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Geophysics,Gravity,Microcomputers

INTRODUCTIONSince he introductionand rapid growth of themicrocomputer ndustry in the mid 1970's, hegeophysicalndustryhasbeenquickto makeus eofthis technology, s microprocessingapabil i tybuiltinto explorationgeophysicalquipment,and morerecently, ssmall tand lonemicrocomputerystemscan be used or interpretation n the office or in thefield.In the field, microcomputers re used to record

andstore urvey ata,on eithercasseter floppydiskdepending n thehardware sed.Onceentered,ielddatacan be manipulatedo givea hard copyoutputeither as l istingor as profi les.Depending n the ypeof survey, he ield compu-ters are also used o carry out the necessary alcula-tionsand corrections,o reduce he raw ielddata otheir final ormat.Examples f this being, eductionof gravitydata,magnetic orrections,lottingof datae . . .* Institute or GeologyandMineralExploration,Geophysical e-partmenl** University f Thessaloniki.oPYKTO: nnOYTO: 461187

A SIMPLEMETHODFORGRAVITY BASENETWORKADJUSTMENTBY THE USEOFA MICROCOMPUTERC.THANASSOULAS*G.A. TSELENTIS*A. ROCCA**

A different approach in adiusting gravity base networks ispresentedby the use of microcomputers.The method used orsolving the correspondingsystem of linear equations s the"Generalized Inverse", and a program wriften in MicrosoftBasic or a PET COMMODORE microcomputer s presentedwith an application example.

As an interpretationalaid, the field computer sused o model th e geophysicalesults n terms ofgeologicmodels o providequantitative alueso theanomaly arameters. ll th e modell ing nd nterpre-tation routines developedare interactiveprogramswhich allow the geophysicistnd geologistmaximuminput to all phasesof the computer manipulation.Thus constraintsbasedon the geologicknowledgecan be input to the computer aided geophysicalinterpretation.This paper presentsan example of the use ofmicrocomputersn the topic of gravity surveypro-blems and morb specificallyon the adjustmentofgravity networks.

GRAVITY NETWORK ADJUSTMENTA gravity network adjustmentconsists f correc-ting eachgravity differencemeasuredbetween woadjacentbases o hat the closingerror alonga closedloop is minimum.That problem was faced by Barton (1929)andRoman (1932),using he method of least squares.

Another approachwas ollowed by Cowles 1938) n

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a similar problem, where he proposeda method ofwritting down a system f simultaneous quationsorcorrection of closing slope data traverses.Thatsystemof equationscan be solved by meansof ananalogousesistanceetwork.An itterativegraphicalmethod for solving the correspondingsystem ofequationswa s proposedby Gibson (1941).Smith(1950) also proposed another graphical methodbasedon Gibson'swork.The use of computer methods n solving thecorresponding ystem of simultaneous quationssimplified he problemof adjusting ravitynetworks.In that field various computer programs n FOR-TRAN 4 exist n the geophysicalomputer ibraries.Recently,Lagios 1984)presented n analysis f aleast squares djustment f a gravitybasestationnetwork and a computerFORTRAN 4 program nwhich the instrumentaldrift has been considered.A reviewof themethods se dof theadjustment fgeodetic etworkswasgivenby Welsch 1979).Thesamemethodsca n be appliedgenerally n gravitybasenetworks.

FUNDAMENTAL THEORYLet us considerhe gravitybasenetworkof figure(1). The values at th e centre of each trianglerepresent he closing error of each loop of the

network.The lettersa, b, c, . . . m, representhecorrections o be applied at each traversebetween

two bases, o that the closingerror of eachclosedloop (triangle) s zero (or minimum).

FIGURE l: Gravity base station network.

From each riangle an equation resultsof the form:a * b * c * c e : 0 ( 1 )

where a. b, c are the corrections to be applied oneach traverse respectively or each triangle, an d cecorresponds o the closingerror of the same riangle.Srgnesare considered to be positive clockwise.

For the gravity network of figure (1 ) the followingsystem of equations (f ig. 2) holds, presented in amatrix form.

oo oooooooooooooooo b

I

t t l

-1 2-o.51.5

-2.5-2.31.91 8

o-1 01o o o o-1 1 ooooooooooo-1 oooo 1 oo o o o o o-1 -1 o1000o o-1 0 0 0 0 1 10 0 0 0o o o o o o o o o-1 0-1

30FIGURE 2: Resultedsystemof linear equailons.

oPYKTO:nnoYTo: 461187

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Since he number of unknowns s greater than theresulted independed linear equations that systemcan't be solved by the classicalmethods. In order tooverride that difficulty the generalized inverse met-hod is used (Mackay, 1981). In that method it ispossibleby usingan itterative algorithm to calculateth e generalized inverse A+(N,M) of an A(N,M)matrix (Moore - Penrose nverse), so that the matrixequation:

A (N ,M) .X (N ,1 ) : H (N ,1 ) (2 )can be solved as

X(N ' l ) : A1 (N ,M) 'H (M '1 ) (3 )This procedure can solve the following three kindof problems:

a) I f N:M and the A(N,M) exists, hen we getan exact solut ion for X(N. l ) .b) If M> N that is the equarionsare more thanth e number of unknowns. then a least -squares solution is given instead.c) If N>M that is the equations are fewer thanthe number of unknowns. then a solution isgiven consistent with th e data (minimumnorm g* inverse).

Th e following results.presented n figure (3 ) havebeen obtained by the method reffered above usingMackav's algorithm on the system of equations off igure (2) .

FIGURE 3: Derived solutionoPYKTO:nOYTO: 61187

PROGRAM USER INSTRUCTIONSThe computerprogramwrittenoriginally n Micro-soft Basic was slightly modified to meet the outputrequirements f the PET - COMMODORE compu-te r operatingsystem.The user of thb program should follow these

instructions:1) Enter number of equationsand number ofunknowns n da ta statement1440.2) Enter the coeficients f each inearequation nsuccessiveata statements, ne statementoreach equation.3) Run the program.: F i Eh * * + 1 ,+ + + i t t t f , t+ + *4 R E t 1 t 1U F :E I1 } IF I i E I IET: F E t 1 + 1r[ r F ] E f l * + t t , t + * ] + t { t { 1 + t1 ! R E NI 4 F E I I i I ] I : F I r : I ' F F I l F I ] ] ' : T : T H T I l I : ' L : . : : L I F E : ] I : E F E I : ] I 1 F : : F , I , I I ] E T I ] i L , I1t P E I 1 ' T ] E B r : H : I I E f I F T H I T F I F I I : . L [ : F : ] l ' E t : , ] : ] 1 P , T : I : EI : ] : i : : I r : 1 I ] E I F I ] rl:i FrEll: a r F E l i TH [ : ! F : F ] E : l F i i l I , 1 | t : , l | E g [ E i L F T i i : r r l : E F E F : r F I E I r ,:l FrEl.l: : REI l 9 ]H i E T HE i l L [ l I IER I : IF L I I IEBF ] ET T L IET ]O I ] : , :- i LEr : T HF I ] T HE I IL I . ] I IE [ : f; F L r I ] I I l I::{ nEt{. l g F r E t l i l : , TH E L I l t E F t : i : : T E l 1 I : : . a r l , , E I , t i T H ; H E n E r t E F r : L l :[ i ] 1 , t i , t _ r : I l l r T H : :: E F E I ] 1 I ] F ' L I T I F J T F l I ] ' : I i I : iF: i , F E l li : PEf r I , i t r i l I rEF a ,F E i r i F l l i r | ::14 FrEfl: rE l F rEr l : l t tL t l I rEF r CrF L [1 ! l :L ] | i: : : REI {l . r r L l t : , E t : t - L : i ! , : , E i i l c i . . _ r F T l , r , . -t . . - 4 r r t r . lq! F rF r l+ 1 F E l l t : 1 1 : i l . t : 1 : r t i l i t 14 r : F E l l4 i EEf t i: 1 ) j : J : t : : t : r r ' :5F FrEl l5 : F rE l l t : : t X : r : ) . : : r t i : r i r : - l5 { F Ef l56 F rE l l ET :5 : r F tE f lr : i j EE i l EF a l l E t { t t t r [F E: - .F : r ] I l i l t ! Tt * L I l l E r i F Ea r , f tT I t { lt-,? FEtlE : q t i E h r r F T B B FE E | T E F t E r r t i r F T E : i T E T E t l f l I r F t , : t 1 l + { i : T : r i : l i6E : F :E l lE : I F i E I ];! FrEfl; : EEI Ii4 t t : r li t18 l : rF E i t l .1I 1 g l : l t l r l1 1: F rE t {121 - rREi l i l t t i E t l E r rB l I :5 I r I | f ' E t . . : .E lt F I l t . f l $1 ,+I : : F E I l1:rcl FEFTIT l il1 - 1 8 l t l ! . i l t . F L t J , t , : l t rl S t r I r i l 1 : i , : | , m ) . ! , i l 1 | ) l l , . t 1 t , : l t . I i . F , t 1 . | ,155 F rE l l1 68 F t E t l * i f l F T F I l E i l ; 5 t E I , I i l i i l l F ! T i j F l t [ I l i l ' , ' 1] r165 F rE l l1 ;e f t E t l i r l F T E F I , lt t l t t F l l i1 i17 5 Pr i rl B l r F t r i I= t T Cr ]190 Fr-rQ ,r:1 TCI tr? r r0 REF T IT , l l r . t )

2 l g l ( J , t ) r X ( l , J )I r5 RErla. e REn +f+ u rq rFAri:El i :F r 'c. ?x E i i ." , ,r . i , r n , r i r L , i E 1 , t s . 1+ +230 I{EXT J.?35 FEtl. . . 4 9I : ! : . , . r . r r . : . L c t r , . , i - r _ .rJ 8 REIiIIH(I)l0 l lE) iT r178 l

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576 REh lf,rtr TF|FCE=T tflr575 REX'-6 T=S39q FOP. .l 1rJ t]t ( JA7 t 1 , l i = Z i l 1 ) - i61f f r=T+Z( , I )a2B t|Evl Ijara'FRI i lT 2r1{rTi4(1 Fnr.r =l TC, 11tlsri' F(rri I:l T,:r 1t :di l l ( l , . It : f ii :7t FoR L=l T0 f li , 1O l t l , I ) = l ) t1 , . ! + ' / iL L l l a ( L , r )f,1B tE)r; Ll ' 10 l E t T , I;1r l l iElT I::i' rl:rr.t I=l T.-r11l ' ; r F' : ,F =l Tn ll: 14 ! ' i l , J =- l l , ; .t ):i B I I E } i T , Ii.:r:i llEllT I?7 8 I F FF i r (T- l HT(T) -1 ) { I ' THE I I , : r nB;gfr IF FE!1(T-i l1T(T))aD THEiln ' in7:?0 rjtrT0 498t-'i5 REtli t t j , ' , :E i l |11 gEFEFT l lTlL T l: ir l I f lTEr]EFl ll fiL15 FrEl ll 1 . l F r : rR= 1 T t l1::i.:rlFIR J=1 T0 hr : r : r ! ( I / J )= !ir4tj Fr:'R L=l TB lil 5 i F t , , t i = 9 1 , r - Y { L L ) l Z { L , . I )rlEA ttE:1T L:r;r:1 ]EXT J:r.. ;1 lE::T Irla,!:rPFr f T,,FFttt{ r:rF lrt1R H=', : lil +I?:t1 tr2t1:1I]'1FIEh .If+ FiEUO!'ESTFTEI1EIIT 1f F1:IF: ULL F'RIHTIil-ITi1'15 lEll'-rlii 6r:rTr:r1::ti: ] : '.TPF]IIIT "EEIIETNLI:T! I} i I ]EF:E': r : r i i Fr . i I l l t'jild FlrF: I=1 Trl 11:r5B FtrR . l : i Ttr l:1110 F. I l lT l , i l , , I : r j: l i i l l lE : iT, I: l i - rB R I I I T : FFt I l t T:rjrn ilElT I!:r5 F]EHl n n i r r 'E f l r i + : H [ r : f l I l 16 Fr : r ] E l -FE X+liati FtEil16 I ( FR I I ] TI r lJ i l FF l lT" i rFr InIl l i lL l l lTFI :1"

; i l ' :1 F r : rF I= l T . l ll1 !9 ! F r : rF : r= 1 T r : r l1 r : 1 : EP P I t I T l l ( L J l,l i r i f l l E : : T I1n ; r : r PF r l l l Tln i i ! PF t l l rT1 ! : r : : r1l E :1T II I t i F F i l lTI I 1 r : r F PI l l T " F F l l I r l JCT ' l ; "I l : r : r F l rF l= l T0 f l1 l : ' l F r : rF r= 1 Tr r fll l l i r F F I I I T F r I - r ) ,l i : t i l t 6 t T rl 1 t : r l r F F l l r T117i r l l E :1T I1 l t } : i F F I i l T1 1._r0 FfrF I = 1 Tr:r Ht-c1i Fr:rF- .r=r T|:r ll1 3 1 f l P F t I l t T i ( L . r ) i13 : t r l l E : :T I1 : : ' ] F ' F J I I T1 . : { i l l l E : i T I1 : i r l FFrIl lT 'a l LL l T l r : r l l : T r : r E1 : ,L rF T I t i l l :1 . l f ! j F l ,F ' l = 1 T r : i ll1 . l i r : r F r I l = r :1l3a-:r,i FIF: ,r=l T0 ll1 ? ! i r F (I ) = F I . ) + ! ' ( L . r I t r H { J )l f i F l l E l l T -l1 : i 1 i f ' F r l l l l I F / I l1 :1 : i l lg l T I1 : r : i r F F 1 i l T ' l r l i lI r [ r l rF Er : ' - l F T ] t ( l : - . I : ', ll1 :1 t r F .F i l T , [_ l x l [F r : rF L l ] l l i l l l i l : I i ' . lI : r 5 . r r ' F r l lT "i F rL - F T EIT B t l l r 0 I r : .EU ,1EI r r . l . : .l : r { , t r F 6F I= 1 T [ ' ]i : i [ f . ! ' : ]= [1 : r t , i r Fr lF r= 1 T t r l l1 ? r i r 1 : r i : 1= i r ' I 4 l i , l l . r l * . E 1J 11 r . r [ l ] E : i r , ll + J t 1 P F r l i l T r l! I ) . H ( I ) H i I l - l t , l I l14 : t1 i l E : :T I1 ; : : r :La r : : r : i J14 : i FEl l1 { : 1 i l i E l l x t r T E 5 T I r i T F 1t +l4:i Ft|l l1 { t i tF T F : , t : r F Ef l l l : r l F Eo l , l i T l r : r l r : F r l l I r l l r l r 1 : rFui l ' l l l r :r l l l l r : i F rF E:T I r : r t1 {1q t1 I IF T F 1 | : ,1 t: t . i J t l t r [ ] . ! : i ' :1 r -1 . -1 . :1 { i : I r i : T F - 1 . r l 1 . 1 t ] , i l . n . [ . f. i : f , ' : 1 . - , 51 + r : i J r i r T F i r i r , ' : r . r 1 . 1 ! t i . ( t 1 . l . r l 1 . 5j + + ! I i R T F r r . i r i r . l , i i , - 1 . ' : i . B . l t r J . 1 t i - : . :1 4 1 : I j F T F n i j ! . f , f . - 1 . - 1 . t 1 . 1 . n i l [ - . i . : i1 r ; i I B T i i i - - l a J . i i : t J . t , l l t t . . : . 1 , t, r ; i r f : T p i a ! i i i l . N i ! 1 i i . 1 1 . !

R E F E R E N C E SBARTON, D.C., 1929,Control and Adiustmentof Sur-veyswith theMagnetomateror the TorsionBalance,

Bulletin of the American Associationof ,PetroleumGeologists, Vol. 13, pp. 1163-1186.COWLES, L., 1938, The adjustment of Misclosures,Geophysics,Vol. 3, pp. 332-339.GIBSON, M.O., 1941, Network Adiustment by Least 'Squares.Alternative Formulation and Solution byIteration, Geophysics,Vol. 6, pp. 168-179'LAGIOS, E., 1984,A Fortran 4 program for a Least -

Squares ravity BaseStationNetworkAdjustment.Computersand Geosciences, ol . 10, No. 2-3, pp'263-276.MACKAY, A., 1981,The Generalizednverse,PracticalComputing, Issue9, PP. 108-110.ROMAN, 1., 19j2, Least Squaresn PracticalGeophlt'sics,Trans.A.I.M.E., Vol' 110,pp. 460-506.SMITH, A.8., 1950,Graphic Adjustment by Least 'Sguares,Geophysics,Vo|. 16, pp' 222-227.WELSCH. W., 1979,A reviewof theAdjustmentof FreeNefworks,Survey Review,XXV, 194,pp. 167-180'

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