Main Menu. characterization using surface reflection seismic data and sonic logs. Summary

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1 Stochastc Sesmc Inverson usng both Waveform and Traveltme Data and Its Applcaton to Tme-lapse Montorng Youl Quan* and Jerry M. Harrs, Geophyscs Department, Stanford Unversty Summary A stochastc approach to sesmc nverson usng the ensemble Kalman flter (EnKF) s proposed. Sesmc depth and tme mage data are used as the nput for EnKF stochastc sesmc nverson. The sonc log s used to estmate source wavelet and create ntal models for the nverson, whch provdes an effcent ntegraton of sonc log data and sesmc data. We use both travel tme and waveform data for the nverson and obtan the absolute sesmc velocty nstead of the relatve mpedance. EnKF can contnuously update the model usng tme-lapse data. A synthetc example s used to demonstrate the possble applcaton to sesmc montorng. Introducton The purpose of sesmc nverson s to recover the subsurface elastc propertes (e.g., acoustc mpedance and velocty) from sesmc data. For example, Oldenburg et al. (983) dscussed the determnstc mpedance nverson; Hass and Dubrule (994) ntroduced a stochastc mpedance nverson; Cao et al. (989) presented an nverson method to estmate background velocty and mpedance smultaneously. Francs (25) and Sancevero et al. (25) compared determnstc and stochastc mpedance nverson usng examples. In general, stochastc sesmc nverson has hgher vertcal resoluton than determnstc nverson. The stochastc sesmc nverson proposed n ths study s an mplementaton of ensemble Kalman flter (EnKF). A complete ntroducton to EnKF can be found n Evensen (27). EnFK can perform lnear and non-lnear stochastc nverson. It can also ntegrate dfferent types of data for the nverson. Takng advantage of these features, we combne waveform data and travetme data for the sesmc nverson. The waveform nverson n our study s a nonlnear nverson. The use of travetme data mproves the estmaton of the absolute sesmc velocty. Ths study s motvated by sesmc montorng for geologcal CO 2 sequestraton. CO 2 sequestraton provdes a possble soluton for reducng the green gas emsson to the atmosphere. For safety and operatonal reasons, we need to montor the contanment of the CO 2 storage n the subsurface. The montorng s a dynamc process. EnKF s naturally sutable for dynamc nverson. We wll use the CO 2 montorng as an example to demonstrate our method, though t can also be used for general statonary reservor characterzaton usng surface reflecton sesmc data and sonc logs. Method Let us consder the sesmc sgnal d recorded at surface that s a functon of subsurface model parameters m. In ths sesmc nverson problem, d s normal ncdence reflecton data obtaned after all necessary sgnal processng, and m s the -D sesmc velocty drectly below the recever. Data d and model m are related through an observaton matrx G for a lnear case: d = Gm, () or a general observaton functon g ncludng non-lnear cases: d = g(m). (2) We want to estmate model m from observed data d by a stochastc nverson procedure mplemented wth the ensemble Kalman flter. We here follow the dervaton n Evensen (23) and apply the general EnFK theory to our problem,.e., jont sesmc nverson usng both waveform and traveltme data. In our case, m s an n-dmensonal model vector composed wth dscretzed -D velocty below the recever; d s an m- dmensonal data vector havng m waveform data ponts and m 2 traveltme data ponts, m=m +m 2. A proper scalng factor s needed to normalze the two types of data. Assume that model m has Gaussan probablty dstrbuton wth mean m and covarance C, and data d also has Gaussan probablty dstrbuton wth mean d and covarance R. We create a model ensemble M = [m,, m N ] (3) that has the mean m and the covarance C, and a data ensemble D = [d,, d N ] (4) that has the mean d and the covarance R. Here, m and d are ensemble members; N s the ensemble sze that should be large enough n order to provde a good approxmaton to the probablty dstrbuton for the model and the data. The EnKF gves the statstcal soluton for a lnear problem shown n equaton as Mˆ = M + K( D GM), (5) T T K = CG ( GCG + R) (6) 95 Downloaded Dec 2 to Redstrbuton subject to SEG lcense or copyrght; see Terms of Use at 95

2 s called Kalman gan. The EnKF soluton for a non-lnear equaton 2 wll be dscussed n next secton. Mˆ s an n N matrx; each column represents a realzaton from the posteror probablty dstrbuton. The average of all columns (or realzatons) forms the soluton for the model estmaton. In a tme-lapse nverson problem, new data are comng contnuously, and the model can be contnuously updated by repeatng the procedure above (equatons 3-5) usng the estmated model obtaned n current step as the ntal model for next tme step. Implementaton We start wth an ntal model m created from pror knowledge, e.g., sonc logs and ther nterpolatons, or just a constant model n the worst case. Then we construct the model ensemble n equaton 3 as m m + = ε ε s an n-dmensonal random vector from Gaussan dstrbuton. Convoluton s used as the observaton functon for waveform data modelng. We calculate reflecton coeffcents from -D velocty and convolve the reflecton profle wth a wavelet extracted from the normal ncdence sesmogram and a sonc log. The observaton functon g n ths study s not a lnear functon, and we cannot drectly use equaton 6, because t s dffculty to fnd an observaton matrx G for ths convoluton modelng operaton. We have to use an observaton matrx-free mplementaton (Mandel, 26) for ths nverson. The model covarance C n equaton 6 can be approxmated by the ensemble covarance as C = AA T /( N ), (7) N A = M E( M) = M m. N = Then model update (equaton 5) can be done wth ˆ T M = M + A( GA) P [ D g( M)], (8) N T P = GA( GA) + R, (9) N and the th column of matrx GA can be obtaned from N [ GA ] = g( m) g( m j ). () N j= For the data ensemble D, we perturb the observed data d and have d = d + γ. Here, γ s an m-dmensonal random vector from Gaussan dstrbuton. Then the data covarance R requred n equaton 9 can be obtaned from the ensemble covarance T R = γγ /( N ). We next apply the procedure above to a synthetc example. An Example of Tme-lapse Sesmc Montorng We have utlzed a smulaton study for sesmc montorng on CO 2 sequestraton n coalbeds. Ths study s part of the Global Clmate and Energy Project (GCEP) at Stanford Unversty. Tme-lapse Models We frst buld a 2-D reservor flow model accordng to the geology and flow parameters of unmneable coalbeds n the Powder Rver Basn. The prmary goal of ths flow smulaton s to create a seres of relatvely realstc CO 2 storage models for montorng tests. For a perod of years, 75 tme-lapse models are generated usng the flow smulator GEM. Varous cases, e.g., CO 2 storage wth or wthout leakage, are smulated. In the coalbed, matrx porosty = 5%, cleat porosty = -5%, matrx permeablty =.5md and cleat permeablty = md. We then convert the flow smulaton results to tme-lapse P-wave velocty models wth the help of a rock physcs model. Fgure shows four velocty models at tme =, 3 months, year, and 3 years. It can be seen that the P-wave velocty decreases due to the CO 2 saturaton. The method dscussed n prevous sectons s appled to these models to test f we can track the CO 2 front usng EnKF A C Dstance(m) Dstance(m) Fgure : Four tme-lapse P-wave velocty modes created based on CO 2 flow smulaton n the coalbeds. A: tme=; B: tme=3 months; C: tme= year; D: tme=3 years. Sesmc Data A fnte dfference method s used to calculate the relatvely realstc sesmc data (served as observed data) for all 4 tme-lapse models. 4 shot gathers are calculated for each B D v(m/s) 96 Downloaded Dec 2 to Redstrbuton subject to SEG lcense or copyrght; see Terms of Use at 96

3 model. The source peak frequency s 5 Hz. Fgure 2 just gves a few samples of the shot gathers calculated usng model D. Prestack depth mgraton s used to mage the calculated sesmc data and one of the resultng depth mages s shown n Fgure 3. The tme mage shown n Fgure 4 s the zerooffset traces. The reflecton waveform n the depth mages plus the reflecton pcks from tme and depth mages are used for jont sesmc nverson. Table lsts the reflectors pcked from depth and tme mages (Fgures 3 & 4) at dstance=5 m, whch s the travetme data used for the jont nverson. Tme (sec) Fgure 2: Samples of the shot gathers calculated usng the fnte dfference Dstance(m) Fgure 3: Depth mage of model D. Table : Samples of traveltme pcks used for the nverson. Reflector Tme (sec) A C Dstance (m) Fgure 5: Tme-lapse velocty models nverted usng EnKF. Models A-D correspond to tme=, 3 months, year, and 3 years, respectvely. Sesmc Inverson wth EnKF Fast forward modelng tools are essental for EnKF nverson, because we have to calculate g(m ) (see equaton ) for each sample of the ensemble that usually has a sze of hundreds. There are two types of forward modelng are nvolved n ths jont nverson. For waveform data, we assume a sonc log s avalable for source wavelet estmaton and use the source wavelet for convoluton modelng. In ths study, we just smply use the true velocty profle for the wavelet estmaton. Constant densty s assumed for mpedance calculaton. The forward modelng n the nverson for traveltme t s a summaton down to a gven reflector,.e., t = 2 /, v s the -D velocty of th depth pxel. v B D Dstance (m) v (m/s) Tme (sec) Dstance (m) Fgure 4: Tme mage of model D. Applyng the procedure descrbed n prevous secton to the observed sesmc data, we obtan the nverted velocty models shown n Fgure 5. In order to see the velocty changes more clearly, the velocty dfference between models B-D and base model A are shown n Fgure 6. A constant ntal model s used n ths test. It can be seen that the overall absolute velocty structure and the velocty drop due to CO 2 njecton are suffcently recovered. Profles n Fgure 7 gve a close comparson between the gven model and the nverted model. Fgure 8 compares the observed (or gven data) and the data calculated wth nverted 97 Downloaded Dec 2 to Redstrbuton subject to SEG lcense or copyrght; see Terms of Use at 97

4 velocty. The gven data and modeled data are vrtually dentcal, though the gven velocty model and the nverted velocty model exhbt some dfference, whch may be caused by the ampltude dstorton n the depth magng. True ampltude magng s very mportant for ths sesmc nverson. 2 Gven Conv. 2 B B C C D D Ampltude Fgure 8: A comparson between observed data (sold lne) and modeled data (dotted lne). Sold lne s sampled from dstance=5m from depth mage and the dotted lne s calculated from nverted velocty at the same locaton Dstance(m) Fgure 6: Velocty dfferences between tme-lapse models B-D and base model A. Left: gven models. Rght: Inverted models dv(m/s) Dstance(m) True Int. Inv. Conclusons The ensemble Kalman flter provdes a powerful tool for stochastc sesmc nverson, especally for dynamc nverson n sesmc montorng. Integratng travetme data nto the nverson makes the estmaton of absolute velocty possble. Waveform data used n the jont nverson gves the hgh resoluton components of nverted velocty. Acknowledgements We would lke to thank the sponsors of GCEP at Stanford Unversty for ther support to ths study. Eduardo Santos, Adeyem Arogunmat, and Tope Aknbehnje helped for the flow smulaton and the creaton of tme-lapse P-wave velocty models from flow models Velocty (m/s) Fgure 7: A comparson between true model (sold black lne) and nverted model (Dashdot blue lne) at dstance=5 m. Dotted yellow lne s the ntal model. 98 Downloaded Dec 2 to Redstrbuton subject to SEG lcense or copyrght; see Terms of Use at 98

5 EDITED REFERENCES Note: Ths reference lst s a copy-edted verson of the reference lst submtted by the author. Reference lsts for the 28 SEG Techncal Program Expanded Abstracts have been copy edted so that references provded wth the onlne metadata for each paper wll acheve a hgh degree of lnkng to cted sources that appear on the Web. REFERENCES Cao, D., W. B. Bevdoun, S. C. Sngn, and A. Tarantola, A smultaneous nverson for background velocty and mpedance maps: Geophyscs, 55, Evensen, G., 23, The ensemble Kalman flter: Theoretcal formulaton and practcal mplementaton: Ocean Dynamcs, 53, , Data assmlaton The ensemble Kalman Flter: Sprnger. Francs, A., 25, Lmtatons of determnstc and advantages of stochastc nverson: CSEG Recorder, February. Haas, A., and O. Dubrule, 994, Geostatstcal nverson A sequental method of stochastc reservor modelng constraned by sesmc data: Frst Break, 2, Mandel, J., 26, Effcent mplementaton of the ensemble Kalman flter: CCM Report 23, Unversty of Colorado at Denver and Health Scences Center. Oldenburg, D. W., T. Scheuer, and S. Levy, 983, Recovery of the acoustc mpedance from reflecton sesmograms: Geophyscs, 48, Sancevero, S. S, A. Z. Remacre, R. S. Portugal, 25, Comparng determnstc and stochastc sesmc nverson for thn-bed reservor characterzaton n a turbdte synthetc reference model of Campos Basn, Brazl: The Leadng Edge, Downloaded Dec 2 to Redstrbuton subject to SEG lcense or copyrght; see Terms of Use at 99