DSpace at VNU: Principal component analysis for field separation

VNU JOURNAL OF SCIENCE, Mathematics - Physics T.xx N03AP 2004 P R IN C IP A L CO M PO NENT A N A LYSIS FO R F IE L D SEPARA TIO N T o n T ic h A i D epartm ent o f Physics, College o f Science, V N U A b s tr 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 ge 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 tio n is o n e o f th e m o s t im p o rta n t p ro b le m s in g e o p h y s ic a l d a ta in te rp re ta tio n F o r po te n tia l 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 a re a s u rve y T he fie ld s e p a tio n b e c o m e s a p ro c e s s o f e s tim a tio n o f low 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 an o m a ly , on th e o n e h a n d , a n d highfre 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 nd 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 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 bo die 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 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 tio n fo r g e o p h y s ic a l in te rp re ta tio n 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 ata have 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 ata suggests th a t s ta tis tic a l techn iqu es o f m u ltiv a ria te an alysis are appropriate P rin c ip a l com ponents a n alysis as a m u ltiv a ria te e xp loratory techniques provides a u sefu l s ta rtin g p o in t fo r fu rth 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 total v a ria tio n o f m u ltiv a ria te observations in to lin e a rly independent com ponents o f decreasing inpotance In th is a rticle , the p rin c ip a l com ponent analysis is u se d fo r fie ld sep aration and in te rg te d d ata processing 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 alysis fo r fie ld separation w hen there are a re al survey data L e t the set o f ran dom valu e s X ] , ,XN be presented by tw o-d im en sion al d ata file (areal surv ey data) fo r the sam e p h y sic a l field, in the form o f m atrix o f N row s a n d n colum ns T h e alg orith m o f fie ld sep aration in clu d e the follow ing operation: C alculation o f the mean for each profile: w here x kj are the observed fie ld d ata fo r th e k th p o in t o f the ith profile TonTichAi C alculation o f covariance for each pair o f profiles bjj = - £ ( X k i - x i)(xkj ~ x j) i j = 2, - Construction o f covariance m atrix B: '»1 fill b|2 I bN2 I w here bii is the observ ed d a ta v a ria n ce fo r the ith p ro file a n d Calculation o f the m axim um eigenvalue ' b u -X b l2 b]N b?',! - baN 6y = bjị solving the m a trix equation b N2 T h e eigenvalues o f th is e q uation ?.J, d e te rm in an t o f (B - /.max by .,XN a re the roots o f the equation, w ith the I) b e in g e q ual to zero Father, it is n ecessary to select the m axim um eigen valu e am ong the o btain ed roots O btaining the m axim um 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: 2a®12 12+ (bj| - ^max)®11 + b ,ia + b 12a 1+ lN a 1N = + b |f (b 22 -? - mils)a ,2 + + b 2Na 1N = b lN a l l + ^2Na 12 + The eigenvector = 1, i =1, 2, a ](a 11ta i 2>-»>&iN) is + (^NN - ^max )®1N - de te rm ine d in term s of n orm alization T h e p h y sical sense of such n o rm a liz a tio n im p lie s th e e xpression of the transfo rm ed d ata at the sam e scale as the p rim a ry fie ld data F inding o f the first principal component Y ] = a'X or = (*11 *12 W e can re g ard th e v alu 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 = , , N) d e te rm in e th e w eight coefficient for each profile E stim ation o f the field component, m atrix expression: chara cteriz ed by m axim u m varia nce, u sin g the P r in c ip a l c o m p o n e n ts a n a ly s is for y ii® n rt'g _ Y ki ~ ,2 (a n , a 12, a 1N) + X i = Y ,„ > y in a l l y n a 12 + x • y u « i N + X N +X1 y I 2a i + X i y +X1 12a 12 + x y i n a 12 + x y I 2a lN + X N • y i na iN + * N , T h e fie ld com pon ent havin g the m axim um variance, e nsures the e stim atio n o f the regional an om aly w h e n X max = ( - % ) ^ A., S ince x ỹ g is the e stim atio 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 atio n o f the local one O n b asic o f th e presented ab ou t alg orirhm , the p rogram fo r c a lc u la tin g regional and local an om alie 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 tistics'D e s c rip tiv e S ta tis tic s ' n = D im ensions[dataO ]; n = n ỊỊIỊỊ: n = n [[2 ; b = Id e n tity M atrix(n2]; D o[x[i] = M ean[dataO [[i]]], (i, n1}] D o(D o[b [[i, j j] = Sum [(dataO[[k, i]] X[i])(data0([k, j]] - x [j]) , {k n1>]/n1 {» n1 }] {j, n1}] d = E igenvectors(b); {d([1]].d ata 0}; d a ta i = T ransp ose [% ].{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 din g -> F alse , C o n to u rs -> 20 F ram eL ab el -> {"x 00 m ", "y.10 m "}, C o n to u rS ty le -> RGBColorJO, 0, 1]J; dt1 = L istC o nto urP lo t[ d a ta i, C o n tou rS din g -> False, C on tou rs -> 20, F ram eL ab el -> {"x 00 m ", "y 0 m "}, C on tou rS tyle -> R G B C olor[0, , 1]J; dt2 = ListC ontourP lotỊdataO - d a ta i, C o n tou rS din g -> F als e , C o n to u rs -> 40 , F ram eL ab el -> {"x 00 m '\ "y-100 m "}, C o n to u rS ty le -> R G BC olorỊO , 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 pram 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, F ig Total Anomalies F 'g -2 Regional Anomalies F ig Local Anomalies TonTichAi C o n c lu s io n s B y u sin g p r in c ip a l com ponents analysis w e m ay e m phasize d iffe re n t com ponents from total an om alie s in d ependence o f o u r in te rp re tation goal T h e m ethod was sim p lifie d to enable e asie r a n d th u s p ossibly g eological in terp retation o f g eoph ysical d ata in d iffe re nt conditions R e fe r e n c e s Ton T ich A i, Mathematica To n T ic h A i, Applied Stephen Wolfram , Tafcev.G.P, for engineer, N ational U niversity Publisher, H an oi 2003 Geophysics U niversity Mathematica Sokolov K.P., M in is try Publisher, Hanoi 1988 A ddison-Wesley Publishing Company, Inc 1988 Geological interpretation o f magnetic anomalies Neilra Leningrad 1981 N ik itin A A., Statistical Processing o f Geophysical Data Electrom agnetic Research Center Moscow 1993 J.'IYoehimczyk and F.Chayes., geology, Vol 10, NO 1,1978 Some Properties o f Principal Component Scores Mathematiacal ... coefficient for each profile E stim ation o f the field component, m atrix expression: chara cteriz ed by m axim u m varia nce, u sin g the P r in c ip a l c o m p o n e n ts a n a ly s is for y... terp retation o f g eoph ysical d ata in d iffe re nt conditions R e fe r e n c e s Ton T ich A i, Mathematica To n T ic h A i, Applied Stephen Wolfram , Tafcev.G.P, for engineer, N ational U... ' n = D im ensions[dataO ]; n = n ỊỊIỊỊ: n = n [[2 ; b = Id e n tity M atrix(n2]; D o[x[i] = M ean[dataO [[i]]], (i, n1}] D o(D o[b [[i, j j] = Sum [(dataO[[k, i]] X[i])(data0([k, j]] - x [j])