GRAVITY GRADIENTS ON THE EARTH’S SURFACE AS DEDUCED FROM SATELLITE ORBITS w. KOHNLEIN

Smithsonian Astrophysical Observatory SPECIAL REPORT 249

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Research in Space Science

S A 0 Special Report No. 249

GRAVITY GRADIENTS ON THE EARTH'S SURFACE

AS DEDUCED FROM SATELLITE ORBITS

W. Kihnlein

August 25, 1967

Smithsonian Institution Astrophysical Ob se rvator y

Cambridge, Massachusetts, 021 38

706-42

TABLE OF CONTENTS

Section Page

ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv

1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

2 HORIZONTAL GRAVITY GRADIENT . . . . . . . . . . . . . . . . . 3

3 VERTICAL GRAVITY GRADIENT . . . . . . . . . . . . . . . . . . . 5

4 NUMERICALRESULTS . . . . . . . . . . . . . . . . . . . . . . . . . . 7

4.1 The Ellipsoidal Gravity Gradient . . . . . . . . . . . . . . . . 7

4 . 2 The Horizontal Gravity Gradient . . . . . . . . . . . . . . . . . 9 4 . 3 The Vertical Gravity Gradient . . . . . . . . . . . . . . . . . . 1 2

5 REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

BIOGRAPHICALNOTE . . . . . . . . . . . . . . . . . . . . . . . . . . 18

i i

LIST O F ILLUSTRATIONS

Figure Page

1 Anomalous horizontal gravity gradient - 2

sec . . . . . . . . . . . . . . . . . . . . . . . . 11

. . . . . . . 2 Anomalous horizontal gravity gradient (zonal pa r t ) 12

3 Anomalous gravi ty gradient along a plumb line (zonal par t ) . . 14

4 Anomalous variation of the gravity along a plumb line

16 - 2 sec . . . . . . . . . . . . . . . . . . . . . . . .

LIST O F TABLES

Table

1

2

Horizontal gravi ty gradients (ellipsoidal). . . . . . . . . . . . . . Vertical gravity gradients (ellipsoidal) . . . . . . . . . . . . . . .

8

9

... 111

ABSTRACT

The variation of gravity on the ea r th ' s surface i s computed in three

mutually perpendicular directions: the horizontal anomalous variations

along the geocentric latitude and longitude curves, and the ver t ica l com-

ponent along the plumb line.

and Gaposchkin's l a tes t harmonic coefficients indicate a correlat ion between

the anomalous ver t ica l gravity gradient and the ea r th ' s continental topography.

The numerical resu l t s obtained f rom Kozai 's

RESUME

Les v a r i i i t i o n s de l a p e s a n t e u r l a s u r f a c e de l a t e r r e s o n t

c a l c u l g e s l e long de t r o i s a x e s t r i r e c t a n g l e s :

h o r i z o n t a l e s anormnles , l e l o n g d e s c o u r b e s de l a t i t u d e e t l o n g i -

t u d e g k o c e n t r i q i i c s , e t la composante v e r t i c a l e , l e long de l a

d i r e c t i o n du f i l plomb. Les r g s u l t a t s numgriques oh tenus

p n r t i r d e s & c e n t s c o e f f i c i e n t s harmoniques de Kozai e t Gaposchkin

i n d i q u e n t l ' e x i s t e n c e d 'une c o r r g l a t i o n e n t r e l e g r a d i e n t v e r t i c a l

anormal de l a pesar i teur e t l a topograph ie d e s c o n t i n e n t s t e r r e s t r e s .

l e s v a r i a t i o n s

KOHCIIEKT

iv

GRAVITY GRADIENTS ON THE EARTH’S SURFACE

AS DEDUCED FROM SATELLITE ORBITS

W. Kohnlein

1. INTRODUCTION

Gravity gradients a r e widely used in geophysics f o r the analysis of com-

plex geological s t ruc tures .

in a gravity gradient f ield even in cases where purely gravimetr ic data some-

t imes fail to show a c lear picture.

a worldwide scale using as gravity information the zonal and nonzonal ha r -

monic coefficients of the geopotential derived by Kozai (1964) and Gaposchkin

(1967). of the same zonal coefficient of second degree.

Mass defects, for example , a r e easi ly recognized

We a r e trying to extend this procedure to

We take as reference an ellipsoidal field of the same potential and

We denote the geopotential as der ived f r o m ar t i f ic ia l satell i tes by

GM U =- r

with

1 t C (3 (cnm cos mA t snm s in mi ) Pnm(sin $1 n=2 m=O

w r 2 2 2 t- ‘Os 4 J 2

GM = product of the gravitational constant and the m a s s of the ear th ,

a = equatorial radius of the ear th ,

r = geocentric radius,

4 = geocentric latitude,

A = geocentric longitude,

Th i s work was supported in pa r t by Grant No. 87-60 f r o m the National Aeronautics and Space Administration.

1

S = harmonic coefficients, ‘nm, n m P (s in 9) = Legendre associated functions, nm

o = ear th’s angular velocity;

and, s imi la r ly , the ellipsoidal potential by

with

r = geocentric ellipsoidal radius, e

K2, K4 = ellipsoidal harmonic coefficients.

Hence we obtain the gravity a t any point f rom

and analogously fo r V. Of course, these express ions a r e valid, in a s t r i c t

sense , only in f r ee space while disturbances, introduced by topographic inter - fe rence a t s e a level, a r e neglected.

To keep the analytical expressions to a minimum, we consider only the

satel l i te potential U, and substitute for U the ellipsoidal potential V when

ne ce s s ar y.

2

2. HORIZONTAL GRAVITY GRADIENT

The tangent plane (horizon) of the geoid (i. e . , U = const a t mean s e a

level) is obtained f r o m equation (1):

and the tangents of the geocentric latitude and longitude curves a r e

a u d r t K d A au = 0 ,

- d r t - d + = O au au .

F

ar aa,

If we introduce these expressions into the total differential of equation ( 3 ) ,

we obtain the components of the horizontal gravity gradient a,ang the geo-

cent r ic latitude and longitude curves on the geoid:

3

Because the a r c lengths of the geocentric latitude and longitude curves can be

wr i t ten

we obtain the variation of gravi ty along the latitude curves (p = const,

and, s imilar ly , the variation of gravity along the longitude curves A = const,

Hence the total amount of the horizontal gravity gradient is readily obtained:

The approximate sign corresponds only to the potential U, while any rotational

symmet r i c potential such as V (substituted instead of U) sat isf ies equation

(11) exactly;

in general do not intersect on the geoid perpendicularly.

deviation i s so small that f o r practical purposes the equal sign can be used in equation (11).

this is because the geocentric latitude and longitude curves

However, the

4

3 . VERTICAL GRAVITY GRADLENT

To obtain the ver t ica l gravity gradient, we s t a r t f r o m equation ( 6 ) ,

and substitute f o r d r / d s , d+ /ds , and dX/ds the expressions of the differential

equation of the plumb line,

-=--Ig d r au d s a r J

2 - - a v / P r z cos + , dX - -

d s ax

which a r e eas

equations (1 4)

ly derived by comparison of the corresponding t e r m s in

and (15):

dx d r dX - = cos + cos A - - r s i n + cos^ - r c o s + s i n 1 - d s d s d s ds ’

dX - r s in+ s i n k &k t r cos + cos X - L!Y = cos + sinX ds d s d s d s ’

5

and

au - 1 au 1 1 au 2 2 a x J

cos + cos h - - t r s in+ cos h - - t r cos + s i n x dx d s - = -

g a r 2 a+ g r cos + g r

r c o s + c o s x i ah au J (15) 1 au d s g a r a = - c o s + s i n h - - t r s i n + s i n h - - -

g r cos 4 2 a+ g r

1 au 1 au 2 a+ - - s i n + - - - r cos+- - a Z

d s g a r - -

g r

2 2 2 2 with dx t dy t dz = ds . Hence the ver t ica l gravity gradient can be

expressed as

6

4. NUMERICAL RESULTS

As a l ready stated, we based the numerical computations on Kozai 's and

However, for comparison and to Gaposchkin's l a tes t harmonic coefficients.

get an idea of the accuracy finally obtained, we also considered, where appro-

priate, King-Hele's (King-Hele and Cook, 1965; King-Hele, Cook, and Scott,

1965) zonal coefficients and Kaula's (1 966) combination solution (obtained

by merging satell i te and surface gravity data) .

The numerical resu l t s a r e presented in two steps:

A. We compute the horizontal and ver t ical gravity gradients of the

ell ipsoidal f ield, and

B. We add to these values correct ions leading to the gravitational field

under consideration.

show immediately the deviations f r o m an idealized field of a body with

ell ipsoidal equipotential surfaces .

In this way we dea l with only smal l numbers , which

4. 1 The Ellipsoidal Gravity Gradient

If, in equation (2), we give the values for the equatorial radius and the

harmonic coefficient of second degree, we can compute K

ell ipsoidal field s t ruc ture (Kohnlein, 1966); o r , in detail,

assuming an 4

a = 6 378 165 m ,

K 2 = C20 = - 1082.645 x

K4 = 0 . 2 4 X 10 ,

(Kozai,1964),

-5

which means that the equipotential sur face V = const has a t s e a level a

flattening of f = 1/298. 25.

7

To obtain the variation of gravity, we only substitute V for U in equations

(9) , ( l o ) , ( l l ) , and (16) . V = const:

Hence we get the horizontal gravity gradient along

0"

= o , ge - +=const

(qhoriz (set - 2

and

- g e

horiz 6 s X=const --

J

i. e . , the gravity of an equipotential ellipsoid va r i e s only along the meridian.

Table 1 gives the numerical values as a function of the geocentric latitude

for the Northern Hemisphere. In the Southern Hemisphere , (6ge/6s)horiz

becomes negative, but the amounts remain unchanged (because of the d i r ec -

tion in which the latitude i s counted).

Table 1. Horizontal gravity gradients (ellipsoidal)

-8 .1, ,,*

E - 8 means X 10 .

90 80 70 60 50 40 30 20 10

0

0 0. 2767E -8" 0. 5206E -8 0.70283 -8 0. 8010E -8 0. 8031E -8 0.7079E -8 0.5264E -8 0. 2805E -8 0

8

To obtain the ver t ical gravity gradient (perpendicular to V = const), we

substitute V into (16):

which leads to negative values both f o r the Northern and Southern Hemispheres

( see Table 2).

of the gravity gradient monotonically dec reases toward the poles.

The strongest variation occurs at the equator,and the amount

Table 2. Vertical gravity gradients

I I

19" I

90 80 70 60 50 40 30 20 10

0

-0.308337E-5 -0. 308350E-5 -0.308388E-5 -0.308446E-5 -0.308518E-5 -0. 3085953-5 -0. 3086673-5 - 0. 308726 E- 5 -0. 308764E-5 - 0. 308778E-5

4. 2 The Horizontal Gravity Gradient

(ellipsoidal)

The components of the horizontal gravity gradient along U = const (with

However, if we at f i r s t U = V) can be obtained f r o m equations (9) and (10).

subt rac t f r o m (6 g / 6 s)x=c oris the ell ipsoidal pa r t 6ge/6s

the components of the "anomalous" horizontal gravity gradient:

we obtain

9

and

Hence the total amount of the horizontal anomalous gravity gradient i s

This anomalous horizontal gravity gradient gives immediately the direction

and magnitude of g rea t e s t variation in the gravity anomalies along U = const

r e f e r r e d to V = const.

in F igu re 1, a s obtained f r o m Kozai's and Gaposchkin's coefficients.

a r e a s of strong local disturbance, like the eas t e rn pa r t of the Antarctic

[ (A6g/6s)hor iz = 0. 34 x a lso have a typical zone-oriented pattern, especially near the North and

South Poles and along latitude + = 60" , e tc . , a s can be seen in the figure.

Hence i t i s natural to consider only the zonal coefficients C

(1) and compare their deviations f r o m the ell ipsoidal pa r t ( s ee Figure 2). F o r

comparison, we also included King-Hele's zonal coefficients, which give a

somewhat smoother curve because of the lower degree (coefficients up to

degree 9) considered. - 2 gradient approaches 0 .08 x c m , i. e . , only a quar te r of the combined

zonal plus nonzonal anomalous effects.

of course , zero, because of the rotational symmet ry of U = U(Cno) with

U = V linked to the ellipsoid.

A vector map 10" X 10" for the whole ea r th is shown

Besides

sec-'1, the northern par t of India, e t c . , we

in U of equation nO

The maximum zonal anomaly of the horizontal gravity

At the poles the zonal anomaly i s , - - -

10

' /

' ,

/ .

/ ,

\ '

1 ' J ' / /

I.

/ \ .

0 E +

+ 8 2 + k - + b 0

b

b

b

+

W +

W + d + k +

b , 'J

b

b 'p

8

b 0

d I

W I

0 s 0 0 !z b u,

8 a,

N I 0 a, m

4

4 I 0 4

X N

k 0 d

.d

h

4: d v

a, k 3 M

ir"

11

IO

KING-HE LE --

-IO 1

Figure 2. Anomalous horitzontal gravity gradient (zonal part) . (i denotes the zonal part) .

The overal l resu l t s obtained from Kaula's coefficients a r e very similar

to those obtained f rom the Kozai-Gaposchkin coefficients, although a d i rec t

comparison is somewhat difficult because of the different degrees and o r d e r s

of harmonic terms used.

4. 3 The Vert ical Gravitv Gradient

The variation of gravi ty along the plumb line is obtained f r o m equation (16) .

S imi la r ly to the horizontal gradient, w e consider only the anomalous effects

re la t ive to the ell ipsoidal field, which lead to

12

If in equation (22) we substitute the potential u = u ( C n o ) with

get the zonal pa r t of the ver t ical variation.

zonal gravity gradient along the plumb line f o r the Kozai and King-Hele ha r -

monic coefficients.

we obtain quite different resu l t s around the North Pole, and then fair ly good

agreement along the mer id ian down to the South Pole (where the amplitudes

a r e twice a s la rge in King-Hele's case a s compared to Kozai's).

gives an est imate of the accuracy at p resent obtained f rom satell i te orbits.

If we plot only the zonewise (latitude dependent) proportion between s e a

and continents (namely, the ear th 's topography) against Kozai's gravity

gradient, we obtain roughly the dotted curve in Figure 3.

tude indicates the excess of water over land, and, analogously, a positive

amplitude would mean that along a cer ta in latitude the land m a s s e s dominate.

The range i s normalized and extends f r o m -1 (only water ) to 1 (only land).

At the North Pole and a t the South Pole both the solid and the dotted lines a r e

c lear ly pointing in the same direction, and a l so the downward slope along the

mer id ian s e e m s to f i t fa i r ly well.

at tr ibuted : 1) to the uncertainty with which the ver t ica l gradient can be

cur ren t ly determined, 2 ) the rough guess of land/ s e a distribution (ignoring

elevations), and 3 ) other geophysical fac tors not considered. However, the

overal l correlat ion between the anomalous zonal gravity gradient along a

plumb line and the m a s s distribution in the ear th ' s c r u s t i s quite obvious f rom

Figure 3.

= V for U, we

Figure 3 shows the anomalous

Apar t f r o m the different degrees (14 and 9, respectively),

This spread

The negative ampli-

The remaining discrepancy can be

1 3

~~

/I - I I I I I

KOZAI

-- KING-HELE

. CONTINENTS vs OCEANS

\’/ \ O0/ \

\ i ’ . .

./I ’x ....... ...... ..... ....... ........ c 4 ................. \SI. v \..

f’ I \ \ p ....

Figure 3. Anomalous gravity gradient along a plumb line (zonal par t ) . (2 denotes the zonal par t ) .

L e t us go one s tep fur ther and consider the combined (zonal and nonzonal)

anomalous vertical gravity gradient.

Kozai-Gaposchkin coefficients the number map shown in Figure 4. element gives the anomalous effect a s a function of its position (+, A ) , relative

to the ellipsoidal field in Table 2.

we find the following character is t ics :

Using equation (22) we obtain for the

Each

If we superimpose the continent contours,

A. Most of the continental a rea is covered with positive gravity gradient

anomalies, i. e. , the variation of gravity along the plumb line is smal le r than

the ell ipsoidal pa r t as a l ready obtained for the zonal resu l t ( see Europe-Asia,

Amer ica , Australia, etc. ).

B. The negative disturbances of the continents a r e strongly correlated

with the worldwide mountain chains, such as the Himalayas, the Rockies,

14

the Andes, e tc . Hence, about 85% of the continental a r e a shows its counter-

p a r t in the anomalous gravity gradient pattern.

Of course , we a l so have, s imi la r to the zonal par t , a r e a s that fit neither

the one nor the other pattern.

pa r t of the Antarctic, the western pa r t of Greenland, e tc .

have a high uncertainty a s f a r as their gravity gradient anomalies a r e con-

cerned.

tion of d i r ec t gravity measurements and satell i te data), we can i n such a r e a s

suddenly find the opposite resu l t s compared to the Kozai-Gaposchkin values.

In br ief , although the correlat ion between the continents and ver t ical gravity

gradient anomalies is c lear ly visible f r o m the mos t recent numerical analyses

(Kozai, 1964; King-Hele and Cook, 1965; King-Hele -- et a l . , 1965; Gaposchkin,

1967; Kaula, l966), we still need grea te r accurac ies for m o r e detailed investi-

gations.

F o r example, this is the case for the eas t e rn

But all these a r e a s

If we take Kaula’s harmonic coefficients (obtained f r o m a combina-

15

16

a v .. 2

E

.r( r-l

P

3 a Id M c 0 Id h

> Id k M a,

r-l

I+

c, .r(

6,

3 w 0

.A c,

.r( d k d > ffl 3 0 d

E" 4 0

4 P)

M 2 iz

c, w d a, c

.r( r-l

c, VI k

.r( w a, 5 3

Id

V

k

c a, V 0 a, M

...

c, .r(

c,

r-l

.r(

c,

.. c, w a, 4

a,

6, c, Id c c

5. REFERENCES

GAPOSGHKIN, E. M.

1967. A dynamical solution for the t e s se ra l harmonics of the geopotential,

and station coordinates using Baker-Nunn data.

Research VII, ed. by R. L. Smith-Rose, S. A. Bowhill, and

J. W. King, North-Holland Publ. Co., Amsterdam, pp. 685-693.

In Space -

KAULA, W. M. 1966. Tes t of satel l i te determinations of the gravity field against gravi-

m e t r y and the i r combination. Publ. No. 508, Inst. Geophys.

Planet. Phys. , University of California.

KING-HELE, D. G., AND COOK, G. E.

1965. The even zonal harmonics of the ea r th ' s gravitational potential.

Geophys. Journ. Roy. Astron. S O C . , vol. 10, pp. 17-29.

KING-HELE, D. G. , COOK, G. E . , ANDSCOTT, D. W.

1965. The odd zonal harmonics i n the ea r th ' s gravitational potential.

Tech. Rep. No. TR-65-23, Royal Aircraf t Establishment,

43 PP. KOHNLEIN, w. J.

1966. The geometr ic s t ruc ture of the ea r th ' s gravitational field. - In

Smithsonian Astrophys. Obs. Spec. Rep. No. 200, ed. by

C. A. Lundquist andG. Veis, vol. 3, pp. 21-101.

KOZAI, Y.

196 4. New determination of zonal harmonics coefficients of the ea r th ' s

gravitational potential.

Rep. No. 165, 38 pp.

Smithsonian Astrophys. Obs. Spec.

17

BIOGRAPHICAL NOTE

WALTER KOHNLEIN received his degrees f r o m the Technological

Institute i n Munich, and the Technological Institute i n Braunschweig, Germany,

in 1958 and 1962, respectively.

He worked fo r a year at Ohio State University, Columbus, Ohio, before

In 1965 he became an associate of the Harvard College joining SA0 i n 1963.

0 bs e rvato r y.

Dr. Kiihnlein's work a t Smithsonian includes studies of the figure of the

e a r t h and its gravitational field by means of ar t i f ic ia l satel l i tes .

18

NO TIC E

Thi.s ser'ies of Special Reports was insti tuted under the supervis ion of Dr . F. L. Whipple, Director of the Astrophysical Observatory of the Smithsonian Institution, shortly a f t e r the launching of the first ar t i f ic ia l ea r th satel l i te on October 4, 1957. Contributions come f r o m the Staff of the Observatory.

First i ssued to ensu re the immediate dissemination of data f o r sa te l - lite tracking, the repor t s have continued to provide a rapid distribution of catalogs of satel l i te observations, orbi ta l information, and pre l imi- n a r y resu l t s of data analyses pr ior to f o r m a l publication in the appro- p r i a t e journals . The Reports a r e a l so used extensively fo r the rapid publication of pre l iminary o r special resu l t s in other fields of a s t r o - physic s .

The Reports a r e regularly distributed to all institutions par t ic i - pating in the U. S. space r e sea rch p r o g r a m and to individual sc ien t i s t s who reques t t hem f rom the Publications Division, Distribution Section, Smithsonian Astrophysical Observatory, Cambridge, Massachuset ts 021 38.