, 1960), Henriksen (1960), Jeffrey8 (1962) and Khan (1969, 
1973). This hydrostatic flattening fh, derived from the second order theory 
(Khan, 1969) is f h  1/299.75. The refiaements of various parameters involved in 
the computation or of the theory a re  not likely to change this figure by more than 
a few parts in lo8. 
A geometric figure for the earth can give misleading ideas about the state 
of stress in the earth's interfor. This is demonstrated in Figure 1 arld 2 which 
show a typical satellite-determined geoid based on a recent ge0potGntla.l solution 
(Lerch, et 4. 1973) referred to the geometric flattening of 1/298.255 and the 
5 
equilibrium flattening of 1/299.75 respectively, The geometric picture is cor- 
rectly represented in Figure 1. It is clear that this picture is significantly dif- 
ferent from that referred to the equilibrium flattening, as in Figure 2. The max- 
imum effect of the difference caused by each reference figure occurs at the poles; 
accordingly polar regions are shown separately in Figure QA and SB and Figure 
4A and 4B. 
The most striking effect of the use of the hydrostatic figure is the emphasis 
which it lays on the great Antarctic negative anomary. Clearly this is one ,I the 
dominant features of the earth's gravitational field an a whole, it is clearly the 
principal reason for the size of the third harmonic of the ez.rth's field. 
Comparison of geoidal undulations over the north and south polar regions 
(Figs. 3A, 3B and 4A, 4B), either with respect to the geometric flattening or 
equilibrium flattening, shows that the south polar regions a re  depressed more 
thm 60 meters relative to the north polar regions. This makes the southern 
hemisphere flatter than the northern hemisphere. 
With this striking representation before one, i t  is natural to try to see for 
example a connection between the Antarctic negative anomaly (also see Kaula, 1972 
and Wang, 1966) and the great shrinkage of the Antarctic ice cap in the Pliocene which 
seems to be indicated by the preliminary conclusions based on Leg 28 of the Deep Sea 
Drilling Project aboard Glomar Challenger (Hayes, et  d., 1973). The geologi- 
cal evidcncc uncovered in this cruise seems to indicatc that extensive glaciation 
began on the Antarctic about early Miocene, resulting in a possible subsidence of 
6 
the continent though some local tectonism as cause of this subsidence carnot be 
discounted entirely. The glacial cycle ceached a peak about 4 to 5 million years 
ago, followed by extensive melting and retreat of ice, probably without subse- 
quent uplift of the Antarctic. Independent stratigraphic studie lr example, 
marine rock types in Yorktown-Duplin formation on the East GI). ret of the United 
States of America which is of Pliocene age and other ma-ine rocks of same age 
in California, Florida and Europe) indicate the possibility of a major sea level 
transgression of about 30 meters about 4 million years ago (Hazel, personal 
communication). If this transgression were to be ascribed to Antarctic ice 
melting, it would generat.? a mass deficiency of approximately 1.1 x 
in the south polar regions. The satellite-determined gravity anomaly referred 
to the equilibrium figure is shown in Figure 5. Integratiun of this anomaly over 
areas with more than 10 milligals gravity deficiency yields a mass deficiency 
of approximately 1.3 x 
grams 
grams. The agreement is as  good as could be expected. 
It is recognized that in the absence of some local tectonism, the hypothesis sug- 
gested above is difficult to reconcile with recer 1 estimates of viscosity for the upper 
and lower mantle (see for example, McConnell, 1968 and O'Connell, 1971) as  well as 
with the work of Goldreich and Toomre (1969). It may suggest that the response of the 
earth to loading is more complex than simple viscous models indicate; and there are 
wide areas (e. g. the Hawaiian Islands) where the response is orders of magnitude 
slower than in the classical Fennoscandian uplift; though local tectonic processes 
have been evoked to explain the slow response in some of these cases. 
7 
Thus it is clearly demonstrated that the geophysical significance of the great 
Antarctic feature is more clearly seen on a map referred to the hydrostatic figure 
of the earth than on on : referred to the best-fitting cllipso!d. 
I Similarly the hydrostatic representation brings out the importance of tine 
geoidal trough in Canada as well as a lesser negatiie anoulaly in the Siberia. 
Outside the polar areas, the hydrostatic representation points to the significance 
of the East Lndies and the Andes, while the older representation laid emphasis cnly on 
the Indian Ocean, the Eastern Pacific, the Bermudas and the Ncrth Atlantic. 
The changes which are  introduced by i hanging the reference figure from a 
geometric flattening of f = 1/2W. 255 to an equilibrium flattening of f = 1/299.75 
are illustrated in Figure 6 for the geoid and Figxire 7 for the gravity anomalies. 
1. Recommended Figures: 
Geodetic uses: f = 1/298.255 
Geophysichl uses: fh = 1/299.75 
Since bcth the geometric a d  equilibrium flattenings rt2onimr-rded above 
are  correct to a few parts in lo8 ,  it is recommended for the sake of consis- 
tency and facility of htercomparison within each field, that the values given 
here should be L bpted and each shauld be used as a fixed value for the ap- 
propriate discipline. 
8 
2. The mass ddiciency inferred from the satellite-determined gravity negative 
over Antarctic region is approximately equal to the ice load removed from 
the region during the Plioceue ice-melting episode. 
3. The third harmonic in earth's shape clearly stems from the Antarctican 
negative anomaly which makes the southern hemiaphere flatter than the 
northern hemisphere. 
ACKNOWLEDGEMENTS 
We are obliged to Drs .  J. E. Hazel, C. R. Bentley, D. W. Hayes and M, 
Turner for useful discussions related to this paper. Thanks hre also due to F. 
Lerch and associates for supplying us with the spherical harmonic coefficients 
of the geopotential solution GEM 6. 
9 
REFERENCES 
1. Bahrendt, J. C., and C. R. Bentley, Antarctic map 'olio series, f d i o  do. 
9, American Geographic21 Society, New York, 1968. 
2. Goldreich, P., and A. Toomi.e, Some rernaLks on polar wandering, J. 
Geophys. Res., 74, 2555-2567, 1969. 
3. Hayes, D. W., Frakes, L. A., Barr&, P., Burns, D. A . ,  Chen, P. H.,  
Ford, A. B., Kanips, A. G., Kemp, E. A., McCollum, D. 'V. ,  Piper, 
D. s. w., Wall, R. E. 'and Webb, P. N., Leg 28 deep sea drililng in the 
southern ocean, Geotimes, 18, 6, 1-9-24, 1973. 
Hazel, 2. E., Personal Commrnjcation. 
5. Henriksen, S. W., The Hydrostatic Flattening of the Earth, .4nn. Int. 
Geophys. Year, 12, 197-198, ?.$GO. 
6. Jeffreys, H., The Eurt;~, Cambridge University Pi'ess, Cambridge, 1962. 
7. Xaula, W. M., Global Gravity and Mantic ConvectLon, TectoAiophysics, 13, 
341, 1972. 
8. Khan, M, A , ,  General Solution of the Problem of Sydrostaatic Equilibrium 
of the Earth, Geophys, J. R., Astro. Soc. 18, 177-178, 1969. 
9. Khan, M. A., Hydrostatic Figure of the Earth. Theory and Results, God- 
dard Space F:ighl Center Rep& No. X-592-73-1.05, 1973, 
10 
10. Lerch, F. J., et  al., Gravitational Field 
personal communication. 
del8 GEM-5 and GEM-6, 
11. McConneU, R. K., Viscosity of the mantle from relaxation time spectra 
3f isostatic adjustment, J. Geophys. Res., 73, 7089-7105, 1968. 
12. Morltz, H, , Paper presented at the 2nd International Symposium on Geodesy 
and Physics af the Earth, Potsdam, 1973, 
13. O'Connell, R. J., Pliestocene glaciation and the viscosity of the lower 
mantle, Geophys. J. R., Astro. Soc. 23, 299-327, 197L 
14. O'Keefe, J. A., Determination of the Earth's Gravitational Field, in H. 
Kallmann (ed.), Space Research 1, 448-457, 1960. 
15. O'Keefe, J. A., N. G. Roman, B. S. Yaplee and A. Eckels, Ellipsoid 
Parameters from Satellite Data, in American Geophysical Union Mono- 
graph No. 4, Washington, 3. C., 1959. 
- 
16. Wmg, C., Earth's Zonal Deformations, J. Geophys. Res. , 71, 1713, 1966. 
11 
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Recommended reference figures for geophysics and geodesy

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