A Generalized Method for Various Data Processing Techniques in Gravity Interpretation- Art%3A10.1007%2FBF00876196

7/30/2019 A Generalized Method for Various Data Processing Techniques in Gravity Interpretation- Art%3A10.1007%2FBF0087

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Page oph , Vol . 114 ( t976) , Bi rkh / iuser Ver lag , Base l . 975

A G e n e r a l i z e d M e t h o d f o r V a r i o u s D a t a P r o c e s s i n g T e c h n i q u e si n G r a v i t y I n t e r p r e t a t i o n

By B . N. P . AGARWAL* and J A G D EO S I N G H *

S u m m a r y - T h e v a r i o u s d a t a p ro ces s i n g t e ch n i q u es (d o w n w ard co n t i n u a t i o n , f i r s t an d s eco n dd e r i v a t i v e s an d t h e i r d o w n w ard co n t i n u a t i o n ) u s ed i n g rav i t y i n t e rp re t a t i o n , a r e an a l o g o u s t o d i f f e r en tt y p es o f l i n ea r f i l t e r i n g o p e ra t i o n s w h o s e t h eo re t i c a l f i l t e r ( am p l i t u d e ) r e s p o n s es can b e d e r i v ed f ro m( u z + v2 )m 2 ex p (d x / ~ - + v2 ) b y s u i t ab l y ch o o s i n g N a n d d , w h e re u an d v a r e an g u l a r f r eq u en c i e s in t w op e rp en d i cu l a r d i r ec ti o n s , d t h e h e i g h t o r d ep t h o f co n t i n u a t i o n i n u n i t o f g r i d i n t e rv a l ; an d N d en o t e s t h eo rd e r o f t h e v e r t i c a l d e r i v a ti v e . B y i n co rp o ra t i n g a m a t h em a t i ca l s m o o t h i n g fu n c t i o n , e -~( "~+ v2) (2 be in gt h e s m o o t h i n g p a ram e t e r ) i n t h e t h eo re t i c a l f il t e r r e s p o n s e fu n c t i o n , i t h a s b ee n p o s s i b le , b y s e l ec t in g as u i t ab l e v a l u e o f s m o o t h i n g p a ram e t e r , t o e s t ab l i s h an ap p ro x i m a t e eq u i v a l en ce o f t h e e f f ec t o f t h e m a t h e -m a t i c a l s m o o t h i n g w i t h t h e i n h e r e n t s m o o t h i n g i n t r o d u c e d , b e c a u s e o f t h e n u m e r i c a l a p p r o x i m a t i o n(ap p ro x i m a t i o n e r ro r ) fo r p r ac t i c a l l y a l l d a t a -p ro ces s i n g t e ch n i q u es . T h i s ap p ro x i m a t e eq u i v a l en ce l e ad st o a g en e ra l i z ed m e t h o d o f co m p u t i n g s e t s o f w e i g h t co e f fi c ien t s fo r v a r i o u s d a t a -p ro ces s i n g t e ch n i q u esf ro m f i l t e r r e s p o n s e m a t ch i n g m e t h o d . S ev e ra l se t s o f w e i g h t co e f fi c ien t s t h u s h a v e b een co m p u t ed w i t hd i f f e r en t s m o o t h i n g p a ram e t e r . T h e am p l i t u d e r e s p o n s e cu rv es o f t h e v a r i o u s ex i s t i n g s e t s o f w e i g h tco e f fi c ien t s h av e a l s o b e en ca l cu l a t ed fo r a s s e s si n g t h e q u a l i t y o f t h e ap p ro x i m a t i o n i n ach i ev i n g t h e d e s i r edf i l te r i n g o p e ra t i o n .

Introduction

In a pap er by AGARWAL and LAL (1972a), an app roxim ate equ iva lenc e of thee f fe c t o f t h e m a t h e m a t i c a l s m o o t h i n g o f t h e t y p e e -~ "2 , wa s e s ta b l is h e d , wh e r eu is a n g u la r f r e q u e n c y - p e r u n i t o f d a t a i n t er v a l a n d / l is c a ll e d t h e s m o o t h i n g p a r a -m e t e r , w i t h t h e in h e r e n t s m o o t h i n g i n t r o d u c e d b e c a u s e o f t h e n u m e r i c a l a p p r o x i m a -t ion invo lved in the ma the m at ica l ana lys i s fo r the ca lcu la t ion o f the va r iou s fi l te r 'swe i g h t c o e ff ic i en t s, e sp e c ia l ly f o r d o w n w a r d c o n t i n u a t i o n . T h e s a m e e q u i v a l e n c e wa sfur the r obse rved for a second ve r t i ca l de r iva t ive wi th the he lp o f one f ie ld exam ple fo rt wo - d i m e n s i o n a l g r a v i ty d a t a b y i n c o r p o r a t i n g a m a t h e m a t i c a l s m o o t h i n g o f t h e ty p ee z(u2+v2)(AGARWAL n d LAL, 1972b). By es tab l i sh ing th i s equ iva lence , i t had beenp o s s i b l e t o d e v e l o p a g e n e r a l iz e d m e t h o d o f c o m p u t i n g o n e - d i m e n s i o n a l s e t s o f we i g h tc o e ff ic i en t s f o r v a r i o u s d a t a p r o c e s s in g t e c h n i q u e s (i.e ., u p w a r d a n d d o wn wa r dc o n t i n u a t i o n , f ir st a n d s e c o n d v e r t i c a l d e r iv a t iv e s a n d t h e ir d o w n w a r d c o n t i n u a t i o n )by an amp l i tude ( fi lt er ) r e sponse matc h ing m etho d (AGARWALand LAL, 1972a,b) .

T h e p u r p o s e o f t h e p r e s e n t p a p e r i s t o e s t a b l is h t h e a b o v e e q u i v a l e n c e fo r t wo -d i m e n s i o n a l g r a v i t y d a t a o p e r a t i o n s a n d s u b s e q u e n t l y , t o d e v e l o p s e v e r a l s e t s o f

* D e p a r t m e n t o f A p p l ie d G e o p h y s ic s , I n d i a n S c h o o l o f M i n e s , D h a n b a d - 8 2 6 0 0 4, I n d i a .


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976 B .N .P . Agarwal and Jagdeo Singh (Pageoph,weight coe f f ic ien t s fo r va r iou s da ta p rocess ing techn iques wi th d i f f e ren t va lues o f 2 inorde r to su i t va r iou s fi e ld r equ i r ements . A co m para t ive s tudy o f the va r ious ex i s t ingse ts o f we igh t coe f f ic ien ts has a l so been m ade in the f r equ ency d om ain for a s sess ingt h e a c c u r a c y i n v o l v e d in e a c h a p p r o x i m a t i o n .

T h e o r yT h r o u g h t h e a p p l i c a ti o n o f t wo - d i m e n s i o n a l F o u r i e r t r a n s f o r m a n a ly s is , AGAR WAL

and LAL (1972C) hav e de r ived th e the oret ica l f i l ter respon se , W~ (u , v , d ) o f th e N t ho r d e r v e r ti c a l d e r iv a t i v e d o wn wa r d c o n t i n u a t i o n t o a d e p t h o f d u n it s o f g ri d s p a c in ga s

o rW N ( u , v, d) = (u 2 + v2)m 2 exp[dx/-u + v2]

WN (p, d ) = pN/2 exp(pd) (1 )wh ere p2 = u : + v2 i s the an gula r r ad ia l f r equ ency pe r un i t o f g r id spac ing .

B y c h o o s i n g a m a t h e m a t i c a l s m o o t h i n g f u n c t i o n , e - zp 2, i n th e f r e q u e n c y d o m a i nfor sup press ing h igh f r equenc ies an d fo l lowing AGARWAL and LAL (1972b), we ca nwr i t e t h e m o d i f i e d fi lt er r e s p o n s e o f th e N t h o r d e r v e r ti c a l d e ri v a t iv e d o wn wa r dc o n t i n u a t i o n f r o m e q u a t i o n (1 ) a s

W~N~ d ) = pNI2 exp [dp - 2p 2] (2)wh e r e t h e m a t h e m a t i c a l s m o o t h i n g o f th e o b s e r v e d d a t a a n d t h e f il te ri n g o p e r a t i o nhave been co m bin ed th rou gh the app l ica t ion of the p rop er t i e s o f l inea r f il te rs .

S ince a l l the ex i s t ing techniques fo r ach iev ing any des i r ed f i l t e r ing opera t ion ,we(P) m a k e s u s e o f a f o r m u l a o f t h e t y p e b y AGARWAL a n d L ~ ( 19 6 9 ):

wc(P) = (1/hN)EWoO(O) + w lO(r l ) + . . . + w,~(r , ) ] , (3)wh e r e Wo, w l . . . . . w , a r e we i g h t c o e ff ic i en t s c o r r e s p o n d i n g t o t h e a v e r a g e g r a v i tyf ie lds 0 ( r ,) , be ing theore t ica l ly ca lcu la ted f rom an in f in ite num be r o f po in t s b u tprac t ica l ly f rom f in ite po in t s on ly o ver c ir c le s o f r ad i i 0 , r l , . . . , r , r e spec tive ly , Theva lues o f r , 's a r e chosen in te rms of the m ul t ip les o f the g r id spac ing h . The f ac tor 1 /hNh a s b e e n i n c o r p o r a t e d f o r c o r re c t i n g t h e d im e n s i o n s i n d e r i v a ti v e s a n d i s o m i t t e d f o rc o n t i n u a t i o n .

I t c a n b e e as i ly p r o v e d t h a t u n d e r a b o v e m e n t i o n e d a s s u m p t i o n s , t h e f il te rr e s p o n s e W~(p) o f e q u a t i o n (3 ) c a n b e wr i t t e n a s

W~(p) = (1/hN)[Wo + W lJo(pr O + . . . + WnJo(pr , )] , (4 )(AGARWAL and LAL, 1969) where Jo i s Besse l f unc t ion of ze ro orde r o f the f ir s t k inda n d p ta k e s t h e v a l u e b e t we e n - r c / h a n d + 2z/h wh i c h i s th e v a l u e o f t h e Ny q u i s tf r equen cy for th e r ange of inves t iga t ion (ZADRO, 1969).


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Vol. 114, 1976) A Generali zed Meth od for Grav ity Data Processing 977Table 1

Weight coefficients for different ring averages in various downward continuation formulae to a depth of oneunit o f grid spacing (h = 1).SmOothing

a r a me te rRad ii \ 2 = 0.01 2 = 0.0 2 2 = 0.03 2 = 0.05 2 = 0.07 2 = 0.09 2 = 0.13\0 + 19.3780 + 17.40 62 + 15.64 56 + 12.67 22 + 10.306 6 + 8.426 8 + 5.751 5h - 4 0 . 0 6 8 0 - 3 4 . 7 2 1 4 - 3 0 . 0 1 3 6 - 2 2 . 2 4 0 0 - 1 6 . 2 6 8 2 - 1 1 . 7 1 4 5 - 5 . 6 9 8 5h~/2 +26.173 5 +22.1 349 +18.62 48 +12.9 528 + 8.7441 + 5.6700 +1.9378h~/5 - 5.0614 - 4.2621 - 3.5865 - 2.5445 - 1.8289 - 1.3564 -0 .8 97 4h~ 0 + 1.0588 + 0.9031 + 0.7736 + 0.5791 + 0.4508 + 0.3698 +0 .2 96 5h lx f~. 5 - 0.4809 - 0.4606 - 0.4439 - 0.4196 - 0.4045 - 0.3957 -0 .3 89 90 +19.5 944 +17. 443 2 +15. 5370 +12.35 21 + 9.8531 + 7.8925 +5.1463h - 40 .8688 - 34 .8519 - 29 .5999 - 21 .0359 - 14 .5654 - 9 .7095 - 3 .4286hx/2 +26.9801 +22.26 18 +18.19 98 +11.72 63 + 7.0118 + 3.6312 -0. 369 8hx/5 - 5.3773 - 4.2950 - 3.3895 - 2.0136 - 1.0873 - 0.4873 +0 .0 84 0h~ /i 0 + 1.2076 + 0.9097 + 0.6647 + 0.3022 + 0.0678 - 0.0776 -0 .2 07 6h lx/iff~.5 - 0.49 86 - 0.3 869 - 0.2961 - 0.1641 - 0.08 07 - 0.03 06 +0 .0 11 9h ~ + 0 .0475 + 0 .0170 - 0 .0078 - 0 .0439 - 0 .0668 - 0 .0807 -0 .0 930h, ~ + 0.0862 + 0.0560 + 0.0316 - 0.0036 - 0.0256 - 0.0386 -0 .0 49 6h~ 3 + 0.0312 + 0.0173 + 0.0061 - 0.0101 - 0.0202 - 0.0262 -0 .0 31 2h lx/~75 + 0.0548 + 0.032 9 + 0.0153 - 0.01 02 - 0.026 0 - 0.03 54 -0 .0 43 3h~ 04 - 0.2572 - 0.2040 - 0.1616 - 0.0992 - 0.0607 - 0.0379 -0 .0 19 1

Weight

Smooth inga r a me te r

Table 2coefficients fo r the different ring average s in various fir st d erivative formulae.

Firstder ivative First der ivative dow nwa rd continuatio n to one unit of gr id spacing with

2 = 0.00 2 = 0.02 2 =0 .0 6 2 = 0.08 2= 0.10 2 = 0.15 2 = 0.200 +3.108 6 + 55.933 +35.701 +28.5 48 +22.8 60 +13. 266 +7.9 30h - 4 .4135 - 134 .945 - 77 .559 - 57 .971 - 42 .816 - 18 .670 - 6 .783h~/2 +2.4649 + 93.595 +48. 734 +33. 942 +22.81 5 + 6.148 -0. 884h~/5 -1 .0 83 7 - 17.156 - 7.881 - 5.080 - 3.124 - 0.683 -0. 16 6h~ 0 +0. 320 6 + 3.329 + 1.518 + 1.007 + 0.669 + 0.300 +0.26 1h lxfi~.5 -0 .3 96 8 - 0.756 - 0.513 - 0.446 - 0.403 - 0.360 -0 .3 580 +2.5243 + 58.643 +36.338 +28.649 +22.625 +12.690 +7.312h -2 .2 216 - 145.066 -79. 932 -58. 340 -41. 933 - 16.512 -4 .4 67hx/2 +0.2362 +103.853 +51.137 +34.313 +21.916 + 3 .956 -3 .2 37h~/5 -0 .1 34 9 - 21.409 - 8.864 - 5.215 - 2.731 + 0.249 +0.8 33h~ 0 -0 .1 67 4 + 5.459 + 2.004 + 1.068 + 0.464 - 0.177 -0. 25 0h ~ -0 .00 51 - 2 .046 - 0 .749 - 0 .409 - 0 .196 + 0 .022 +0.046h ~ - 0 .0896 + 0 .474 + 0 .121 + 0 .030 - 0 .028 - 0 .086 - 0 .094h, ,~ -0 .0 45 3 + 0.500 + 0.149 + 0.060 + 0.004 - 0.051 -0 .0 58h~ 3 -0 .0 29 2 + 0.221 + 0.060 + 0.019 - 0.007 - 0.032 -0 .0 35h 1x~25 -0 .0 40 0 + 0.354 + 0.100 + 0.035 - 0.005 - 0.045 -0 .0 50h, ,~ -0 .0 27 2 - 0.983 - 0.365 - 0.207 - 0.110 - 0.012 -0 .0 01


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5/8

V o l . 1 1 4 , 1 97 6) A G e n e r a l i z e d M e t h o d f o r G r a v i t y D a t a P r o c e s s i n g 979Ca lcu& t ion o f we igh t coe ff ic i en t s, w .

L e t t h e n u m b e r o f u n k n o w n we i g h t c o e ff ic i en t s i n e q u a t i o n (4 ) b e n + 1 f o rapp roxim at ing any des i r ed f il t er r e spo nse ; then fo r ca lcu la t ing the we igh t coe ff icien t sw. ' s , the l e ft -hand s ide o f equ a t ion (4) is rep laced by the ap pro pr ia te mo di f ied fi lt e rr esponse hav ing a su i tab le va lue of 2 . Th en n 4- 1 se t s o f equa t ion s a r e dev e lop ed bycons ide r ing n + 1 equ ispace d f r equenc y in te rva l s in be tween and inc lud ing 0 ( ze ro)a n d rc/h. T h e s o l u t i o n o f t h e r e s u lt in g s i m u l t a n e o u s e q u a t i o n s y i el d s t h e we i g h t

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A m p l i t u d e r e s p o n s e s o f t h e s e ts o f w e i g h t c o e f fi c ie n ts f o r d o w n w a r d c o n t i n u a t i o n o f : ( 1 ) T h e o r e ti c a l;( 2 ) B a r a n o v ; ( 3 ) H e n d e r s o n ; ( 4 ) G r a n t a n d W e s t ; a n d ( 5 ) P e t e r s , a l l f o r d = 1 a n d ( 6) P e t e r s f o r d = 2 .


6/8

980 B. N. P . Agarwal and Jagdeo Singh (Pageoph ,c o e f f ic i e n ts w , . T a b l e s 1 t o 3 g iv e th e w e i g h t c o e f f i c ie n t s t h u s d e r i v e d f o r d o w n w a r dc o n t i n u a t i o n ( d = h --- 1 ), f ir s t d e r i v a t i v e a n d f i rs t a n d s e c o n d d e r i v a t i v e d o w n w a r dc o n t i n u a t i o n ( d = 1) w i t h v a r i o u s v a lu e s o f t h e s m o o t h i n g p a r a m e t e r 4 .

Calculation of ilter (amplitude) responseE q u a t i o n (4 ) h a s b e e n u s e d t o c a l c u l a te t h e a m p l i t u d e r e s p o n s es o f t h e v a r i o u s

e x i s ti n g s et s o f w e i g h t co e f f ic i e n ts f o r d o w n w a r d c o n t i n u a t i o n , f ir s t d e r i v a t i v e a n d i tsd o w n w a r d c o n t i n u a t i o n w h i c h a r e s h o w n i n F ig s . 1 t o 3 .

Discussion o f resultsF r o m F i g . 1 , i t is c le a r t h a t t h e a m p l i t u d e r e s p o n s e s o f t h e s e t s o f w e i g h t c o e f f ic i e n ts

by PETERS (1949), GRANT an d WEST (1965), an d to so m e ex ten t b y HENDERSON (!960) ,s h o w a c o n s i d e r a b l y p o o r m a t c h w i t h t h e t h e o re t ic a l r e sp o n s e o f t h e d o w n w a r dco nt inu a t io n (d = 1 ), wh ereas t he r e spo nse o f BARANOV'S (1953) se t is qu i t e c lose in

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j o RADIANS/G RID INTERVALFi g u r e 2Am pli tud e responses of the sets of weight coefficients for f i rs t derivat ive of: (1) Th eore t ical ; (2) Bara nov ;and (3 ) Henderson .


7/8

V o l . 1 14 , 1 97 6) A G e n e r a l i z e d M e t h o d f o r G r a v i t y D a t a P r o c e s s i n g 9 8 1

t h e o r y , e x c e p t a t t h e h i g h c u t - o f f f r e q u e n c y r e g i o n w h e r e m i s m a t c h i n g is c o n s id e r a b le .T h e a s c e n d i n g n a t u r e o f t h e r e sp o n s e c u r v e s o f B a r a n o v a n d He n d e r s o n w i t h i n c re a s ei n f re q u e n c y c a n b e a p p r o x i m a t e d b y m o d i f ie d f il te r re s p o n s e c u r v e s h a v in g s m o o t h i n gparam ete r s o f roug hly 0 .045 an d 0 .13 r espec t ive ly . S imi la r ly , the r e spon se of the we igh tc o e ff ic i en t s b y G r a n t a n d W e s t c a n a l s o b e a p p r o x i m a t e d b y a s u i t a b l e v a l u e o f 2 . T h er e s p o n s e o f P e t e r s ' c o ef fi c ie n ts c a n n o t b e a p p r o x i m a t e d b e c a u s e o f t h e h i g h o s c i l la t o r yna tu re and be ing we l l be lo w r espon se leve l 1 ov er a cons ide rab ly la rge f r equencyr e gi o n . T h e r e fo r e , a n e q u i v a l e n c e b e t we e n t h e m a t h e m a t i c a l s m o o t h i n g a n d t h ei n h e r e n t s m o o t h i n g , d u e t o n u m e r i c a l a p p r o x i m a t i o n , h a s c le a rl y b e e n e s t ab l is h e d ,thus l ead ing to a genera l ized me tho d for com pu t ing severa l s e t s o f we igh t coe f fic ien t sf o r a d o w n w a r d c o n t i n u a t io n o p e r a ti o n .

The am pl i tude r espon se o f Baran ov ' s s e t o f we igh t coe f fic ien t s fo r f ir s t de r iva t ives h o ws a n o s c i l l a t o ry n a t u r e a b o u t t h e t h e o r e t i c a l r e s p o n s e , wh e r e a s t h e r e s p o n s e o fH end er so n ' s s e t , tho ug h r i s ing sm oo th ly wi th an inc rease o f f r equency , i s a lway s les sthan the theore t ica l r e sponse wi th cons id e rab le dep ar tu r e a t h igh f r equenc ies (F ig . 2 ).T h e a m p l i t u d e r e s p o n s e s o f t h e s e t s o f we i g h t co e f fi c ie n ts o f B a r a n o v a n d He n d e r s o nf o r fi rs t d e r i v a ti v e d o w n w a r d c o n t i n u a t i o n t o a u n i t d e p t h o f g ri d s p a c in g ( t h o u g hr is in g u n i f o r m l y w i t h a n i n c r e as e o f f r e q u e n c y ) , s h o w l a rg e m i s m a t c h i n g w i t h t h etheore t ica l r e spo nses (F ig . 3 ) . Also , the r e spo nse o f Ba rano v ' s coe f fic ien t s i sc o n s i d e ra b l y m o r e t h a n t h a t o f H e n d e r s o n 's .

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|

I !

A m p l i t u d e r e s p o n s e s o f t h e s e ts o f w e i g h t c o e ff ic i en t s fo r f i rs t d e ri v a t iv e d o w n w a r d c o n t i n u a t i o n :T h e o r e t i c a l ( 1 ) d = 1 ; B a r a n o v , ( 2 ) d = 8 9 n d ( 3 ) d = 1 ; a n d H e n d e r s o n , ( 4 ) d = 1 a n d ( 5 ) d = 2 .


8/8

982 B . N . P . A g a r w al a n d J a g de o Singh

Conclusions

Frequency ana l ysi s o f the ex i s t ing set s o f we i ght coe f fi c ient s for dow nw ard con -t inuat ion, f irs t derivative and i t s dow nw ard con t inua t ion, h as revealed that theresponses o f the se ts o f B aranov are m ore ap prox i mate to the theore t ica l re sponses incom par i son w i th the se ts o f Hen derson and Peters . T he nature o f the am pl ituderesponses o f the set s o f we i ght coe f f ic i ent s by H enderson and B aranov are such thatthey can be app roxim ated by ch oo s ing sm ooth ing parameters 2 ~- 0 .13 and 2 -~ 0 .045respec ti ve ly i n the mod i f ied f ilt er response funct i on o f do wn wa rd cont i nuat i on , w i thd = 1 , whi ch e s tabl i shes the approx i mate equi va l ence be tween the mathemat i ca lsmoothi ng and the i nherent smoothi ng due to numer i ca l approx i mat i on . T hi s hasl ead to the deve l opm ent o f a genera l ized metho d o f com put i ng set s o f we i ght co -ef ficients, by wh ich i t i s poss ib le to ap proxim ate the respo nses o f the majori ty of theexis t ing sets of weigh t coef f ic ients for various data proc ess ing techniques .

T he mater i a l presented here forms a par t o f the Ph. D . thes i s ent i t l ed "Someinterpretat ional techniques for gravity m easurem ents" by the firs t author, w hich wa ssubmi t t ed to Indi an Schoo l o f Mi nes , Dhanbad on 22 October 1973 .

R E F E R E N C E SB. N. P . AGARWALan d T . LAL (1969), Calculation of he second vertical derivative ofgravityfield, P u r e a n d

A p p l i ed G eo p h y s i c s 76 , 5-16 .B. N . P. AGARWAL an d T. LAL (1972a), Application o f freq uen cy analysis in two d im ensiona l gravityi n te rpre ta t i on , G e o e x p l o r a t i o n i0 , 9 1 -1 0 0 .B. N. P. AGARWALand T . LAL (1972b) , A generalised method of computing second derivative ofgra vityf ield,Geophysical P ro s p ec t i n g 20 , 385-394 .B. N . P. AGARWAL an d T. LAL (1972C), Calculation of the vertical gradient of the 9ravity field using th eFou r i e r r ans f o rm , Geo physical Prospecting 20 , 448-458 .V. BARANOV 1953), Calcul du gradient vertical du champ de gravitO on du champ magn~tique, MeasurO d Iasurface du sol, Geo physical Prospecting 1, 171-191.F. S. GRANT an d G. F. WEST (1965), I n te rpre ta t i on theory in applied Geophysics, M c G r a w Hill, New Y o r k ,583pp .R. C. HENDERSON (1960), A com pr ehensive system of automatic computation in magnetic and gravityi n te rpre ta t i on , Geophysics 25 , 569-585 .

L. J. PETERS (1949), The d irect approach to magnetic interpretation and its practical application, Geophysics14 , 290-319 .M . B. ZADRO (1969), An ideal isotropic bidimensional ilt er a nd its application in the nterpretat ion of gravityanomalies, S t u d i a G eo p h . e t G eo ed . 13 , 239-251 .(R ece i v ed 1 8 t h S ep t em b er 1 9 7 5)

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