Deconvolution and correlation-based interferometric redatuming by wavefield inversion

Transcription

1 Delft Unversty of Tecnology Deconvoluton and correlaton-based nterferometrc redatumng by wavefeld nverson Barrera Paceco, Dego; Sclecer, J.; van der Neut, Joost; Wapenaar, Kees DOI.9/segeab.6 Publcaton date 7 Document Verson Accepted autor manuscrpt Publsed n Proceedngs of te 87t SEG annual meetng, expanded abstracts Ctaton (APA) Barrera Paceco, D., Sclecer, J., van der Neut, J., & Wapenaar, K. (7). Deconvoluton and correlaton-based nterferometrc redatumng by wavefeld nverson. In A. Ma Popovc, & S. Fomel (Eds.), Proceedngs of te 87t SEG annual meetng, expanded abstracts (pp. 6-). (SEG Tecncal Program Expanded Abstracts 7). SEG. ttps://do.org/.9/segeab.6 Important note To cte ts publcaton, please use te fnal publsed verson (f applcable). Please ceck te document verson above. Copyrgt Oter tan for strctly personal use, t s not permtted to download, forward or dstrbute te text or part of t, wtout te consent of te autor(s) and/or copyrgt older(s), unless te work s under an open content lcense suc as Creatve Commons. Takedown polcy Please contact us and provde detals f you beleve ts document breaces copyrgts. We wll remove access to te work mmedately and nvestgate your clam. Ts work s downloaded from Delft Unversty of Tecnology. For tecncal reasons te number of autors sown on ts cover page s lmted to a maxmum of.

2 Deconvoluton and correlaton-based nterferometrc redatumng by wavefeld nverson Dego F. Barrera, J. Sclecer, J. van der Neut and K. Wapenaar *DEP/UNICAMP & INCT-GP, IMECC/UNICAMP & INCT-GP and Delft Unversty of Tecnology SUMMARY Sesmc nterferometry s a metod to retreve Green s functons for sources (or recevers) were tere are only recevers (or sources, respectvely). Ts can be done by correlatonor deconvoluton-based metods. In ts work we present a new approac to reposton te sesmc array from te eart s surface to an arbtrary datum at dept usng te one-way recprocty teorems of convoluton and correlaton type. Te redatumng process s done n tree steps: retrevng te downward Green s functon for sources at te eart s surface and recevers at te datum, retrevng te correspondng upward Green s functon, and (c) retrevng te reflected upward wavefeld for sources and recevers at te datum. Input for steps and are te surface data and wavefelds smulated n a velocty model of te datum overburden. Step (c) uses te responses of steps and as nput data n te convoluton-based nterferometrc equaton. Te metod accounts for nomogenetes n te overburden medum, tus reducng antcausal events and artefacts as compared to a purely correlaton-based procedure. INTRODUCTION Sesmc nterferometry s a tecnque tat allows to retreve Green s functons for sources at postons were only recevers are avalable (or vce versa). Te classc redatumng procedure correlates surface sesmc data wt tose acqured at dept. Ts correlaton-based metod as been well studed n te lterature, e.g., by Wapenaar et al. (8), Scuster (9), Curts (9), Wapenaar et al. (), van der Neut (). Sesmc nterferometry by deconvoluton s an alternatve to te classcal correlaton-based sceme. Tere are many stuatons were te deconvolutonal form s more convenent tan te correlaton based metods. One of te man advantages of te deconvoluton-based procedure s ts nerent compensaton for te propertes of te source wavelet. Anoter mportant advantage s tat te underlyng teory does not requre te assumton of a lossless medum (Slob and Wapenaar, 7). In ts work, we combne deconvoluton-based nterferometry wt nverse wavefeld extrapolaton to derve an alternatve procedure for retrevng te total wavefeld at te datum. Inverse wavefeld extrapolaton s a concept used to descrbe te process of retrevng te focusng functons at an arbtrary datum by means of retropropagaton of te wavefeld recorded at te eart s surface (van der Neut et al., b). In our proposed procedure, we use a convenent form of nverse wavefeld extrapolaton to determne te down- and upward propagatng Green s functon at te datum for a pont source at te eart s surface. Ts down- and upgong Green s functon s ten used to retreve te complete wavefeld at te datum, free from nteractons wt te part of te medum above te redatumng level. Usng ts new approac, we reduce te nfluence of te overburden n te form of multples, spurous events of te Green s functons, and antcausal events. Te only requred nformaton for te proposed tecnque s a velocty model of te datum overburden. Ts model s used to smulate te vertcal dervatve of te transmtted wavefeld to te datum and te truncated overburden-scattered wavefeld at te surface and ts cossrespondng vertcal dervatve. RECIPROCITY THEOREMS OF CONVOLUTION AND CORRELATION TYPE We consder two states A and B n te Helmoltz equaton n order to calculate te recprocty teorem of te convoluton type. We assume tat nsde some volume V, bot states ave te same propertes,.e., r A (x) =r B (x) = and c A (x) = c B (x)=c(x). Snce te states dffer only n te source, correspondng wavefelds p A and p B must satsfy apple ˆpA apple ˆpB + w c (x) ˆpA = Ŝ A, () + w c (x) ˆpB = Ŝ B. () Here denotes te varable densty, ˆp te pressure feld, c(x) s te spatally varyng wave velocty and Ŝ(x,w) s a source term. Equatons () and () allow to deduce te recprocty teorem of convoluton type as ˆp B ˆp A ˆp A ˆp B ˆnds = V V ˆp A Ŝ B ˆp B Ŝ A dv, were V s te boundary of V and ds s te nfntesmal element on V. A completely analogous analyss can be carred out startng at te complex conjugate of equaton () togeter wt equaton (). Replacng te wavefeld ˆp A and te source term Ŝ A n te above dervaton by ter complex conjugates ˆp A and Ŝ A, were te superscrpt denotes te complex conjugate, results n ˆp B ˆp A ˆp A ˆp B ˆnds = V ˆp A Ŝ B ˆp B Ŝ A () dv. V Ts s te recprocty teorem of correlaton type, wc s ndependent of te actual sape of volume V. Consderng tat are no sources nsde volume V, te rgt-and sdes of equatons () and () vans. To analyze te closed-surface ntegral, we consder a cylndrcal sape, were te surface V s decomposed nto tree parts. We note tat at nfnte radus, te ntegral over te cylnder mantle does not contrbute (Wapenaar and Berkout, 989). Denotng te top part of te cylnder surface as tat V = (x,x,x ) R x = x and ()

3 Deconvoluton and correlaton-based nterferometrc redatumng te bottom surface as V = (x,x,x ) R x = x, we can rewrte expressons () and () as ( ˆpB ˆp A ˆp A ˆp B ) ˆn dx dx = V () ( ˆpB ˆp A ˆp A ˆp B ) ˆn dx dx. V and ( ˆpB ˆp A ˆp A ˆp B ) ˆn dx dx = V ( ˆpB ˆp A ˆp A ˆp B ) ˆn dx dx. V Accordng to Wapenaar and Berkout (989) te total wave feld can be decomposed n down- (+) and up-gong ( ) consttuents, as (6) ˆp(x,w)= ˆp + (x,w)+ ˆp (x,w). (7) Substtuton of decomposton (7) n expressons () and (6) wt te normal vectors at surfaces V and V as ˆn =(,, ) and ˆn =(,,), respectvely, we obtan te one-way recprocty teorems of convoluton and correlaton type, respectvely (Wapenaar and Fokkema, 6), tat are wrtten convenently as and V V V V ˆp B ˆp A + ˆp A ˆp B + dx dx = ˆp B ˆp A + ˆp A ˆp B + dx dx. ˆp B ˆp A ˆp A + ˆp B + dx dx = ˆp B + ˆp A + + ˆp B ˆp A dx dx. DOWN- AND UPWARD GREEN S FUNCTIONS Usng te recprocty teorems of convoluton and correlaton type, t s possble to retreve te up- and downward propagatng wavefelds at an arbtrary datum n dept. For ts purpose, we stll need an addtonal relatonsp, prevously derved by van der Neut et al. (a). To derve ts relatonsp n our notaton, we start agan at two states, A and B (ndcated by superscrpts A and B) n te frequency-space doman (see Fgure ). In state A, we consder a monopole pont source postoned mmedately above surface V. In ts stuaton, te vertcal dervatve of te downgong wavefeld at te surface can be expressed as ˆp A + = d(x x A )d(x x A ) (Wapenaar et al., ). Te valdty regon of ts expresson n state A s lmted by surfaces V and V. Between tese surfaces, te medum may be arbtrarly nomogeneous. Above V (8) (9) State A State B Fgure : Two wavefeld states n an nomogeneous overburden. State A s used to descrbe te transmtted wavefeld from te surface and t responses recorded at te datum and at te surface. State B s used to descrbe te total wavefeld takng nto account all events propagatng n te medum. and below V we consder omogeneous alfspaces wtout a free surface (Fgure ). In state B, we consder te same nomogeneous medum between surfaces V and V as n state A. Above V, we stll consder a omogeneous medum alfspace wtout a free surface, and below V we now consder a scatterng body. Te source n state B s also a pont source mmedately above surface V, suc tat te vertcal dervatve of te downgong wavefeld can be represented as ˆp B + = d(x x B)d(x x B ) (Wapenaar et al., ). In bot states A and B, we consder te wavefeld decomposton nto up- and downgong consttuents n analogy to equaton (7). An analyss of te pyscal stuaton n bot states allows for an nterpretaton of all propagatng events at every surface n Fgure, resultng n Table. Table : Analyss of te up- and downgong wavefelds at surfaces V and V n states A and B, respectvely. Surface Drecton Wavefeld n State A Wavefeld n State B V + pont source n x A pont source n x B V - Ĝ A (x,w;x A ) Ĝ B (x,w;x B ) V + Ĝ A + (x,w;x A ) Ĝ B + (x,w;x B ) V - Ĝ B (x,w;x B ) Substtuton of te wavefeld expressons from Table n te one-way recprocty teorem of convoluton type n equaton (8) leads to r(x A ) ĜB (x A,w;x B ) V r(x B ) ĜA (x B,w;x A )= ĜB (x,w;x B ) Ĝ A +(x,w;x A )d x, () were we ave also used tat te vertcal wavefeld dervatves at te pont sources are gven by delta functons as mentoned above. Equaton () s an expresson tat allows for least-squares nverson to retreve te up-gong Green s functon Ĝ B (x,w;x B ) n State B (.e., for te complete nomogeneous medum) f we know te terms r(x B ) ĜA (x B,w;x A ) and ĜA + (x,w;x A ) n State A (wc uses only a model of te nomogenetes n between te surfaces V and V ). In analogy to ts analyss, we can replace te wavefeld ex-

4 Deconvoluton and correlaton-based nterferometrc redatumng pressons of Table n te one-way recprocty teorem of correlaton type n equaton (9). Ts leads to ĜB (x,w;x B ) Ĝ A (x,w;x A )d x+ V r(x B ) ĜA + (x B,w;x A )= () ĜA + (x,w;x A ) Ĝ B +(x,w;x B )d x. V Equaton () allows us to calculate te vertcal dervatve of te downgong Green s functon Ĝ B + (x,w;x B ) at te datum, f we know te complex conjugate of te transmtted wavefeld ĜA + (x,w;x B ) from te eart s surface untl te datum n state A. We also need to know te complex conjugate of te downgong wavefeld r(x B ) ĜA + (xb,w;x A ) over te surface V and te vertcal dervatve of te complex conjugate of te truncated upgong wavefeld ĜA (x,w;x A ) n state A. REDATUMING Te above equatons allow for retrevng te down- and upgong consttuents of te Green s functons for sources at te surface and recevers at te datum. Terefore, we stll need an equaton to redatum te sources. For ts purpose, we note tat eac trace n te output gater Ĝ B (x,w;x B ) can be nterpreted as te stack of a convoluton gater, wc s obtaned by deconvoluton of eac trace n te reflecton response ˆR n (x,w;x ) at a fxed source pont x at te datum wt eac trace of te vertcal dervatve of te downgong consttuent Ĝ B + (x,w;x B ) for a fxed source poston x B at surface and recever postons x at te datum. Matematcally, te deconvoluton-based redatumng equaton can be expressed n te case of varable densty as r(x ) ĜB (x,w;x B )= r(x ) ˆR n (x,w;x ) Ĝ B +(x,w;x B )d x. V Fgure explans ts lnear nverson problem grapcally. = () Fgure : Sketc explanng te lnear nverson problem (). Once te one-way Green s functons responses at te datum ave been determned n te prevous steps, te reflected wavefeld at te datum can be calculated by nverson. Equaton () completes te procedure proposed n ts work. wt ts equaton, t s possble to retreve te total wavefeld at te datum ˆR n (x,w;x ) usng any nverson metod (n ts work we used stablzed least-squares nverson), f we know te up- and downward Green s functon consttuents r(x ) ĜB (x,w,x B ) and ĜB + (x,w;x B ). RESULTS In ts numercal example, we demonstrate and nterpret te man result of ts work. To do so, we used a orzontallayered velocty model wt a wdt of 8 km and a dept of km contanng veloctes between.8 km/s and. km/s. Te datum s located at m dept. Te acquston models as sources spaced at m and te same number of recevers for eac sot located at te eart s surface and n te datum. Usng tese data and te true velocty model of te datum overburden, we determne te down- and upward wavefeld consttuents for recevers at dept, and ten te redatumed wavefeld wt sources and recevers at te datum. For te redatumng, we needed to smulate te followng wavefelds n te overburden model: te transmtted wavefeld from te eart s surface at te datum n m n dept and ts correspondng vertcal dervatve, and te truncated reflected wavefeld wt source and recevers at te eart s surface and ts correspondng vertcal dervatve. For comparson, we smulated te full wavefeld (Fgure a) wt sources at te surface and recevers at te datum. Te vsble events are labeled wt numbers n order to dentfy and nterpret tem. Fgure b and c sow te ray pats of te down- and upgong events, respectvely, and te correspondng labels. To facltate te nterpretaton, we use green arrows for downgong events and red arrows for upgong events. Ts also elps us to compare te respectve events wt te nverted down- and upgong wavefeld consttuents sown below Dstance (Km) Fgure : Full syntetc sesmc wavefeld labeled wt te ray pats of ts down- and (c) upgong consttuents at te datum n m dept. An mportant aspect of all steps of our numercal nverson,.e., te retreval of te down- and upward Green s functons, and te full reflected wavefeld for sources and recevers at datum, s tat we actually nvert a pont spread functon (PSF). Usng a damped least-squares sceme to retreve te nverse of te PSF, we tested dfferent values of te regularzaton parameter e n te nverse process. We used for e dfferent percentage values wt respect to te maxmum value of te PSF. Fgures,, and 6 sow te nverson results for te percentages:.% and.%. Fgure sows te Green s functon Ĝ B + (x,w;x B ) at te datum n m dept, nverted usng equaton (), for te ndcated values of te regularzaton parameters. In bot sectons, te knematc propertes and te relatve ampltudes ncely matc tose of te downward propagatng events n Fgure a. (c).... Dept (Km).... Dept (Km)

5 Deconvoluton and correlaton-based nterferometrc redatumng Fgure : Downward responses obtaned by PSF nverson accordng to equaton () wt dfferent values of e:.% and.%. In a correspondng way, we used te one-way recprocty teorem of convoluton type, equaton (), to retreve te upgong Green s functons Ĝ B (x,w;x B ) by means of stablzed leastsquares nverson (Fgure ). We observe tat, as desred, only te upgong Green s functons consttuents were retreved. A comparson to Fgure a reveals correct postonng. Also te dynamc propertes of te nverted events correspond well to tose n te modelled secton Fgure 6: Redatumng responses wt dfferent values of e n te PSF nverson of equaton ():.% and.%. te exact trace, we notce tat wt ncreasng e, te nstabltes decrease but te wavelet dstorton ncreases. Normalzed ampltude exact model =% =.% =.% =.% Fgure : Upward responses obtaned by PSF nverson accordng to equaton () wt dfferent values of e:.% and.%. Fnally, Fgure 6 sows te source-recever redatumed wavefelds at te datum, recovered by nvertng equaton (). We observe tat te redatumed wavefelds recover te desred prmary reflecton events wt slgtly dfferent qualty. Wle te larger values of e do a better job of suppressng artfacts, tey also lead to a stronger dampng of te desred wavefeld at larger offsets and a certan dstorton of te wavelet. Te qualty of te nverson result as a functon of te regularzaton parameter can be better assessed n Fgure 7, wc compares te central trace of eac redatumng response for four dfferent values of e to te modeled trace at te same locaton. Wle all events are knematcally ncely matced wt.. Fgure 7: Central traces of te redatumng responses n Fgure 6 compared wt te central trace of te exact model. CONCLUSIONS In ts work, we ave derved a new nterferometrc procedure to calculate te down- and upward Green s functons wt sources at te eart s surface and recevers at a datum n dept. It makes use of nverse wavefeld extrapolaton. Wt ts metodology t s possble to retreve only te down- and upward propagatng consttuents at an arbtrary focusng surface wtout antcausal events and wtout artfacts f te model of te overburden s known. Combnng te down- and upward Green s functon retreved by wavefeld nverson wt te conventonal verson of deconvoluton-based nterferometrc redatumng, we were able to retreve te reflected wavefeld at te datum, as demonstrated n a smple syntetc-data example. As a major advantage, tere s no nfluence of antcausal events and no causal nteractons wt te overburden n te fnal responses, wc were removed wt te nverse wavefeld extrapolaton n te frst and second step of te redatumng process. ACKNOWLEDGMENTS Ts work was kndly supported by te Brazlan researc agences CAPES and CNPq as well as Petrobras and te sponsors of te Wave Inverson Tecnology (WIT) Consortum.