Research Article Green s Theorem for Sign Data

Transcription

1 Internatonal Scholarly Research Network ISRN Appled Mathematcs Volume 2012, Artcle ID , 10 pages do: /2012/ Research Artcle Green s Theorem for Sgn Data Lous M. Houston The Unversty of Lousana at Lafayette, Lafayette, LA , USA Correspondence should be addressed to Lous M. Houston, houston@lousana.edu Receved 14 March 2012; Accepted 19 Aprl 2012 Academc Edtors: A. Bellouqud and M.-H. Hsu Copyrght q 2012 Lous M. Houston. Ths s an open access artcle dstrbuted under the Creatve Commons Attrbuton Lcense, whch permts unrestrcted use, dstrbuton, and reproducton n any medum, provded the orgnal work s properly cted. Sgn data are the sgns of sgnal added to nose. It s well known that a constant sgnal can be recovered from sgn data. In ths paper, we show that an ntegral over varant sgnal can be recovered from an ntegral over sgn data based on the varant sgnal. We refer to ths as a generalzed sgn data average. We use ths result to derve a Green s theorem for sgn data. Green s theorem s mportant to varous sesmc processng methods, ncludng sesmc mgraton. Results n ths paper generalze reported results for 2.5D data volumes n whch Green s theorem apples to sgn data based only on tradtonal sgn data recovery. 1. Introducton In certan cases, data, consstng of coherent sgnal and random nose, s summed repeatedly n order to enhance the sgnal and reduce the nose. It has been found that when the sgnal-tonose rato s between 0.1 and 1, f only the sgns of the data are retaned pror to summaton, then the sgnal can be recovered 1. We refer to ths type of data as sgn data. Sgn data has advantages n sgnfcantly reducng the amount of space needed to record the data. The reducton n space can amount to a rato of 20 to 1 bts of data storage 1. In the sesmc ndustry, t has been dscovered that most of the data processng methods used on regular data are also effectve on sgn data 2. Many of these data processng methods are dependent on Green s theorem. In partcular, Krchhoff mgraton s dependent on Green s theorem. Mgraton s a sesmc process that corrects data coordnates whch are dstorted by attrbutes of the sesmc experment. In accordance wth Green s theorem, we show that an ntegral of an operaton on sgn data over a volume s equal to an ntegral of an operaton of the orgnal data over a surface. Ths result generalzes the work of Houston and Rchard 3 n whch, based only on tradtonal sgn data recovery, t was shown that Green s theorem s satsfed when the sgn data encompasses a 2.5D volume. 2.5D data s data whch s three-dmensonal but only has

2 2 ISRN Appled Mathematcs sgn (sgnal + unose) sgnal + unose sgn(sgnal + unose) sgnal + unose 0 sgnal x Fgure 1: A synthetc sgnal sn c x/3 e x/15 2 and the computer-generated unform random nose used to examne sgn-bt ampltude recovery. Ths test has nose of unt magntude a 1 and a sgnal-to-nose rato of one. Shown from bottom to top s the sgnal, the sgnal plus nose, the sgn of the sgnal plus nose, the average over 200 teratons of sgnal plus nose, and the average over 200 teratons of the sgn of sgnal plus nose. varatons along two dmensons. In ths paper, we fnd that Green s theorem s satsfed for sgn data when the data volume s largely symmetrc. For cases n whch the data volume s of arbtrary shape we have derved a varant of Green s theorem. 2. A Generalzed Sgn Data Average It was shown by O Bren et al. 1 that there s a recovery of ampltude from sgn data n the presence of unform random nose. If the random nose X has ampltude a, the sgnal s desgnated by the functon f k, and the number of teratons s M, then we can wrte f k a sgn ( ) f k X, M 2.1 where sgn ( y ) 1, y > 0 0, y 0 1, y < Fgure 1 llustrates the data recovery process ncludng sgn data.

3 ISRN Appled Mathematcs 3 Multply both sdes of 2.1 by the functon g k : g k f k a g k sgn ( ) f k X. M 2.3 Now sum both sdes of 2.3 over the k ndex: g k f k a g k sgn ( ) f k X. M k k 2.4 If we allow k, then 2.4 becomes g f a g sgn ( ) f X M 2.5 or g f a g sgn ( f X ). 2.6 It s clear that f f f and g g, then 2.6 becomes f a sgn ( ) f X, M 2.7 whch s essentally 2.1. A contnuous verson of 2.6 mght be wrtten as g v f v dv a g v sgn ( f v v ) dv. 2.8 An argument for the consstency of 2.8 s as follows. Let f v f. Then we have f a g v sgn ( f v ) dv g v dv. 2.9 Now ntegrate over all values of v: f a g v sgn( f v ) dv g v dv Let g v be a unform probablty densty, ρ v.

4 4 ISRN Appled Mathematcs That mples g v dv 1, 2.11 and 2.10 becomes f a ρ v sgn ( f v ) dv, 2.12 whch was shown by Houston et al A Generalzed Average for Sgn Data Dervatves Consder the followng nth order forward fnte dfference 5 : n ( ) Δ n h f v 1 n f v n h, wth n n!/! n!. If we make the varable, v dscrete by choosng a small real nterval, q and wrtng f f ( q ), 3.2 where s an nteger ndex, then 3.1 becomes Δ n h f n ( ) 1 n f ( q n h ) Because the fnte dfference s a lnear operator, we can use 2.6 to derve g Δ n h f a g Δ n h sgn( f X ). 3.4 Equaton 3.4 takes nto account the fact that g ( q ) f ( q lh ) a g ( q ) sgn ( f ( q lh ) X ), 3.5 where l s an arbtrary ndex.

5 ISRN Appled Mathematcs 5 Snce d n lm dvn h 0 Δ n h h n, suggests the followng equaton: g v dn f v dv a g v dn dvn dv n sgn( f v v ) dv. 3.7 Let d n /dv n d n /du n and f v f u. Then from 3.7 we have d n a g v d n /du n sgn ( f u v ) dv f u. 3.8 dun g v dv Once agan, ntegrate over all v,letg v ρ v,and 3.8 becomes d n f u a ρ v sgn ( f u v ) dv, dun 3.9 whch was shown by Houston et al The Applcaton to Green s Theorem Equaton 3.7 mples the specal case: g v d2 f v dv a g v d2 dv2 dv sgn( f v v ) dv Employng three varables n 4.1 yelds g v 1,v 2,v 3 2 a g v 1,v 2,v 3 2 f v 1,v 2,v 3 dv 1 dv 2 dv 3 sgn ( f v 1,v 2,v 3 v ) dv1 dv 2 dv 3, 4.2 whch can be smplfed f g g v 1,v 2,v 3, f f v 1,v 2,v 3,anddV dv 1 dv 2 dv 3 : g 2 fdv a g 2 sgn ( f v ) dv. 4.3

6 6 ISRN Appled Mathematcs Summaton over the varables yelds g 2 fdv a g 2 sgn ( f v ) dv. 4.4 Consequently, 4.4 can be wrtten as g 2 fdv a g 2 sgn ( f v ) dv. 4.5 If we make the mappng g v 2 g, v then 2.8 becomes f 2 gdv a sgn ( ) 2 f v gdv. 4.7 Summaton over the varables yelds f 2 gdv a sgn ( ) 2 f v gdv, 4.8 whch can be wrtten as f 2 gdv a sgn ( ) 2 f v gdv. 4.9 Subtractng 4.9 from 4.5 yelds ( ( g 2 f f g) 2 dv a g 2 sgn ( ) f v sgn ( ) ) 2 f v gdv Green s theorem s ( S U 1 U 2 n U 2 ) U ) 1 ds (U 1 2 U 2 U 2 2 U 1 dv. n V Consequently, we can wrte a varant of Green s theorem for sgn data as S ( g f n f g ) ( ds a g n 2 sgn ( ) f v sgn ( ) ) 2 f v g dv

7 ISRN Appled Mathematcs 7 If the spatal ntervals are unform, that s, v 1 a 1,b 1, v 2 a 2,b 2, v 3 a 3,b 3 and a 1,b 1 a 2,b 2 a 3,b 3, then 4.12 becomes S ( g f n f g ) ( ds a g 2 sgn ( ) ( ) ) f v sgn f v 2 g dv. n 4.13 In the case of 4.13, we see that Green s theorem s satsfed by replacng the functon f wth sgn data and dvdng the surface ntegral by the nose ampltude. Therefore, all processng of real data based on Green s theorem wll be effectve for sgn data wth an assocated varance. We also note that the effectveness of Green s theorem on sgn data s enhanced by operatng over a symmetrc volume. When the volume s not symmetrc, we can apply the varant of Green s theorem gven by The 2.5D Case Now let us consder the specal case for whch the functons f and g have only twodmensonal varaton. That s, f f v 1,v 2 g g v 1,v Wthout usng a generalzed sgn data average, ths mples ( a g 2 sgn ( ) ( ) ) ( ) f v sgn f v 2 g dv b 3 a 3 g 2 f f 2 g dv 1 dv or S ( g f n f g ) ( ) ds b 3 a 3 g 2 f f 2 g dv 1 dv 2. n 5.3 Equaton 5.3 s Green s theorem for sgn data when the data encompasses a 2.5D volume and s consstent wth results reported n Houston and Rchard Computatonal Tests Equaton 2.6 derves from a weghted average of the functon y sgn f X. We can thus wrte the expectaton value of y as E Y g y g. 6.1

8 8 ISRN Appled Mathematcs We fnd that E Y g f a g. 6.2 The varance, Var Y, can be wrtten as Var Y E Y E Y Ths can be reduced to a smpler form by usng the fact that ( E Y 2) Thus, we have Var Y 1 E Y We can demonstrate 2.6, g f a g sgn ( f X ), 6.6 computatonally. Let f sn ( q ), g cos ( q ). 6.7 Consequently, we want to demonstrate that cos ( q ) sn ( q ) a cos ( q ) sgn ( sn ( q ) X ). 6.8 We can compute the average percentage error β as ( cos( q ) sn ( q ) a cos( q ) sgn ( sn ( q ) ) ) 2 X β 100 ( cos( q ) sn ( q ))

9 ISRN Appled Mathematcs 9 Table 1: Twenty trals whch compare the values of the ndcated expressons for M teratons per tral. The trals are dvded nto ten trals for two dfferent nose ampltudes. Recall that M.LetM and q M 1 cos q sn q a M 1 cos q sgn sn q X a β %, Var Y a β %, Var Y We should see a correlaton between β and Var Y. The results of ths comparson are shown n Table Conclusons Usng the results of sgn data sgnal recovery leads to a dervaton of a generalzed sgn data average. Extendng these results to ncorporate dervatves leads to a varant of Green s theorem for sgn data. We fnd that Green s theorem drectly apples to sgn data when the data volume s symmetrc and the surface ntegral s dvded by the nose ampltude. A specfc applcaton of ths result s that Green s theorem apples to sgn data when the data volume s 2.5D and a generalzed sgn data average s not requred. Acknowledgment Dscussons wth Maxwell Lueckenhoff are apprecated.

10 10 ISRN Appled Mathematcs References 1 J. T. O Bren, W. P. Kamp, and G. M. Hoover, Sgn-bt ampltude recovery wth applcatons to sesmc data, Geophyscs, vol. 47, no. 11, pp , N. A. Anstey, Sesmc Prospectng Instruments, Gebruder Borntraeger, Berln, Germany, 2nd edton, L. M. Houston and B. A. Rchard, The Helmholtz-Krchoff 2.5D ntegral theorem for sgn-bt data, Geophyscs and Engneerng, vol. 1, no. 1, pp , L. M. Houston, G. A. Glass, and A. D. Dymnkov, Sgn data dervatve recovery, ISRN Appled Mathematcs, vol. 2012, Artcle ID , 7 pages, W. G. Kelley and A. C. Peterson, Dfference Equatons: an Introducton wth Applcatons, Academc Press, San Dego, Calf, USA, 1st edton, 1991.

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