Petack igation of obique efecto Petack conideation fo te igation of obique efecto Jon C. Bancoft and Cae P. Uenbac ABSTRACT A eo-offet tacked ection tat contain an obique efecto can be igated uccefuy by aiing te igation veocity. Wat petack poceing can be pefoed to ipove te diffacted enegy on te tacked ection? A pecia oveout coection can be appied to enance te obique efection on te tacked ection and odified petack igation opeato ay ao be ued to enance te obique efection. Te benefit of tee pecia pocee wi depend on te inea etent and ange of te obique efecto. Satifactoy eut can ao be obtained wen tacking wit te RMS veocity and wen tacking wit an RMS veocity tat i inceaed by te invee of te coine of te ange of obiquity. INTRODUCTION A pecia -D pottack igation of obique efecto wa decibed by Fenc 975 in wic te igation veocity V ig i inceaed ove te RMS veocity V a given in equation wee i te ange of obiquity, a iutated in Figue. V V ig co, -D ine y Figue. A pepective view of a -D eiic ine above an obique efecto. CREWES Reeac Repot Voue
Bancoft and Uenbac It i aued tat te ediu beow te eiic ine i oiontay ayeed and tat tave tie ay be etiated uing RMS veocitie. Figue i a pan view of Figue, and iutate te eo offet aypat fo diffeent uface ocation. Te aypat ae confined to vetica pane tat ae noa to te obique efecto. Te efection point on te efecto ove away fo te vetica pane of te eiic ine to aintain a noa efection equied by eo offet and to find te iniu tavetie a condition of Feat pincipe. Obique efecto Seiic ine Vetica pane of aypat Figue. Pan view of an obique efecto owing eo-offet aypat tat ae confined to vetica pane. If te ange of obiquity wee eo, ten a efection woud occu on te efecto beow te vetica pane of te eiic ine. Conequenty, te efection point wi beave a a conventiona cattepoint and poduce te conventiona ape of a diffaction, defined in tie igation a T V T, wee T i te two-way tavetie, T te vetica two-way tavetie fo te efecto, and te oionta ditance fo te cattepoint to te coocated ouce and eceive. Wen te ange of obiquity i not eo, te efection point ove away fo te eiic ine, and te oionta ditance fo te efection point to te coocated ouce and eceive becoe co giving a new diffaction ape defined by, CREWES Reeac Repot Voue
Petack igation of obique efecto T co T V, te equation decibed by Fenc 975, and give te igation veocity defined by equation. Te igation of obique efecto i not obviou wen poceing data a te geoety of te ubuface i uuay not known. If it appea tat a potion of te igated ection i unde-igated i.e. te appeaance of un-coaped diffaction, ten a nube of igated ection ay be poduced, eac foed wit an inceae in te pecentage of te RMS veocity. Te poce of igating wit inceaed veocitie continue unti a atifactoy igated iage i obtained. Softwae a been deveoped in wic a a window potoe aound te aea of inteet ay be igated in ea tie by iding a veocity ba. Once te optiu igated iage a been obtained, te RMS and igated veocitie can be ued to etiate te agnitude of, te ange obiquity. Te actua ign of te ange i not deteined, but a nube of -D ine in te aea ay ten be ued to ap tee ange to etiate te ocation of te obique efecto. Ti etod a been ued uccefuy to ap canne and and faut. In one poceing cae, a difficut aea of a ection wit an appaent cyce kip, wa igated wit a veocity of *V, to poduce a cea iage of a faut, wit an ange of obiquity of appoiatey 7 degee. Figue ow two igation at % and % of te RMS veocitie, wee te RMS veocitie wee etiated uing a DMO poce. Te % iage appea acceptabe, oweve, wen eaining te % iage, a faut degee becoe evident in te cente of te ection. % % a b Figue. Two igated iage, a wit % V and b % V. CREWES Reeac Repot Voue
Bancoft and Uenbac Note tat odifying te igation veocitie to iage obique efection i ony vaid fo tat pat of te ection tat contain te obique efection. Ote pat of te ection wi be ove igated. A fina igated ection ay be foed wit a cut and pate etod tat incude te obique iage wit te oigina igation. Te above dicuion a been confined to tacked data tat tend to appoiate a eo-offet ection. Te eo-offet appoiation beak down wen te efecto ae dipping and o wen te efecto ae obique to te -D ine. Te foowing dicuion evauate te ape of a petack efection fo a oionta obique efecto, ten evauate te eut of tacking fo a pottack igation and fouate option fo petack igation. PRESTACK CONSIDERATIONS OF OBLIQUE REFLECTORS Conide a pan view of te obique efecting uface in Figue. Note te ocation A wee te eiic ine and te obique efecto inteect. Te CMP gate at A contain ouce-eceive ocation identified by te quae bace. A aypat wi be yetica in te vetica pane beow te eiic ine wit a efecto point tat i ocated beow A. Te oveout on te offet tace wi appea to ave a cattepoint at A, and ave te RMS veocitie aociated wit an otogona efecto. Refection fo a CMP gate ocated at B wi be conideaby diffeent fo te efection at A. At B, te eo offet efection a a igation veocity V ig defined in equation. Sa offet fo B wi ave a efection point tat i vey coe to te eo offet point and wi teefoe ave a geoety tat i iia to eo offet and wi ave a oveout veocity coe to V ig. Fo age offet, te efecto point wi ove aong te efecto towad A. Te ocation of te efection point wa difficut to define and wi be decibed in an appendi. Obique efecto Seiic ine A B Figue. Pan view of an obique efecto owing eo-offet aypat tat ae confined to vetica pane. CREWES Reeac Repot Voue
Petack igation of obique efecto Te petack tavetie uface of efection fo an obique efecto fo CMP ocation and offet defined by i defined by equation A of te appendi and i own in Figue 5a. Ti ape i iia to a Ceop pyaid Figue 5b fo a cattepoint at A o a noa efecto. Figue 5c contain a pot of bot uface fo copaion. Te petack tave tie uface defined by equation A i te uation uface equied fo a petack Kicoff igation. - - - - - - - - - a b - - - - - - c Figue 5. Petack tavetie uface of a an obique efecto, b Ceop pyaid, and c a cobined dipay. Note in Figue 5c te diffeence in te eo offet ypeboa defined by V fo te cattepoint, and V ig fo te obique efecto. It ay ao be aued tat te offet cuvatue toug te peak ae te ae and ave a oveout veocity V. Te oveout wit aiu dipaceent can be obeved at te eft ide of te figue. Note te Ceop pyaid oveout i quite fat nea eo offet and equie a oveout veocity tat i odified by te coeponding dip on te eo offet diffaction. Te oveout on te obique efecto uface ORS appea oe ypeboic wit a owe veocity. Many attept wee ade to appoiate te ORS incuding a Tayo eie epanion about. Soe of tee eut ae contained in te appendi, but te two ipet uface, uing tacking veocitie of V and V /co ae own beow in Figue 6. Figue 6a ow te tacking uface of conventiona poceing tat ue V fo conventiona NMO coection aong wit te defined ORS identified wit te aow. Ti uface i defined by equation 5, CREWES Reeac Repot Voue
Bancoft and Uenbac co T V V T. 5 Te fit two te on te igt-and ide of equation 5 tat invove T and define te eo-offet tie a a function of. Te tid te invoving i te conventiona oveout te. Note te cuvatue on te eft ide of Figue 6a i too ow. Equation 6 ue te obique ange to odify te oveout veocity and i own in Figue 7b. co co T T V V. 6 Now te veocity atc on te eft ide of Figue 6b i uc bette a pedicted eaie. Howeve, wen te back of Figue 6b i viewed in Figue 6d, te eo dipaceent veocity i in eo. - - - - - - - - - a b - - - - - - - - - - c d - Figue 6. Cobined pot of te obique efecto uface wit a conventiona oveout, and b wit oveout veocitie uing V ig defined in equation. Te ea view of a and b ae own in pat c and d epectivey. CREWES Reeac Repot Voue
Petack igation of obique efecto COMMENTS AND CONCLUSIONS. Petack igation of obique efecto can be accopied wit a uation uface defined by equation A and conveted to tie uing te RMS veocity.. Te aiu offet ued in te figue aue an offet fou tie te dept of te efecto. Ti offet i quite age fo conventiona poceing. Typica poceing ay iit te offet to be copaabe to te dept of te efecto. Te aiu igation dipaceent i ao quite age and coepond to igation dip up to 75 degee.. Stacking obique efection data wit conventiona oveout veocitie V wi poduce a diffaction tat i ony focued nea te ape. Te age dipaceent of te diffaction wi not be focued.. Ue of a tacking veocity V ig tat i odified fo te obiquity efecto wi poduce a diffaction tat i focued on te fank of te diffaction, but wi not focu at te ape. 5. Te above dicuion a aued tat te obique efecto i inea, poiby fo a faut. Howeve, if te efecto i not inea, uc a tat fo a buied ive canne, ten ony tat pat of te efecto coe to te eiic ine o a offet oud be ued. In ti cae, ue of te conventiona V veocitie ay poduce te bet ovea tack of te diffaction. 6. Moe cope oveout function, baed on a Tayo eie appoiation to te obique efecting uface ay be uefu if a offet of ae ued ee appendi. REFERENCES Fenc, W. S., 975, Copute igation of obique eiic efection pofie, Geopyic,. 96-98 CREWES Reeac Repot Voue
Bancoft and Uenbac APPENDIX Te at fo deiving te tave tie fo an obique efecto i deived in ti appendi. A nube of appoace wee taken baed on te pyica geoety of te pobe. Te ony etod tat wa uccefu wa uing Feat pincipe tat educe to te aypat wit iniu tave tie. Te ubuface ode ued aued a contant veocity of unity V aowing te tavetie to be etiated a ditance. Te dept of te cattepoint wa ocated at a dept, wit te aiu dipaceent and offet of. Te ouce ocation i defined by, y,, te eceive ocation by, y,, te efection point on te obique ine, y,, te ange of obiquity, and te eative ope of te efecto in te -y pane i tan-π/. Given a ouce ocation defined by te dipaceent, and te offet, te engt of aypat to any point, y on te obique efecto i defined by equation A. Obique efecto Seiic ine y -π/, y X Figue A. Pan view of an obique efecto owing offet aypat. and te engt of te eceive ay i [ y ] [ y ], A. A Te tota engt of te aypat cobine equation A and A to give te doube quae-oot equation A, CREWES Reeac Repot Voue
Petack igation of obique efecto CREWES Reeac Repot Voue [ ] [ ] y y. A Conveting te y paaete in A to uing te inea equation y we get [ ] [ ]. A Taking te deivative of equation A wit epect to and equating it to eo wi define a vaue fo tat give te iniu engt of te aypat. i.e. [ ] [ ] d d d d } {. A5 d d. A6 Soving ti equation fo wa difficut and MATHEMATICA wa ao unabe to obtain a oution wit te equation in ti fo. Te epeion wa put in te fo. A7 and eac ide quaed o tat it coud ten be eaanged to a quadatic function in. ] [. A8 Soving ti give two oution, ony one of wic i coect. An invaid oution i intoduced by quaing equation A7. Te coect oution fo i. A9 Te vaue fo y i obtained fo te inea equation y. Now te efection point, y, i defined, and te vaue of i ten ineted into equation A to give te engt of te aypat. Ti yied an epeion fo,,,. A new epeion fo i ten obtained by ubtituting and, wee i now undetood to be te offet ee te two definition of in Figue A. Ti eut in te foowing epeion: [ ] A
Bancoft and Uenbac A nube of attept wee initiated to ipify ti equation. One of te oe uccefu attept wa uing MATHEMATICA to epe,,, a a powe eie in. Sipifying te eut into a t ode appoiation gave te foowing eut: A Te nd ode appoiation fo te above equation A i potted in Figue A. - - - - - Figue A. Suface pot of nd ode Tayo eie epanion fo equation A. Te uface dipayed in Figue A i definitey infeio to te ipe uface uing conventiona NMO coection wit V and V ig. It oud be noted tat te initia MATHEMATICA Tayo eie epanion fied appoiatey page. Odd ode te woud eventuay ipify to eo, and te even ode te ipified to toe given above. CREWES Reeac Repot Voue