How much does the migration aperture actually contribute to the migration result?

Transcription

1 Migraion aperure conribuion How muc does e migraion aperure acually conribue o e migraion resul? Suang Sun and Jon C. Bancrof ABSTRACT Reflecion energy from a linear reflecor comes from e inegran oer an aperure, ofen described by e Fresnel zone. Wiin e Fresnel zone, e diffracion energy consruciely builds e reflecion energy. To ge rue reflecion ampliude, a migraion aperure a is wice e Fresnel zone size mus be considered. Tis paper deries e size of e presack Fresnel zone as a funcion of alf e source-receier offse. I ealuaes ow e size of e migraion aperure affecs e migraion resul, en esablises an accepable minimum migraion aperure for orizonal reflecors. For bo zero-offse and offse daa, wice e Fresnel zone size is e minimum migraion aperure o presere rue ampliude. A migraion aperure a is larger does no improe e migraion resul. INTRODUCTION A Fresnel zone is defined as e inercep wen a sperical wae peneraes a plane o a dep of alf a waeleng (Claerbou, 1985). Te Fresnel zone idenifies e area a conribues energy o a migraed race and is sown in Figure 1. Te migraion aperure in Kircoff summaion meod mus be larger an e Fresnel-zone size o ge e rue reflecion ampliude. I is worwile o inesigae e relaionsip beween e presack Fresnel-zone size and e presack migraion aperure, o maximize e signal and minimize e noise. I is also necessary o esablis a minimum aperure o presere e rue ampliude. τ/ Summaion yperbola Fresnel zone τ/ FIG. 1. Fresnel zone for zero-offse secion. CREWES Researc Repor Volume 13 (1) 573

2 Sun and Bancrof Sun (1998) concluded a e minimum migraion aperure sould be e ineral were e difference beween e raelimes of refleced and poin-diffraced rays equals e duraion of e recorded seismic pulse. He sowed zero-offse numerical examples o confirm is resuls. Sun s minimum migraion aperure is wice e Fresnel zone size definiion. In is paper, numerical examples sow a e minimum migraion aperure using wice e Fresnel-zone size also guaranees e presack migraion resuls in an offse secion. OFFSET AND FRESNEL ZONE For presack migraion e Kircoff summaion raelime cure in an offse secion is deermined by e double-square-roo (DSR) equaion as: = 4 4 ( x ) ( x ), (1) were is alf e source-receier offse, is wo-way erical raelime, is RMS elociy a e scaerpoin, and x is e surface disance beween e midpoin and e scaerpoin. For a orizonal reflecor, according o Claerbou s definiion menioned aboe, we deried e Fresnel zone size as a funcion of alf source-receier offse in an offse secion using DSR equaion. Te DSR equaion can be rewrien as: = 4( x 4x From Figure, we included e offse Fresnel-zone displacemen x f using equaion () as: ( τ ) = ( 4 τ ) = 4( x f ) 4xf were is e raelime a e gien alf-offse, and τ is e period of e waele. We can sole for e Fresnel zone radius x f, giing: ) () (3) 574 CREWES Researc Repor Volume 13 (1)

3 Migraion aperure conribuion x f = τ 4 4 For zero-offse, i.e. =, e Fresnel zone radius is exacly e same as Seriff s (198) definiion: 4 (4). = τ x f (5) Te summaion cure and e Fresnel zone in an offse secion are sown in Figure S R Scaer poin Zero-offse Consan offse τ/ Fresnel zone x f Summaion cure FIG.. Fresnel zone for offse secion. Assuming no frequency decay in e source waele, e ariaion of e Fresnel zone radius, erses e alf source-receier offse for orizonal reflecor, is sown in Figure 3. As e Fresnel zone radius increases wi alf source-receier offse, e migraion aperure sould also increase wi alf source-receier offse. CREWES Researc Repor Volume 13 (1) 575

4 Sun and Bancrof FIG. 3. Fresnel zone radius aries wi alf source-receier offse. FRESNEL ZONE IN CHEOPS PYRAMID In e pre-sack olume (x,, ) e DSR equaion describes e raelime from a scaerpoin as a surface a is referred o as Ceops pyramid (Claerbou, 1985). To migrae energy back o a scaerpoin, e presack migraion sums all e energy along e Ceops pyramid. Howeer e reflecion energy, i.e. e specular energy from a linear reflecor, comes from an area proporional o e size of e Fresnel zone. For orizonal reflecor e ig-frequency specular energy of e scaerpoin forms a yperbola as sown in Figure 4a by a solid cure. Te dased lines sow e band-limied Fresnel zone as a funcion of e alf source-receier offse in Ceops pyramid. Te offse Fresnel zone can be seen more clearly by e conours in e plan-iew of e Ceops pyramid as sown in Figure 4b. 576 CREWES Researc Repor Volume 13 (1)

5 Migraion aperure conribuion (a) (b) FIG. 4. Specular energy and Fresnel zone of a orizonal reflecor in a) Ceops pyramid in (x,, ), and b) conours of Ceops pyramid in (x, ). Te sape of e Fresnel zone can also be illusraed by e inersecion of Ceops pyramid and e presack surface of a orizonal reflecor: a yperbolic cylinder. Raising e Ceops pyramid by alf period τ/ produces an inersecion a maps e Fresnel zone as illusraed in Figure 5. Te fron iew is sown in (a) and a iew from aboe is sown in (b). (a) x (b) FIG. 5. Presack Fresnel zone for a orizonal reflecor defined by inersecing e yperbolic cylinder wi Ceops pyramid, a) fronal iew and b) from aboe. CREWES Researc Repor Volume 13 (1) 577

6 Sun and Bancrof FRESNEL ZONE AND MINIMUM MIGRATION APERTURE Migraion is used no only for srucure correcion bu also for esimaing e rue reflecion ampliude. To ge e rue reflecion ampliude, bo e migraion aperure and is locaion are e deerminae facors. Howeer for a orizonal reflecor, e angen poin, i.e. e reflecion poin, is e same poin as e scaerpoin. In is case e migraion aperure is cenred a e scaerpoin. For a dipping reflecor, e angen poin on e diffracion cure idenifies e migraion aperure. Wa is e minimum migraion aperure really needed o migrae a poin o ge e rue reflecion ampliude? Tradiionally we use a migraion aperure a is as large as possible, assuming i can ge e bes migraion resul. In fac e migraion aperure need no o be so large o ge e reflecion ampliude from linear reflecor. In some cases e minimum migraion aperure sould guaranee e rue reflecion ampliude. A larger migraion aperure as no improemen on e migraion resul (for a linear reflecor). Numerical examples sow e minimum presack migraion aperure, o presere e rue ampliude, sould be wice e size of e Fresnel zone defined in e aboe secions. NUMERICAL EXAMPLES For a simple case we es e migraion aperure assuming a ere is only wo inerfaces in e subsurface wic as an AVO effec. Te middle layer is wi some gas in i. Figure 6 sows e elociy and Poisson s raio of model. Assuming an inciden plane wae, we use Zoeppriz equaion o calculae e reflecion coefficien and e pase of e refleced signal. Tess confirm a e minimum migraion aperure for zero-offse migraion and e consan offse migraion sould be e double size of Fresnel zone size. FIG. 6. Two-dimensional geology model used in e numerical example. Resuls of upper layer Te following ree figures sow ow e migraion aperure conribues o e migraion resul. Te ampliude firs goes up o is peak en goes down and sabilizes. 578 CREWES Researc Repor Volume 13 (1)

7 Migraion aperure conribuion Noice e sable poin: as Figure 8 sows, e migraion aperure a e sable poin is jus double e size of e Fresnel zone. (a) (b) FIG. 7. Migraion aperure relaes o Fresnel zone of upper layer. a) Zero-offse; b) Half-offse equals 5m. FIG. 8. Fresnel radius aries wi alf-offse of upper layer. Resuls of lower layer Te following ree figures sow ow e migraion aperure conribues o e migraion resul. Again, for e lower layer, e ampliude also goes up o is peak CREWES Researc Repor Volume 13 (1) 579

8 Sun and Bancrof en goes down and sabilizes. Noice e sable poin: as Figure 1 sows, e migraion aperure a e sable poin is jus double e size of Fresnel zone. (a) (b) FIG. 9. Migraion aperure relaes o Fresnel zone of lower layer. a) Zero-offse; b) Half-offse equals 5m. FIG. 1. Fresnel radius aries wi alf-offse of lower layer CONCLUSIONS Te Fresnel zone for presack daa was defined o aid in idenifying e presack migraion aperure. 58 CREWES Researc Repor Volume 13 (1)

9 Migraion aperure conribuion To ge rue reflecion ampliude, e minimum presack migraion aperure sould be e wice of e Fresnel zone size in bo e zero-offse secion and e offse secion. A smaller presack migraion aperure a equals e size of e Fresnel zone may produce a saisfacory image. ACKNOWLEDGEMENTS We ank e sponsors of e CREWES projec and NSERC for supporing is researc. REFERENCES Claerbou, J.F., 1985, Imaging e Ear's Inerior, Blackwell Scienific Publicaions, Aailable oer Inerne. Seriff, R.E., 198, Nomogram for Fresnel-zone calculaion, Geopysics, V 45, P Sun, J., 1998, On e limied aperure migraion in wo dimensions, Geopysics, V. 63, P CREWES Researc Repor Volume 13 (1) 581