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
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)
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
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)
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
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)
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
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)
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. 968-97. Sun, J., 1998, On e limied aperure migraion in wo dimensions, Geopysics, V. 63, P. 984-994 CREWES Researc Repor Volume 13 (1) 581