DSpace at VNU: Principal component analysis for field separation

A ddison-W esley Publishing Company, Inc.. J.'IYoehimczyk and F.Chayes., Some Properties o f Principal Component Scores.. Mathematiacal geology, Vol.

P R IN C IP A L CO M PONENT A N A LYSIS FO R F IE L D SEPARA TIO N

A b s t r a c t T h e a r tic le c o n ta in s d iffe re n t te c h n iq u e s o f g e o p h y s ic a l da ta

p ro c e s s in g b y u s in g s o ftw a re M a th e m a tic a F ie ld s e p a ra tio n is o n e o f th e fie ld s , w h e n th e re a re o b s e rv a tio n a l da ta fo r th e b o th th e p ro file s u rv e y a n d fre q u e n c y c o m p o n e n t, i.e , th e re g io n a l a n o m a ly , o n th e o n e h a n d , a n d hig h- fre q u e n c y fie ld c o m p o n e n t, i.e th e re s id u a l o r lo c a l a n o m a ly o n th e o th e r

ha n d

P rin c ip a l c o m p o n e n ts a n a ly s is fo r fie ld s e p a ra tio n p ro v id e s im m e d ia te in s ig h t

in to th e s tru c tu re o f fie ld d a ta a n d is a p p lie d fo r m o d e lin g th e a n o m a ly fie ld

th a t c o n ta in s d iffe re n t b o d ie s

T h e c a lc u la tio n p ro c e s s is re a liz e d b y u s in g th e c o m p u te r a lg e b ra ic sys te m (m a th e m a tic a ) T h e re s u lt o f re s e a rc h is u s e d in g e o p h y s ic a l fie ld s e p a ra tio n

fo r g e o p h y s ic a l in te rp re ta tio n

1 I n t r o d u c t io n

T r a d it io n a l in te rp re ta tio n s o f th e geophysical d a ta ha ve cocentrated on one o r two preselected va ria b le s o r fu nction s o f the variables How ever, the m u ltiv a ria te s tru ctu re o f the d a ta suggests th a t s ta tis tic a l techn iqu es o f m u ltiv a ria te an a lysis are appropriate

P r in c ip a l com ponents a n a lysis as a m u ltiv a ria te e xp lora tory techniques provides a

u sefu l s ta rtin g p o in t fo r fu r th e r investigations It m ay also p ro vid e in s ig h t in to the geological processes u n d e rly in g th e data It is a m ethod fo r decom posing the tota l v a ria tio n

o f m u ltiv a ria te observations in to lin e a r ly independent com ponents o f decreasing inpotance

In th is a rticle , the p r in c ip a l com ponent ana lysis is u se d fo r fie ld sep a ra tion and

in te rg ra te d d a ta processing

1 A p p l i c a t io n o f t h e p r i n c i p a l c o m p o n e n t a n a ly s ic : F i e l d s e p a r a t io n f o r a r e a l

s u r v e y d a ta

C o n s id e r th e a p p lica tio n o f th e p rin c ip a l com ponent a n a lysis fo r fie ld separation

w hen there are a re a l survey data L e t the set o f ran dom va lu e s X ] , ,XN be presented by

tw o-d im en sion al d a ta f ile (area l surv ey data) fo r the sam e p h y sic a l field, in the form o f

m a trix o f N row s a n d n colum ns T h e a lg orith m o f fie ld sep a ra tion in clu d e the follow ing operation:

C alculation o f the mean for each profile:

T o n T i c h A i

D epartm ent o f Physics, College o f Science, V N U

w here x kj are the observed fie ld d a ta fo r th e k th p o in t o f the ith profile

1

2 Ton Tich Ai

C alculation o f covariance for each p a ir o f profiles

bjj = - £ ( X k i - x i)(xkj ~ x j) i j = 1.2,

-Construction o f covariance m atrix B:

fill b|2 '»1

I bN2 I

w here bii i s the observ ed d a ta v a ria n ce fo r the ith p ro file a n d 6y = bjị

C alculation o f the m a x im u m eigenvalue /.max by solving the m a trix equation

' b u -X bl2 b]N b?',! - baN

.1 b N2

T h e eigenvalues o f th is e q uation ?.J, ,XN a re the roots o f the equation, w ith the

d e te rm in a n t o f (B - I) b e in g e q ua l to zero Father, it is n ecessary to select the m a xim um eigen valu e am ong the o bta in ed roots

O btaining the m a x im u m eigenvector o f m atrix B, w h ic h correspon d to the Ảmux , w ith the a id o f th e set o f equations:

(bj| - ^max)®11 + b ia®12 + + b |f

b12a

+ b ,2a 12+ + b lN a 1N =

1 + (b22 -? - mils)a,2 + + b2Na 1N =

b lN a l l + ^2Na 12 + + (^NN - ^max )®1N - 0

T h e eigenvector a ](a11ta i2>-»>&iN) is de te rm ine d in term s o f n orm a liza tion

= 1 , i =1, 2, T h e p h y sica l sense of such n o rm a liz a tio n im p lie s th e e xpression of the tra nsfo rm ed d a ta at the sam e scale as the p rim a ry fie ld data

F inding o f the first p rin cip a l component Y ] = a 'X o r

= (*11.*12

W e ca n re g a rd th e v a lu e s Y ) K (k = 1,2, ,n) as the w e ig h t coe fficien ts fo r each point

o f the fie ld data In th is connection, the valu es a ! j (i = 1 , 2 , N) d e te rm in e th e w eight coefficient for each profile

E stim ation o f the field component, cha ra cteriz ed by m a xim u m varia nce, u sin g the

y i i ® n + X 1 y n a12 + x2 • y u « iN + XN rt'g _

ki ~

Y,2

(a n , a 12, a 1N) + X i =y I2a i 1 + X i y12a12 + x2 y I2a lN + X N

Y ,„ >

y in a l l + X 1 y i na 12 + x2 • y i na iN + * N ,

T h e fie ld com pon ent ha vin g the m a xim um varia nce, e nsures the e stim a tio n o f the regional an om a ly w h e n X max = ( 7 0 - 9 0 % ) ^ A., S ince x ỹ g is the e stim a tio n o f the regional anom aly, th e n the d iffe re nce xjjj0 = Xjji - x£jg w ill be the e stim a tio n o f the loca l one

O n b asic o f th e presented ab ou t a lg orirhm , the p rogram fo r c a lc u la tin g regional and local an om a lie s o f p o te n tia l fie ld is m ade by a u th o r in lan gu ag e “M a th e m a tica ”:

« S ta tis tic s 'D e s c rip tiv e S ta tis tic s '

n = D im e nsion s[d ata O ];

n1 = n Ị ỊIỊỊ :

n 2 = n [[2 ;

b = Id e n tity M atrix(n2];

D o [x [i] = M ea n[d ataO [[i]]], (i, n1}]

D o (D o [b [[i, j j] = Sum [(dataO [[k, i]]

-X [i])(da ta 0([k, j]] - x [j]) , {k n1>]/n1 {» 1 n1 }] {j, 1 n1}]

d = E ig en vecto rs(b );

{d([1 ]].d a ta 0 };

d a ta i = T ra n sp o se [% ].{d |[1 ]]};

Do[Do[data1([i,j]]= data1[[i j]] + x[i], 0, n1}J, {i, n2}J

dto = ListC ontourP lotfdataO , C on tou rS ha din g -> F alse , C o n to u rs -> 20

F ra m e L a b e l -> {"x 1 0 0 m ", "y.1 0 0 m "},

C o n to u rS ty le -> RGBColorJO, 0, 1]J;

dt1 = L is tC o n to u rP lo t[

d a t a i, C o n to u rS h a d in g -> False , C o n to u rs -> 20, F ra m e L a b e l -> {"x 1 0 0 m ",

"y 1 0 0 m "}, C o n to u rS tyle -> R G B C o lo r[0, 0 , 1]J;

d t2 = L istC o nto urP lo tỊd ata O - d a ta i, C o n to u rS h a d in g -> F a ls e , C o n to u rs ->

4 0 , F ra m e L a b e l -> {"x 1 0 0 m '\ "y-1 00 m "}, C o n to u rS ty le -> R G B C olo rỊO , 0, 1J];

M o d e l i n g d i f f e r e n t f i e l d s e p a r a t io n s T o d em onstrate the fie ld separation a b ility

o f the m ethod, in th is article, the m odel o f three spheres o f d iffe re n t pra m e te rs is selected

T h e re su lts o f c a lcu la tio n are p resented in fig u res 1, 2, 3

Ton Tich Ai

B y u sin g p r in c ip a l com ponents ana lysis w e m ay e m phasize d iffe re n t com ponents from total an om a lie s in d ependence o f o u r in te rp re ta tion goal T h e m ethod wa s sim p lifie d

to enable e a sie r a n d th u s p ossibly g eological in terp retation o f g eoph ysical d a ta in d iffe re nt conditions

R e fe r e n c e s

1 Ton T ich A i, Mathematica for engineer, N ational U niversity Publisher, H an oi 2003

2 To n T ic h A i, Applied Geophysics U niversity M in is try Publisher, H anoi 1988

3 Stephen Wolfram , Mathematica. A ddison-W esley Publishing Company, Inc 1988

4 Tafcev.G.P, Sokolov K.P., Geological interpretation o f magnetic anomalies. Neilra Leningrad 1981

5 N ik itin A A., Statistical Processing o f Geophysical Data. Electrom agnetic Research Center Moscow 1993

6 J.'IYoehimczyk and F.Chayes., Some Properties o f Principal Component Scores. Mathematiacal geology, Vol 10, NO 1,1978

2 C o n c lu s io n s