Full Waveform Inversion Using an Energy-Based Objective Function with Efficient Calculation of the Gradient

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1 Full Wavform Invrsion Using an Enrgy-Basd Objctiv Function with Efficint Calculation of th Gradint Itm yp Confrnc Papr Authors Choi, Yun Sok; Alkhalifah, ariq Ali Citation Choi Y, Alkhalifah (217) Full Wavform Invrsion Using an Enrgy-Basd Objctiv Function with Efficint Calculation of th Gradint. 79th EAGE Confrnc and Exhibition 217. Availabl: Eprint vrsion Post-print DOI / Publishr EAGE Publications BV Journal 79th EAGE Confrnc and Exhibition 217 Rights Archivd with thanks to 79th EAGE Confrnc and Exhibition 217 Download dat 23/11/218 7:5:39 Link to Itm

2 W P1 16 Full Wavform Invrsion Using An Enrgy-Basd Objctiv Function with Efficint Calculation of th Gradint Y. Choi* (King Abdullah Univrsity of Scinc & chnology),. Alkhalifah (King Abdullah Univrsity of Scinc & chnology) Summary Full wavform invrsion (FWI) using an nrgy-basd objctiv function has th potntial to provid long wavlngth modl information vn without low frquncy in th data. Howvr, without th back-propagation mthod (adjoint-stat mthod), its implmntation is impractical for th modl siz of gnral sismic survy. W driv th gradint of th nrgy-basd objctiv function using th back-propagation mthod to mak its FWI fasibl. W also rais th nrgy signal to th powr of a small positiv numbr to proprly handl th nrgy signal imbalanc as a function of offst. Exampls dmonstrat that th proposd FWI algorithm provids a convrgnt long wavlngth structur modl vn without low-frquncy information, which can b usd as a good starting modl for th subsqunt convntional FWI. 79 th EAGE Confrnc & Exhibition 217 Paris, Franc, Jun 217

3 Introduction Full wavform invrsion (FWI) has bn widly studid and dvlopd to dlinat accurat dscription of th subsurfac contnt. Nvrthlss, FWI still suffrs from th local minima problm mainly rsulting from th cycl-skipping btwn th obsrvd and modlld data. h availabl frquncy low nough in th data or a starting modl clos nough to th tru on can hlp FWI convrg to an accurat solution. In gnral, howvr, th frquncy band of sismic data is usually not low nough and obtaining a good starting modl is not a trivial task as it rquirs using laborat approachs such as travl-tim tomography, migration vlocity analysis, and tc. Rcntly, many studis of FWI hav bn dvotd to solving th cycl-skipping problm in th cas of a lack of low frquncy information. Choi and Alkhalifah (215) unwrappd th phass at rlativly high frquncy in th frquncy-domain and invrtd th unwrappd phas to avoid th cycl-skipping problm. Wu t al. (214) suggstd FWI basd on th nvlop objctiv function, whr th nvlop of sismic trac yilds artificial low frquncy componnts which hlp FWI construct a long wavlngth modl. On th othr hand, Jon t al. (214) proposd th nrgy objctiv function for FWI to construct a macro-vlocity modl, which can b usd as a good starting modl for a subsqunt convntional FWI. h nrgy (or nrgy signal) of a sismic trac is dfind as an intgration of squard wavfild with a shifting tim window. hy showd that th nrgy objctiv function has a convx shap fr of local minima and thir FWI approach gnrats a convrgnt long wavlngth vlocity modl. Howvr, to calculat th gradint of th nrgy objctiv function, thy xplicitly calculat th partial drivativ of th nrgy signal using a finit-diffrnc mthod, which maks thir FWI approach impractical spcially for th modl siz of a gnral xploration sismic survy. In this abstract, w driv th gradint of th nrgy objctiv function using th back-propagation mthod (adjoint-stat mthod) to mak th proposd FWI practical for th modl siz of a gnral sismic survy. W also provid thortical insights into th fasibility of th nrgy objctiv function. On th othr hand, th nrgy signal has singularly strong amplitud (nrgy) nar zrooffst bcaus of th intgration of th squard wavfild, thrfor, th convntional last-squar misfit function dos not proprly handl th nrgy signal. Jon t al. (214) took th logarithm of th nrgy signal to addrss th faturs of th nrgy signal. W propos raising th nrgy signal to th powr of a vry small positiv numbr (.g., 1-5 ) to proprly handl it in FWI. In th xampls, w apply th proposd FWI with th back-propagation mthod to th low-cut filtrd synthtic data gnratd in a salt-dom modl. Exampls dmonstrat th fasibility and potntial of FWI using th nrgy objctiv function spcially in cas of a lack of low frquncy information. Enrgy signal h nrgy of a sismic trac is originally dfind as () 2 whr u( ) is a sismic trac with tim variabl and stimat th nrgy signal (Jon t al. 214) as u d, (1) 2 t is th rcording tim. On th othr hand, w u ( t) u( ) d. (2) h nrgy signal in quation 2 is an intgration of th squard sismic trac with shifting tim window. If th shifting tim window is rplacd with th Havisid stp function, th nrgy signal is xprssd as a convolution of th Havisid stp function and squard trac: 2 u ( ) ( ) ( ) t H t u d, (3) whr u( ) is a causal signal. h Havisid stp function in quation 3 plays th rol of a filtr, which amplifis vry low frquncy componnts. On th othr hand, th squard wavfild in 79 th EAGE Confrnc & Exhibition 217 Paris, Franc, Jun 217

4 quation 3 is xprssd as an auto-convolution in th frquncy-domain, which yilds artificial low frquncy componnts. hrfor, artificial vry low frquncy componnts dominat in th nrgy signal in quations 2 and 3. Figur 1 shows th original low-cut filtrd trac, squard trac, Havisid stp function, nrgy signal, and thir corrsponding frquncy-domain spctra. Although th original trac is low-cut filtrd blow 5 Hz (Figurs 1a and 1), th squard trac includs artificial low frquncy componnts (Figurs 1b and 1f). h spctrum of Havisid stp function xponntially dcrass as frquncy incrass (Figur 1g), thus th nrgy signal includs mainly artificial vry low-frquncy componnts (Figurs 1d and 1h). Figur 2 shows th low-cut filtrd sismogram and its corrsponding nrgy signal sismogram with diffrnt clipping rangs. h nrgy signal has singularly vry strong amplitud nar zro offst (Figurs 2b and 2c), thus th last squar misfit function cannot proprly handl th nrgy signal sismogram through FWI. (a) (b) (c) (d) () (f) (g) (h) Figur 1 h first row shows th tim-domain tracs and th scond row is thir corrsponding frquncy-domain spctra: th (a, ) original low-cut filtrd trac, (b, f) its squard trac, (c, g) Havisid stp function, and (d, h) nrgy signal. (a) (b) (c) Figur 2 h (a) filtrd sismogram and (b, c) corrsponding nrgy signal sismogram with diffrnt clipping rangs. Objctiv functions using nrgy signal and gradint calculation As an altrnativ to th convntional misfit function, Jon t al. (214) took th logarithm of th nrgy signal to addrss th faturs of th nrgy signal sismogram. In addition to th logarithm, w suggst raising th nrgy signal to th powr of vry small positiv numbr. h objctiv functions using logarithm and powr for th nrgy signal ar writtn as 79 th EAGE Confrnc & Exhibition 217 Paris, Franc, Jun 217

5 2 1 u () t 2 E log ln dt 1 sr, 2 and Epowr u ( ) ( ) d ( t) t d t dt sr, 2, (4) rspctivly, whr and ar th nrgy signals of th modlld and obsrvd wavfild, rspctivly, and is a powr of nrgy signal. h logarithm and th powr with a vry small numbr in quation 4 can mitigat th ffct of th singularly vry strong amplitud nar zro offst in Figur 2b. W st th valu of as 1-5 in th xampls. h gradints of th objctiv functions in quation 4 can b obtaind by taking th drivativ of th objctiv functions with rspct to th kth modl paramtr: E log u( t) 1 u ( ) ln t Epowr u () t 1 dt p and u ( ) ( ) ( ) k sr, pk u ( t) d ( t) t u t d t dt p. (5) k sr, pk Jon t al. (214) xplicitly calculatd th partial drivativ of nrgy signal ( ) using th finit-diffrnc mthod, which is vry xpnsiv to calculat and maks FWI impractical for th modl siz of gnral sismic survy. In this abstract, w driv th gradint of th abov objctiv functions using th back-propagation mthod in a similar way to rvrs-tim migration. h partial drivativ of nrgy signal in quation 5 is xprssd in a vctor form from quation 2 as u () t d () t Τ 1 u 2 n j u j u j 2 u j u j u j k j k j k j k u () t u p p p p, (6) whr th subscript j stands for th tim sampl lmnt and n is th numbr of total tim sampls of vctor. By xpanding ach lmnt in quation 6, th partial drivativ of nrgy signal can b rwrittn as u1 u1 u1 u1 Τ Τ u 2 u2 u u u 2 2 A, whr A u3 u 3, pk pk u n (7) whr u is a vctor form of th original modlld wavfild. Finally, th gradints in quation 5 ar xprssd in vctor/matrix form as E log u Ar log, whr pk sr, p 1 u() t r log t, u ( ) ( ) t d t (8) E powr u Ar powr, whr pk sr, p 1 powr r t u t u t d t. (9) From quations 8 and 9, w can fficintly calculat th gradints using th back-propagation mthod (or adjoint-stat mthod), which is almost th sam as th convntional FWI (or RM) xcpt for adjoint sourcs givn by th nw rsidual sismograms and Ar. u Ar log powr p k Exampls W gnrat synthtic data from th SEG/EAGE salt modl shown in Figur 3a and filtr out th data blow 3 Hz. h maximum frquncy of data is 8 Hz. W plac 37 shots at a dpth of 2 m and 77 rcivrs at th sam dpth. On of sismograms is shown in Figur 2a. h starting vlocity modl is linarly incrasing with dpth (Figur 3b). h logarithmic objctiv function provids similar rsults to that of th powr objctiv function, thus w display only rsults of th powr objctiv function to sav spac in this abstract. h proposd FWI algorithms gnrat long wavlngth structurs in th invrtd modl and th gradint (Figurs 4a and 4b) vn though th data lacks low frquncy information. h subsqunt convntional FWI starting from th prvious invrtd modl in Figur 4a gnrats a quit good convrgnt vlocity modl dlinating wll both th insid and th shap of salt (Figur 4c), whras th on starting from th initial modl in Figur 3b provids poor rsults (Figur 4d). h xampls dmonstrat that th proposd FWI algorithm gnrats a convrgnt long wavlngth vlocity modl, which can b usd as a good starting modl for th subsqunt FWI. 79 th EAGE Confrnc & Exhibition 217 Paris, Franc, Jun 217

6 (a) Figur 3 h (a) SEG/EAGE salt modl and (b) starting modl for FWI. (b) (a) (b) (c) (d) Figur 4 h (a) invrtd modl obtaind from FWI using th nrgy-basd objctiv function, (b) gradint at th first itration, and subsqunt convntional FWI rsults starting from th (c) prvious invrtd modl and (d) initial modl in Figur 3b. Conclusions FWI using th nrgy objctiv function has a potntial to provid long wavlngth structur information vn without low frquncy in th data, sinc th nrgy signal mainly includs artificial low frquncy componnts. Howvr, without th back-propagation mthod (or adjoint-stat mthod), FWI using th nrgy objctiv function is impractical for th modl siz of a gnral sismic survy. W driv th gradint of objctiv function using th back-propagation mthod to mak th proposd FWI practical. h nrgy signal has singularly vry strong amplituds nar zro offst, in which th convntional misfit function dos not work. W propos raising th nrgy signal to th powr of a small positiv numbr in addition to taking th logarithm to proprly dal with th nrgy signal imbalanc. Numrical xampls show that th proposd FWI algorithm with th back-propagation mthod provids a convrgnt long wavlngth structur modl vn without low frquncy information, which can b usd as a good starting modl for th subsqunt convntional FWI. Acknowldgmnts W ar gratful to King Abdullah Univrsity of Scinc and chnology for financial supports. For computr tim, this rsarch usd th rsourcs of th Supr Computing Laboratory at KAUS in huwal, Saudi Arabia. Rfrncs Choi, Y. and Alkhalifah,. [215] Unwrappd phas invrsion with an xponntial damping. Gophysics, 8, R251-R264. Jon, S., Ha, W., Kwon, J. and Shin, C. [214] Full wavform invrsion using th nrgy objctiv function. 76th EAGE Confrnc & Exhibition, Expandd Abstracts, Amstrdam, u E16 2. Wu, R.S., Luo, J. and Wu, B. [214] Sismic nvlop invrsion and modulation signal modl. Gophysics, 79, WA13-WA th EAGE Confrnc & Exhibition 217 Paris, Franc, Jun 217