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Thermal nature of the source of the Indian gravity anomaly
Journal of Volcanology and GeothermalResearch, 10 (1981) 293--298 Elsevier Scientific Publishing Company, Amsterdam -- Printed in Belgium
293
THERMAL NATURE OF THE SOURCE OF THE INDIAN GRAVITY ANOMALY
Yu.A. TARAKANOV and T.N. CHEREVKO Institute of Physics of the Earth, Academy o f Sciences of the U.S.S.R., Moscow (U.S.S.R.) Soviet Geophysical Committee, Academy of Sciences of the U.S.S.R., Moscow (U.S.S.R.) (Revised version receive d March 12, 1981 )
ABSTRACT Tarakanov, Yu.A. and Cherevko, T.N., 1981. Thermal nature of the source of the Indian gravity anomaly. J. Volcanol. Geotherm. Res., 10: 293--298. The non-hydrostatic part of the second zonal harmonic term has been included in the Indian gravity anomaly. The interpretation of the potential, gravity and deflection of the vertical has been produced by a new method. The depth of the source is 580 km and the solution satisfiesthe temperature gradient of l°K/km. The ratio ~ Vp/Ap > 0 shows the thermal nature of the density distribution near 670 kin.
INTRODUCTION T h e I n d i a n gravity a n o m a l y is the largest o n e k n o w n . It is s i t u a t e d in t h e I n d i a n O c e a n n e a r Shri L a n k a c e n t r e d at 5°N, 79°E (Kahle et al., 1 9 7 8 , p. 713) and covers the I n d i a n - A u s t r a l i a n p l a t e a n d t h e Asian c o n t i n e n t a l m o s t c o m pletely. T h e sign o f t h e a n o m a l o u s p o t e n t i a l and gravity is negative. T h e I n d i a n a n o m a l y has b e e n i n t e r p r e t e d in several ways. D z i e w o n s k i et al. ( 1 9 7 7 ) f o u n d f r o m analysis o f n e a r l y 700,00.0 P-wave travel-times t h e v e l o c i t y a n o m a l i e s A Vp b e l o w t h e I n d i a n a n o m a l y . It was d e t e r m i n e d t h a t t h e e x p a n s i o n o f the v e l o c i t y d i s t u r b a n c e in a series o f spherical h a r m o n i c s is c o m p l e t e t o 3 X 3 only. I t was f o u n d t h a t a negative s o u r c e lies b e l o w the I n d i a n a n o m a l y at 6 7 0 - - 1 1 0 0 k m d e p t h and a positive b l o c k at 1 1 0 0 - - 1 5 0 0 k m d e p t h . T h e m e a n a n o m a l y o f t h e v e l o c i t y in b o t h shells is 0.03 k m / s . T h e a n o m a l i e s in t h e shell at 1 5 0 0 - - 2 2 0 0 k m d e p t h are u n c e r t a i n and zero in t h e shell at 2 2 0 0 - - 2 9 0 0 k m d e p t h . O n l y the results o f the inversion for v e l o c i t y a n o m a l i e s in t h e m a n t l e b e l o w 1 1 0 0 k m were a s s u m e d c o r r e c t . T h e negative sign o f t h e r a t i o / x V p / A p at 1 1 0 0 - - 1 5 0 0 k m d e p t h c o u l d be r e l a t e d to c h e m i c a l d i f f e r e n c e s or to i n d i r e c t effects of convection. V i n n i k et al. ( 1 9 7 8 ) a n a l y s e d t h e seismological and g r a v i m e t r i c a l d a t a f o r the Indian a n o m a l y . A positive a n o m a l o u s c o m p r e s s i o n a l v e l o c i t y was f o u n d
0377-0273/81/0000--0000/$02.50
© 1981 Elsevier Scientific Publishing Company
291 from teleseismic travel times in the Pamirs and Tien Shan. A positive velociD~ a n o m a l y was f o u n d in the shell at 1 1 0 0 - - 1 8 0 0 k m depth. The positive block c o m p e n s a t e s a p p r o x i m a t e l y 2/3 of the gravity field of the negative source al 6 7 0 - - 1 1 0 0 km depth. The ratio A V p / A p at 1 1 0 0 - - 1 8 0 0 k m d e p t h has a positive sign. T h e thermal nature of the density distribution from 800 to 1800 km is a m o r e preferable feature tVinnik et al., 1978). In this p a p e r the n o n - h y d r o s t a t i c part o f the' second zonal iaarmonic coefficient is n o t included in the gravity a n o m a l y , ttowever, this term has been used in the velocity anomalies by Dziewonski et al. (19771. The n o n - h y d r o s t a t i c pm't of the second zonal h a r m o n i c term has a very particular significance in the interpretation of the ratio A Vp/_:x~,. INTERPRETATION OF TIlE GRAVITY FIELD Let us consider a right-angle co-ordinate system, in which the x-axis points n o r t h and the z-axis is in the direction o f the e x t e r n a l n o r m a l to the xy-plane. The origin of the co-ordinate system is the mass centre Q, o f the a n o m a l o u s b o d y (Fig. 1 }. The gravitational potential, gravity and d e f l e c t i o n of the vertical are k n o w n at a p o i n t P ( x , y , z ) . The e x p a n s i o n of the potential and its derivatives into series in the vicinity of the mass centre Q is expressed as ( T a r a k a n o v and Cherevko, 1979): T -- (Gin/r){1 Tx
Tz
.
( G m x l H ) ~1
.
.
"dx 3T
+4 ] ( ~ q ~ " R 2 l ( a q 2 / R 2 j l ( r 2 / R 2 } ] (
3[(ceq2/R2)/(r2/R2)](1 2
+
.
1
.
( G m z / r 3 ) ( 1 + :9 [ ( c ~ q 2 / R 2 ) / ( r 2 / R 2 ) ] ( 1 -
3Z2/r2) " - ~ 2 )} az~/r
5z2/3r2)}
3z P
i
/x
" 5s0~ ' ,
~2
Oi /
Q1
~Tft
,
"oo~,o i ]
7=
/'
,/
Fig. 1. T h e c o - o r d i n a t e s y s t e m s and c h a r a c t e r i s t i c s of t h e s o u r c e o f t h e I n d i a n gravity
anomaly.
(1~
295 where G is the gravitational constant, m is the anomalous mass, r is the distance between the mass centre Q and the observation point P, aq2/R 2 is a
coefficient of the form. This coefficient of the form depends on the moments of the inertia A and C, mass of the body m and mean radius of the Earth R : (2)
aq2/R 2 = (C - A)/mR 2
w h e r e the d y n a m i c f l a t t e n i n g a and the radius o f t h e inertia q o f t h e b o d y are: a = (C-A)/C
(3)
q2 = C / m
T h e m o m e n t s o f the inertia C relative t o t h e z-axis and A relatively to the x- a n d y - a x e s d e p e n d on the mass d i s t r i b u t i o n o f the source: d
= ~rnk(x ~+z~)
= Zmk(y~+z~)
(4)
C = ~mk(x~+y~)
w h e r e x k , Y k , z k are the c o - o r d i n a t e s o f t h e mass rn k in a p o i n t k. T h e values T x a n d T z in a p o i n t P on a s p h e r e are given b y (Fig. l ) : = T S cos& - T R sinA
Tx
Tz
= - T s sinA - T R cosA
(5)
w h e r e T s is the t a n g e n t i a l derivative o f T, T R = - ~ T / ~ R is t h e radial derivative o f t h e a n o m a l o u s p o t e n t i a l , A is an angle distance b e t w e e n t h e p o i n t s P a n d Q. T h e c o - o r d i n a t e s y s t e m s x, z and S, R lie in the s a m e vertical plane. In f o r m u l a e (1) and (4): x/R
= sinA
z/R
= (hQ/R) + cosA -1
r2/R 2 = 1 + ( 1 - h v / R )
(6) 2 - 2(1-
hv/R)cosA
w h e r e hQ is t h e d e p t h of the mass c e n t r e ( F i g . l ) . F o r m i n g the r a t i o T x / T z a n d T z / T we m a y o b t a i n t h e e q u a t i o n o f t h e depths:
F(hQ) = 0
(7)
where: F(hv)
= (zT/r 2Tz)+
(1 + ½ [ ( ¢ ~ q 2 / R 2 ) / ( r 2 / R 2 ) ] ( 1 - 3 z 2 / r 2 ) } / ( 1 +
~9 [ ( a q 2 / R 2 ) / ( r 2 / R 2 ) ] {1 - 5 z 2 / 3 r 2 ) } aq2/R 2 = ~(r2/R2)(zTx - xTz)/[xTz(1
- 5 z 2 / r 2) - 3 z T x ( 1
- 5z2/3r2)]
T h e mass of t h e source is derived f r o m t h e first e q u a t i o n (1): rn = ( r T / V ) / ( l
+ ½ [ ( u q 2 / R 2 ) / ( r 2 / R 2 ) ] (1 - 3 z 2 / r 2 ) )
(9)
T h e c o e f f i c i e n t o f t h e f o r m (2) d o e s n o t d e t e r m i n e t h e f o r m o f t h e b o d y c o m p l e t e l y . L e t us t a k e t h e source as a t h i n h o m o g e n e o u s shell w h o s e side
296 length (in degrees) is 20. Then the d i f f e r e n c e b e t w e e n t h e d e p t h s o f the g e o m e t r i c centre h O and the mass centre hQ o f the spherical cap is (Fig. 1): A h = h 0 - hQ -- -±R(cqt2R /2(1)/,
- hQ/R)
(10)
The radius o f the spherical cap is given b y (Fig. 1): 0 = arc cos{(1 - ( h Q / R ) + ( A h / R ) ] / [ 1 - ( h Q / R ) - ( A h / R ) ] i
(111
The full f o r m of the a n o m a l o u s b o d y can be f o u n d as a c l o u d o f points. Let us assume the b o d y as a sum o f the mass m k in the p o i n t s xk,Yk,Z k. We can o b t a i n the n u m e r i c a l values mh,x~,yk, zk using the m o m e n t s o f the mass: m = ~ mk k=0 f'2
I1
mxQ = ~ mkx k = 0 k=l
II
myQ = ~ mkY k = 0 k=l
mzo
= ~ mkz k =0 k=l
tl
B+C-2A
mk(2x~-
y~-
z~)
~ mk(2y~k=l
x~-
z~)
= k=l II
C + A-
2B :
(12)
n
A+B-2C
= k=l n
D = ~ mkxkYk k=l
E = ~ mkxkzk k=l
n
F =
mkYkZk h=l
and so on. T h e r e are 2 N + 1 i n d e p e n d e n t spherical h a r m o n i c s which are h o m o g e n e o u s N p o l y n o m i a l s of degree N. The sum o f the m o m e n t s o f t h e mass is ~ (2i + 1) i=O if we have N h a r m o n i c s o f the e x p a n s i o n of a p o t e n t i a l . We m a y find Z
+ ~ (2i + 1 numerical values m k , x k , Yk, zk. The co-ordinates o f the i=0 mass m0 at the p o i n t Q are x0 = yo = zo = 0. The a g r e e m e n t b e t w e e n the n u m b e r o f the points where the anomalies should be measured, the q u a n t i t y o f the anomalies (T, TS, TR, TSS, T R R , TSR and so on) and the n u m b e r u n k n o w n m o m e n t s o f the mass given b y expressions (12) was m a d e b y T a r a k a n o v and C h e r e v k o (19'/9).
297 The numerical interpretation was applied to the gravity data given by a set o f spherical harmonic coefficients of the potential up to the 16th order and degree (Smith et al., 1976). The six pair points, symmetrical relative to the centre anomaly and lying at the same meridian of 76°E, were chosen for gravity calculation. The horizontal distance of these points lies in t he range of A = (+10 °) - (+15 °) due to the m axi m um and minimum of the deflection of the vertical which are situated in this range. The mean values characteristic of the source of the Indian anomaly (GEM-6) are:
= 670 km
+ < A h >
= 580 km
= 0.025
= - 9 0 km
0 = 15 °
O ( R - h Q ) = 1500 km
(13)
The non-hydrostatic part of the second zonal harmonic term has been obtained by Jeffreys (1963). THERMAL CONSIDERATIONS To some e x t e n t an u n e x p e c t e d result of our calculation is a drop of the d e p t h o f the mass centre and geometric centre. The velocity anomaly in the range o f 6 7 0 - - 1 1 0 0 km is A Vp = - 0 . 0 3 km/s (Dziewonski et al., 1977). A decrease o f the density m a y be assumed in the range 580--670 km. Then we have A V p / A p > 0 and the thermal n a t u r e of the anomaly in this shell should be adopted. The post-spinel phase transformation may be at a dept h of 570 km if the t e m p e r a t u r e gradient in the range 4 3 0 - 6 0 0 km is nearly l ° K / k m (Zharkov, 1978), The geometric dept h of the source in the Indian Ocean is 580 km and we can assume the increase of the t em pe r at ur e and the positive sign of the ratio ~ V p / A p . Vinnik et al. (1978) have found that the t em perat ure at 1100--1800 km depth is lower. The dipolar mass and temperature distribution may show an indirect effect o f convection. ACKNOWLEDGEMENTS The authors would like t o thank Dr. V.P. Trubitsyn for stimulating discussions and some th ought f ul comments.
REFERENCES Dziewonski, A.M., Hager, B.H. and O'Connell, R.J., 1977. Large-scale heterogeneities in the lower mantle. J. Geophys. Res., 82: 239--255. Jeffreys, H., 1963. On the hydrostatic theory of the figure of the earth. Geophys. J. R. Astron. Soc., 8: 196--202.
298 Kahle, H.-G., Chapman, M. and Talwani, M., 1978. Detailed 1 ' × 1° gravimetric Indian Ocean1 geoid and comparison with GEOS-3 radar altimeter geoid profiles. Geophys. J. R. Astron. Soc., 55: 703--720. Smith, D.E., Lerch, F.J., Marsh, J.G., Wagner, C.A. Kolenkiewicz, R. and Khan, M.A., 1976. Contribution to national geodetic satellite program by Goddard space flight center. J. Geophys. Res., 81: 1 0 0 6 - 1 0 2 6 . Tarakanov, Yu.A. and Cherevko, T.N., 1979. The interpretation of the largest gravity anomalies of the Earth. Bull. Acad. Sci. USSR, Earth Phys., 4: 25--43. Vinnik, L.P., Lukk, A.A., Mirzokurbonov, M., Tarakanov, Yu.A. and Cherevko, T.N., 1978. Sources of biggest geoid undulations according to the seismic and gravitational data. Proc. (Dokl.)Acad. Sci. USSR, 2 4 1 : 7 8 9 792. Zharkov, V.N.. 1978. The Interior of the Earth and Planets. Nauka, Moscow, 192 pp.
293
THERMAL NATURE OF THE SOURCE OF THE INDIAN GRAVITY ANOMALY
Yu.A. TARAKANOV and T.N. CHEREVKO Institute of Physics of the Earth, Academy o f Sciences of the U.S.S.R., Moscow (U.S.S.R.) Soviet Geophysical Committee, Academy of Sciences of the U.S.S.R., Moscow (U.S.S.R.) (Revised version receive d March 12, 1981 )
ABSTRACT Tarakanov, Yu.A. and Cherevko, T.N., 1981. Thermal nature of the source of the Indian gravity anomaly. J. Volcanol. Geotherm. Res., 10: 293--298. The non-hydrostatic part of the second zonal harmonic term has been included in the Indian gravity anomaly. The interpretation of the potential, gravity and deflection of the vertical has been produced by a new method. The depth of the source is 580 km and the solution satisfiesthe temperature gradient of l°K/km. The ratio ~ Vp/Ap > 0 shows the thermal nature of the density distribution near 670 kin.
INTRODUCTION T h e I n d i a n gravity a n o m a l y is the largest o n e k n o w n . It is s i t u a t e d in t h e I n d i a n O c e a n n e a r Shri L a n k a c e n t r e d at 5°N, 79°E (Kahle et al., 1 9 7 8 , p. 713) and covers the I n d i a n - A u s t r a l i a n p l a t e a n d t h e Asian c o n t i n e n t a l m o s t c o m pletely. T h e sign o f t h e a n o m a l o u s p o t e n t i a l and gravity is negative. T h e I n d i a n a n o m a l y has b e e n i n t e r p r e t e d in several ways. D z i e w o n s k i et al. ( 1 9 7 7 ) f o u n d f r o m analysis o f n e a r l y 700,00.0 P-wave travel-times t h e v e l o c i t y a n o m a l i e s A Vp b e l o w t h e I n d i a n a n o m a l y . It was d e t e r m i n e d t h a t t h e e x p a n s i o n o f the v e l o c i t y d i s t u r b a n c e in a series o f spherical h a r m o n i c s is c o m p l e t e t o 3 X 3 only. I t was f o u n d t h a t a negative s o u r c e lies b e l o w the I n d i a n a n o m a l y at 6 7 0 - - 1 1 0 0 k m d e p t h and a positive b l o c k at 1 1 0 0 - - 1 5 0 0 k m d e p t h . T h e m e a n a n o m a l y o f t h e v e l o c i t y in b o t h shells is 0.03 k m / s . T h e a n o m a l i e s in t h e shell at 1 5 0 0 - - 2 2 0 0 k m d e p t h are u n c e r t a i n and zero in t h e shell at 2 2 0 0 - - 2 9 0 0 k m d e p t h . O n l y the results o f the inversion for v e l o c i t y a n o m a l i e s in t h e m a n t l e b e l o w 1 1 0 0 k m were a s s u m e d c o r r e c t . T h e negative sign o f t h e r a t i o / x V p / A p at 1 1 0 0 - - 1 5 0 0 k m d e p t h c o u l d be r e l a t e d to c h e m i c a l d i f f e r e n c e s or to i n d i r e c t effects of convection. V i n n i k et al. ( 1 9 7 8 ) a n a l y s e d t h e seismological and g r a v i m e t r i c a l d a t a f o r the Indian a n o m a l y . A positive a n o m a l o u s c o m p r e s s i o n a l v e l o c i t y was f o u n d
0377-0273/81/0000--0000/$02.50
© 1981 Elsevier Scientific Publishing Company
291 from teleseismic travel times in the Pamirs and Tien Shan. A positive velociD~ a n o m a l y was f o u n d in the shell at 1 1 0 0 - - 1 8 0 0 k m depth. The positive block c o m p e n s a t e s a p p r o x i m a t e l y 2/3 of the gravity field of the negative source al 6 7 0 - - 1 1 0 0 km depth. The ratio A V p / A p at 1 1 0 0 - - 1 8 0 0 k m d e p t h has a positive sign. T h e thermal nature of the density distribution from 800 to 1800 km is a m o r e preferable feature tVinnik et al., 1978). In this p a p e r the n o n - h y d r o s t a t i c part o f the' second zonal iaarmonic coefficient is n o t included in the gravity a n o m a l y , ttowever, this term has been used in the velocity anomalies by Dziewonski et al. (19771. The n o n - h y d r o s t a t i c pm't of the second zonal h a r m o n i c term has a very particular significance in the interpretation of the ratio A Vp/_:x~,. INTERPRETATION OF TIlE GRAVITY FIELD Let us consider a right-angle co-ordinate system, in which the x-axis points n o r t h and the z-axis is in the direction o f the e x t e r n a l n o r m a l to the xy-plane. The origin of the co-ordinate system is the mass centre Q, o f the a n o m a l o u s b o d y (Fig. 1 }. The gravitational potential, gravity and d e f l e c t i o n of the vertical are k n o w n at a p o i n t P ( x , y , z ) . The e x p a n s i o n of the potential and its derivatives into series in the vicinity of the mass centre Q is expressed as ( T a r a k a n o v and Cherevko, 1979): T -- (Gin/r){1 Tx
Tz
.
( G m x l H ) ~1
.
.
"dx 3T
+4 ] ( ~ q ~ " R 2 l ( a q 2 / R 2 j l ( r 2 / R 2 } ] (
3[(ceq2/R2)/(r2/R2)](1 2
+
.
1
.
( G m z / r 3 ) ( 1 + :9 [ ( c ~ q 2 / R 2 ) / ( r 2 / R 2 ) ] ( 1 -
3Z2/r2) " - ~ 2 )} az~/r
5z2/3r2)}
3z P
i
/x
" 5s0~ ' ,
~2
Oi /
Q1
~Tft
,
"oo~,o i ]
7=
/'
,/
Fig. 1. T h e c o - o r d i n a t e s y s t e m s and c h a r a c t e r i s t i c s of t h e s o u r c e o f t h e I n d i a n gravity
anomaly.
(1~
295 where G is the gravitational constant, m is the anomalous mass, r is the distance between the mass centre Q and the observation point P, aq2/R 2 is a
coefficient of the form. This coefficient of the form depends on the moments of the inertia A and C, mass of the body m and mean radius of the Earth R : (2)
aq2/R 2 = (C - A)/mR 2
w h e r e the d y n a m i c f l a t t e n i n g a and the radius o f t h e inertia q o f t h e b o d y are: a = (C-A)/C
(3)
q2 = C / m
T h e m o m e n t s o f the inertia C relative t o t h e z-axis and A relatively to the x- a n d y - a x e s d e p e n d on the mass d i s t r i b u t i o n o f the source: d
= ~rnk(x ~+z~)
= Zmk(y~+z~)
(4)
C = ~mk(x~+y~)
w h e r e x k , Y k , z k are the c o - o r d i n a t e s o f t h e mass rn k in a p o i n t k. T h e values T x a n d T z in a p o i n t P on a s p h e r e are given b y (Fig. l ) : = T S cos& - T R sinA
Tx
Tz
= - T s sinA - T R cosA
(5)
w h e r e T s is the t a n g e n t i a l derivative o f T, T R = - ~ T / ~ R is t h e radial derivative o f t h e a n o m a l o u s p o t e n t i a l , A is an angle distance b e t w e e n t h e p o i n t s P a n d Q. T h e c o - o r d i n a t e s y s t e m s x, z and S, R lie in the s a m e vertical plane. In f o r m u l a e (1) and (4): x/R
= sinA
z/R
= (hQ/R) + cosA -1
r2/R 2 = 1 + ( 1 - h v / R )
(6) 2 - 2(1-
hv/R)cosA
w h e r e hQ is t h e d e p t h of the mass c e n t r e ( F i g . l ) . F o r m i n g the r a t i o T x / T z a n d T z / T we m a y o b t a i n t h e e q u a t i o n o f t h e depths:
F(hQ) = 0
(7)
where: F(hv)
= (zT/r 2Tz)+
(1 + ½ [ ( ¢ ~ q 2 / R 2 ) / ( r 2 / R 2 ) ] ( 1 - 3 z 2 / r 2 ) } / ( 1 +
~9 [ ( a q 2 / R 2 ) / ( r 2 / R 2 ) ] {1 - 5 z 2 / 3 r 2 ) } aq2/R 2 = ~(r2/R2)(zTx - xTz)/[xTz(1
- 5 z 2 / r 2) - 3 z T x ( 1
- 5z2/3r2)]
T h e mass of t h e source is derived f r o m t h e first e q u a t i o n (1): rn = ( r T / V ) / ( l
+ ½ [ ( u q 2 / R 2 ) / ( r 2 / R 2 ) ] (1 - 3 z 2 / r 2 ) )
(9)
T h e c o e f f i c i e n t o f t h e f o r m (2) d o e s n o t d e t e r m i n e t h e f o r m o f t h e b o d y c o m p l e t e l y . L e t us t a k e t h e source as a t h i n h o m o g e n e o u s shell w h o s e side
296 length (in degrees) is 20. Then the d i f f e r e n c e b e t w e e n t h e d e p t h s o f the g e o m e t r i c centre h O and the mass centre hQ o f the spherical cap is (Fig. 1): A h = h 0 - hQ -- -±R(cqt2R /2(1)/,
- hQ/R)
(10)
The radius o f the spherical cap is given b y (Fig. 1): 0 = arc cos{(1 - ( h Q / R ) + ( A h / R ) ] / [ 1 - ( h Q / R ) - ( A h / R ) ] i
(111
The full f o r m of the a n o m a l o u s b o d y can be f o u n d as a c l o u d o f points. Let us assume the b o d y as a sum o f the mass m k in the p o i n t s xk,Yk,Z k. We can o b t a i n the n u m e r i c a l values mh,x~,yk, zk using the m o m e n t s o f the mass: m = ~ mk k=0 f'2
I1
mxQ = ~ mkx k = 0 k=l
II
myQ = ~ mkY k = 0 k=l
mzo
= ~ mkz k =0 k=l
tl
B+C-2A
mk(2x~-
y~-
z~)
~ mk(2y~k=l
x~-
z~)
= k=l II
C + A-
2B :
(12)
n
A+B-2C
= k=l n
D = ~ mkxkYk k=l
E = ~ mkxkzk k=l
n
F =
mkYkZk h=l
and so on. T h e r e are 2 N + 1 i n d e p e n d e n t spherical h a r m o n i c s which are h o m o g e n e o u s N p o l y n o m i a l s of degree N. The sum o f the m o m e n t s o f t h e mass is ~ (2i + 1) i=O if we have N h a r m o n i c s o f the e x p a n s i o n of a p o t e n t i a l . We m a y find Z
+ ~ (2i + 1 numerical values m k , x k , Yk, zk. The co-ordinates o f the i=0 mass m0 at the p o i n t Q are x0 = yo = zo = 0. The a g r e e m e n t b e t w e e n the n u m b e r o f the points where the anomalies should be measured, the q u a n t i t y o f the anomalies (T, TS, TR, TSS, T R R , TSR and so on) and the n u m b e r u n k n o w n m o m e n t s o f the mass given b y expressions (12) was m a d e b y T a r a k a n o v and C h e r e v k o (19'/9).
297 The numerical interpretation was applied to the gravity data given by a set o f spherical harmonic coefficients of the potential up to the 16th order and degree (Smith et al., 1976). The six pair points, symmetrical relative to the centre anomaly and lying at the same meridian of 76°E, were chosen for gravity calculation. The horizontal distance of these points lies in t he range of A = (+10 °) - (+15 °) due to the m axi m um and minimum of the deflection of the vertical which are situated in this range. The mean values characteristic of the source of the Indian anomaly (GEM-6) are:
= 670 km
= 580 km
= - 9 0 km
0 = 15 °
O ( R - h Q ) = 1500 km
(13)
The non-hydrostatic part of the second zonal harmonic term has been obtained by Jeffreys (1963). THERMAL CONSIDERATIONS To some e x t e n t an u n e x p e c t e d result of our calculation is a drop of the d e p t h o f the mass centre and geometric centre. The velocity anomaly in the range o f 6 7 0 - - 1 1 0 0 km is A Vp = - 0 . 0 3 km/s (Dziewonski et al., 1977). A decrease o f the density m a y be assumed in the range 580--670 km. Then we have A V p / A p > 0 and the thermal n a t u r e of the anomaly in this shell should be adopted. The post-spinel phase transformation may be at a dept h of 570 km if the t e m p e r a t u r e gradient in the range 4 3 0 - 6 0 0 km is nearly l ° K / k m (Zharkov, 1978), The geometric dept h of the source in the Indian Ocean is 580 km and we can assume the increase of the t em pe r at ur e and the positive sign of the ratio ~ V p / A p . Vinnik et al. (1978) have found that the t em perat ure at 1100--1800 km depth is lower. The dipolar mass and temperature distribution may show an indirect effect o f convection. ACKNOWLEDGEMENTS The authors would like t o thank Dr. V.P. Trubitsyn for stimulating discussions and some th ought f ul comments.
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298 Kahle, H.-G., Chapman, M. and Talwani, M., 1978. Detailed 1 ' × 1° gravimetric Indian Ocean1 geoid and comparison with GEOS-3 radar altimeter geoid profiles. Geophys. J. R. Astron. Soc., 55: 703--720. Smith, D.E., Lerch, F.J., Marsh, J.G., Wagner, C.A. Kolenkiewicz, R. and Khan, M.A., 1976. Contribution to national geodetic satellite program by Goddard space flight center. J. Geophys. Res., 81: 1 0 0 6 - 1 0 2 6 . Tarakanov, Yu.A. and Cherevko, T.N., 1979. The interpretation of the largest gravity anomalies of the Earth. Bull. Acad. Sci. USSR, Earth Phys., 4: 25--43. Vinnik, L.P., Lukk, A.A., Mirzokurbonov, M., Tarakanov, Yu.A. and Cherevko, T.N., 1978. Sources of biggest geoid undulations according to the seismic and gravitational data. Proc. (Dokl.)Acad. Sci. USSR, 2 4 1 : 7 8 9 792. Zharkov, V.N.. 1978. The Interior of the Earth and Planets. Nauka, Moscow, 192 pp.