Using Gaussian Elimination for Determination of Structure Index in Euler Deconvolution

Transcription

1 Austrli Jourl of Bsic d Applied Scieces, 4(1): , 010 ISSN Usig Gussi Elimitio for Determitio of Structure Idex i Euler Decovolutio 1 Rez.toushmli d M.ghbri 1 Islmic zd uiversity,hmed brch Islmic zd uiversity, khormbd brch Abstrct: Euler s homogeeity reltio hs ttrcted spordic iterest from geophysicists over the yers. The choice of structurl idex i Euler homogeeity equtio remis vexig problem, becuse structures re poorly imged d depths re bised if the wrog idex is used for y give feture. I this pper we ssume tht structurl idex is oe of my ukows d with solvig system of equtio with Gussi elimitio fid vlue of structure idex. Key words: structure idex, Euler Decovolutio, Gussi Elimitio 1)Euler decovolutio: Euler s homogeeity reltio hs ttrcted spordic iterest from geophysicists over the yers. It my be std succictly i the form T T T ( x xo) ( y yo) ( zzo) NT x y z Where x0,y0,z0 is the positio of source whose totl field T is detected t (x,y,z).the totl field hs regiol or bckgroud vlue B. where T, T d T represet first-order derivtive of x y z the mgetic (grvity) field log the x-, y- d z- directios, respectively, N is kow s structurl idex d relted to the geometry of the mgetic(grvity) source(si:thompso,198). Tkig ito ccout bse level for the regiol mgetic field (B) equtio (1) c be rerrged d writte s T T T T T T x y z NB x y z NT., x y z x y z Assigig the structurl idex (N) system of lier equtios c be obtied d solved for estimtig the loctio d depth of the mgetic d grvity body. The choice of structurl idex i euler homogeeity equtio remis vexig problem, becuse structures re poorly imged d depths re bised if the wrog idex is used for y give feture (Reid et l., 1990). I this pper we ssumed N=v, B=0 d obti the vlu of N with solvig system of lier equtio by Usig Gussi Elimitio. ) Gussi Elimitio: -1) How is Set of Equtios Solved Numericlly?: Oe of the most populr techiques for solvig simulteous lier equtios is the Gussi elimitio method. The pproch is desiged to solve geerl set of equtios d ukows Correspodig Author: Rez.toushmli, Islmic zd uiversity,hmed brch E-mil:Atomicboy_rez@yhoo.com 6390

2 x x x... x b x x x x b Gussi elimitio cosists of two steps 1. Forwrd Elimitio of Ukows: I this step, the ukow is elimited i ech equtio strtig with the first equtio. This wy, the equtios re reduced to oe equtio d oe ukow i ech equtio.. Bck Substitutio: I this step, strtig from the lst equtio, ech of the ukows is foud. -)Forwrd Elimitio of Ukows: I the first step of forwrd elimitio, the first ukow, x 1 is elimited from ll rows below the first row. The first equtio is selected s the pivot equtio to elimite x 1. So, to elimite x 1 i the secod equtio, oe divides the first equtio by 11 (hece clled the pivot elemet) d the multiplies it by 1. This is the sme s multiplyig the first equtio by 1 / 11 to give x1 1x... 1x b Now, this equtio c be subtrcted from the secod equtio to give x x b b or x... x b where This procedure of elimitig x 1, is ow repeted for the third equtio to the th equtio to reduce the set of equtios s x 3x3... x b 3x 33x3... 3x b 3... x x x b

3 This is the ed of the first step of forwrd elimitio. Now for the secod step of forwrd elimitio, we strt with the secod equtio s the pivot equtio d s the pivot elemet. So, to elimite x i the third equtio, oe divides the secod equtio by (the pivot elemet) d the multiply it by. This is the sme s multiplyig the secod equtio by 3 / 3 d subtrctig it from the third equtio. This mkes the coefficiet of x zero i the third equtio. The sme procedure is ow repeted for the fourth equtio till the th equtio to give x 3x3... x b 33x3... 3x b 3 3x3... x b The ext steps of forwrd elimitio re coducted by usig the third equtio s pivot equtio d so o. Tht is, there will be totl of - 1 steps of forwrd elimitio. At the ed of - 1 steps of forwrd elimitio, we get set of equtios tht look like x 3x3... x b 33x3... 3x b x b -3)Bck Substitutio: Now the equtios re solved strtig from the lst equtio s it hs oly oe ukow. x b ( 1) ( 1) The the secod lst equtio, tht is the ( - 1) th equtio, hs two ukows: x d x -1, but x is lredy kow. This reduces the ( - 1) th equtio lso to oe ukow. Bck substitutio hece c be represeted for ll equtios by the formul i1 i1 bi ij xj j i 1 xi for i 1,,,1 i1 d ii 639

4 x b ( 1) ( 1) 3)Usig Gussi Elimitio i Euler Decovolutio T=[ ]; x=[1 3 4]; y=[ 4 6 8]; T x T y T z =[ ] =[ ] =[ ] Iput the coefficiets for 4 equtios 1) Doig forwrd-elimitio phse: colum 1 of 4 Foud the first row tht hs ozero i colum V X Y Z = V X Y Z = V X Y Z = V X Y Z = ) Doig forwrd-elimitio phse: colum 1 of 4 Divided etire row 1 by Now subtrct times row 1 from row V X Y Z = V X Y Z = V X Y Z = V X Y Z = ) Doig forwrd-elimitio phse: colum 1 of 4 Now subtrct times row 1 from row 3 V X Y Z = X Y Z = V X Y Z = V X Y Z = ) Doig forwrd-elimitio phse: colum 1 of 4 Now subtrct times row 1 from row 4 V X Y Z = X Y Z = X Y Z = V X Y Z =

5 5) Doig forwrd-elimitio phse: colum 1 of 4 Doe with colum 1. Move o to ext colum V X Y Z = X Y Z = X Y Z = X Y Z = ) Doig forwrd-elimitio phse: colum of 4 Foud first row o or fter row tht hs ozero i colum V X Y Z = X Y Z = X Y Z = X Y Z = ) Doig forwrd-elimitio phse: colum of 4 Divided etire row by Now subtrct times row from row 3 V X Y Z = X Y Z = X Y Z = ) Doig forwrd-elimitio phse: colum of 4 Now subtrct times row from row 4 V X Y Z = Y Z = X Y Z = ) Doig forwrd-elimitio phse: colum of 4 Doe with colum. Move o to ext colum V X Y Z = Y Z = Y Z = ) Doig forwrd-elimitio phse: colum 3 of 4 Foud first row o or fter row 3 tht hs ozero i colum 3 V X Y Z = Y Z = Y Z = ) Doig forwrd-elimitio phse: colum 3 of 4 Divided etire row 3 by Now subtrct times row 3 from row 4 V X Y Z = Y Z = Y Z = ) Doig forwrd-elimitio phse: colum 3 of 4 Doe with colum 3. Move o to ext colum V X Y Z = Y Z = Z =

6 13) Doig forwrd-elimitio phse: colum 4 of 4 Foud first row o or fter row 4 tht hs ozero i colum 4 V X Y Z = Y Z = Z = ) Doig forwrd-elimitio phse: colum 4 of 4 Divided etire row 4 by Doe with colum 4. Tht ws the lst colum, move o to bck-elimitio V X Y Z = Y Z = Z = ) Doig bck-elimitio phse: colum 4 Subtrct times row 4 from row 3 V X Y Z = Y Z = Z = ) Doig bck-elimitio phse: colum 4 Subtrct times row 4 from row V X Y Z = Y = Z = ) Doig bck-elimitio phse: colum 4 Subtrct times row 4 from row 1 V X Y Z = X Y = Y = Z = ) Doig bck-elimitio phse: colum 3 Subtrct times row 3 from row V X Y = X Y = Y = Z = ) Doig bck-elimitio phse: colum 3 Subtrct times row 3 from row 1 V X Y = X = Y = Z = ) Doig bck-elimitio phse: colum Subtrct times row from row 1 V X = X = Y = Z =

7 Doig bck-elimitio phse: colum 1 All doe! V = X = Y = Z = With the clculted vlue of the structurl idex (V=N) we hve chieved our gol d we do t eed specific geologicl ssumptio to determie the structurl idex. If we lso hve more poits this method will Exteded d we c use this method. Coclusio: The choice of structurl idex i euler Decovolutio remis vexig problem, becuse structures re poorly imged d depths re bised if the wrog idex is used for y give feture. I this pper we ssume tht structurl idex is oe of my ukows d with solvig system of equtio with Gussi elimitio fid Vlue of structure idex. With the clculted vlue of the structurl idex we hve chieved our gol d we do t eed specific geologicl ssumptio to determie the structurl idex. If we lso hve more poits this method will Exteded d we c use this method. REFRENCES Golub, H. Gee, V Lo, F. Chrles, Mtrix Computtios (3rd ed.), Johs Hopkis, ISBN Reid, A.B., J.M. Allsop, H. Grser, A.J. Millett, I.W. Somerto, Mgetic iterprettio i three dimesios usig Euler Decovolutio. Geophysics, 55(1): Reid, A.B., [1] GETECH, Euler Decovolutio, Pst, Preset d Future: A Review,c/o Deprtmet of Erth Scieces, Leeds Uiversity, Leeds, Uited Kigdom Thompso, D.T., 198. EULDPH: A ew techique for mkig computer-ssisted depth estimtes from mgetic dt. Geophysics, 47(1): Preuss, W.H., et l., 199. Numericl Recipes i Fortr, d editio Cmbridge Uiversity Press. 6396