Th P5 13 Elastic Envelope Inversion SUMMARY. J.R. Luo* (Xi'an Jiaotong University), R.S. Wu (UC Santa Cruz) & J.H. Gao (Xi'an Jiaotong University)

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1 -4 June 5 IFEMA Madrd h P5 3 Elastc Envelope Inverson J.R. Luo* (X'an Jaotong Unversty), R.S. Wu (UC Santa Cruz) & J.H. Gao (X'an Jaotong Unversty) SUMMARY We developed the elastc envelope nverson method. he envelope of the wavefelds carres ultra low frequency nformaton whch can be used to recover the large-scale component of the model, and the ntal model dependence of waveform nverson can be reduced. We derved the msft functon and the correspondng gradent operator for elastc envelope nverson. Numercal tests usng synthetc data for the Marmous II model proved the valdty and feasblty of the proposed approach. he nverted p-wave velocty and s-wave velocty from the combned (EI+WI) envelope nverson plus waveform nverson ndcated that t can delver much mproved results compared wth regular elastc full waveform nverson. 77 th EAGE Conference & Exhbton 5 IFEMA Madrd, Span, -4 June 5

2 -4 June 5 IFEMA Madrd Introducton Elastc full waveform nverson (FWI) has been nvestgated by many authors ( Mora, 987; Crase et al., 99; Sears et al., ). However, as n acoustc stuaton, good ntal models are always needed n elastc full waveform nverson n order to get good nverson results. o generate low-frequency sgnal below 5Hz s very expensve, so effort has been focused to the recovery of long wavelength background wthout very low frequences. ravel tme nverson and mgraton velocty analyss are two tradtonal methods n ths area. In recent years, Shn and Cha (8) has developed Laplace doman full waveform nverson whch can gve a smooth background model from an naccurate ntal model. Zhou et al. (997) combned traveltme and waveform nverson. Bond and Almomn (4) combned full waveform nverson wth wave equaton mgraton velocty analyss. Lu et at. () proposed the normalzed ntegraton method. Wu et al. (3, 4) developed envelope nverson (EI) method whch can gve a very smooth background structure. Most of the approaches above are developed n the acoustc stuaton. However, the real earth s elastc. In ths paper, we extend the envelope nverson method to elastc stuaton. We frst ntroduce the elastc envelope nverson method and derve the gradent for the elastc envelope nverson, then we use the Marmous II model to show the valdty of ths method. race envelope and Hlbert transform Frst we ntroduce what s the envelope of a sgnal. We extract envelope by takng the ampltude after the analytcal sgnal transform usng Hlbert transform. A sgnal havng no negatve-frequency components s called and analytc sgnal whch can be constructed from a real sgnal f () t and tlbert transform H{ f( t )}, () he envelope of f () t can then be derved by, et () f () t H{ f()}, t () Envelope nverson for the elastc stuaton he -D sotropc elastc wave equaton can be expressed as, j u f, j t x j j j M u ( u u ). kk,, j j, Where u s the th component of the dsplacement vector and j s the j component of the j stress tensor. s the densty; and are Lamé coeffcents. f s the body force and M s the tracton. he p-wave velocty and s-wave velocty are gven by, v p, v s. (4) In the elastc envelope nverson, we defne the msft functon as follow, ( m ) [ e ( t)] [ e ( t)] dt, syn obs 4 (5) where m s the model parameter, e and e are the envelope of the th component of the syn obs synthetc wavefeld and the observed wavefeld respectvely. Usng Hlbert transform, the above equaton can be wrtten as, (m) 4 [s (t)] [ (t)] [u (t)] [u H (t)] (3) dt= E 4 dt (7) 77 th EAGE Conference & Exhbton 5 IFEMA Madrd, Span, -4 June 5

3 -4 June 5 IFEMA Madrd where s and u are the th component of the synthetc wavefeld and the observed wavefeld respectvely, s and u are the correspondng Hlbert transforms. E s the nstant H H envelope data resdual. Here we appled squared envelope, because t has better performance n long-wavelength background recovery (Wu et al. (4); Luo et al. (4)). Gradent calculaton for elastc envelope nverson We frst assume the densty s a constant. If we consder and as the model parameters. Calculatng the dervatve of the msft functon wth and, we get, = E (t) H{E s E (t) H{E s s (t) (t)} (t)} dt J r env, s (t) dt J r env. where J s v p, r env E s (t) H{E (t)}. (9) J s v s from equaton (8) we see that the gradent for the elastc envelope nverson has exactly the same form as that n the conventonal elastc full waveform nverson (Mora, 987), except that the resdual vector r env s dfferent. he gradent for the elastc envelope nverson can be obtaned usng the backpropagaton method as follows, (8) () where means backpropagaton wavefelds whch s obtaned usng the envelope resdual vector r env as the source wavelet. Usng equaton (4) we can get the gradent wth respect to the velocty as follows, () because r env s related to the effectve envelope resdual not the sesmc data resdual, the low frequency nformaton n the envelope can be lower than the lowest frequency n the source wavelet. hs low frequency nformaton can be used to construct the large-scale background structure. he low-frequency nformaton coded n the envelope s not drectly from the source wavelet. Examples We show the valdty of ths method usng the Marmous II model (Fgure ). he source wavelet we used s the Rcker wavelet wth the domnant frequency of Hz, and the low frequences below 5 Hz s cut off from the source wavelet, so there wll be no low frequency nformaton n the data. Fgure shows the ntal model for nverson. We used -D lnear ntal model. Startng from these ntal models, we nvert the p-wave velocty and s-wave velocty usng envelope nverson. Fgure 3 shows the envelope nverson results. 77 th EAGE Conference & Exhbton 5 IFEMA Madrd, Span, -4 June 5

4 -4 June 5 IFEMA Madrd From the results, we can see the large-scale component of the model. Usng the results n Fgure 3 as the new ntal model, we perform elastc waveform nverson. Fgure 4 shows the fnal nverson results (EI+FWI results). As comparson, we also performed the conventonal elastc full waveform nverson startng drectly from the lnear ntal model. Fgure 5 shows the nverson results. From the comparson we can see that the combned EI+FWI can delver much better results than the conventonal FWI method. Fgure he Marmous II model. vp; vs. Dstance (km) Dstance (km) Fgure Lnear ntal model. vp; vs. Dstance (km) Dstance (km) Fgure 3 Envelope nverson results usng the low-cut source. vp; vs. Dstance (km) Dstance (km) Fgure 4 EI+FWI results usng the low-cut source wavelet. vp; vs. 77 th EAGE Conference & Exhbton 5 IFEMA Madrd, Span, -4 June 5

5 -4 June 5 IFEMA Madrd Dstance (km) Dstance (km) Fgure 5 he conventonal FWI result usng the low-cut source. vp; vs. Conclusons We extended the envelope nverson method to the elastc stuaton. Envelope carres ultra low frequency nformaton whch can be sued to estmate the large scale component of the model structure, so that the ntal model dependence of waveform nverson can be reduced. hs s demonstrated by the Marmous II model tests, where we nverted the p-wave velocty and s-wave velocty and used -D lnear gradent startng model. he elastc envelope nverson s ndependent of the frequency band of the source wavelet. he model parameter can be nverted whle the low frequency s removed from the source wavelet. Acknowledgements We thank Xao-b Xe for helpful dscussons. hs research s supported by the WOPI Research Consortum at the Unversty of Calforna, Santa Cruz, USA. References Mora, P. [987] Nonlnear two-dmentonal elastc nverson of multoffset sesmc data. Geophyscs, 5, -8. Crase, E., C. Wdeman, M. Noble, and A. arantola, [99] Nonlnear elastc waveform nverson of land sesmc reflecton data. Journal of geophyscal research, 97, Sears,.J., P.J. Barton, and S.C. Sngh, [] Elastc full waveform nverson of multcomponent ocean-bottom cable sesmc data: Applcaton to Alba Feld, U. K. North Sea. Geophyscs, 75, R9- R9. Shn, C, and Y.H. Cha, [8] Waveform nverson n the Laplace doman. Geophys. J. Int. 73, Zhou, C., G.. Schuster, S. Hassanzadeh, and J.M. Harrs, [997] Elastc wave equaton traveltme and waveform nverson of crosswell data. Geophyscs, 6, Bond, B. and A. Almomn, [4] Effcent and robust waveform-nverson workflow: omographc FWI followed by FWI. SEG echncal Program Expanded Abstracts 4, Lu, J., H. Chaurs, and H. Calandra, [] he normalzed ntegraton method an alternatve to full waveform nverson? Near surface 7 th European Meetng of Envronmental and Engneerng Geophyscs, B7. Wu, R.S., J. Luo, and B. Wu, [3] Ultra-low-frequency nformaton n sesmc data and envelope nverson. 83 rd Annual Internatonal Meetng, SEG, Expanded Abstracts, [4] Sesmc envelope nverson and modulaton sgnal model. Geophyscs, 79, WA3- WA4. 77 th EAGE Conference & Exhbton 5 IFEMA Madrd, Span, -4 June 5