Resolution and stability analysis of offset VSP acquisition scenarios with applications to fullwaveform

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1 Resolution and stability analysis of offset VSP acquisition scenaios with applications to fullwavefom invesion I. Silvestov, IPGG SB RAS, D. Neklyudov, IPGG SB RAS, M. Puckett, Schlumbege, V. Tcheveda, IPGG SB RAS Summay In this pape, we pesent a methodology that can be used to analyze diffeent acquisition scenaios with applications to full-wavefom invesion. The method is based on singula value decomposition (SVD) of the lineaized fowad map. We focus on the paticula case of an offset vetical seismic pofile (VSP) suvey. Using a 2D isotopic model, we analyze efficiency of full-wave invesion (FWI) fo a specific VSP acquisition scenaio when pessue measuements ae caied out in the well. We compae esolution and stability of invesion fo this scheme with typical multicomponent VSP data invesion. Intoduction New challenges faced in exploation geophysics lead to the necessity fo finding new ways to incease the infomational content and eliability of obsevations. Effective geophysical data pocessing outines ae needed to extact valuable infomation about the eath. In the seismic exploation community, it is commonly ecognized that one of the most pomising methods is full- wavefom invesion (FWI). In the case of VSP data pocessing, FWI povides a unique method to quantify elastic paametes and esevoi popeties in the boehole vicinity, including thei distibution away fom the boehole, and also below the total depth of the well. Fequency band, acquisition geomety, and natual noise within the boehole limit the esolving ability of any invesion pocedue. These paametes should be quantified initially to ensue that the invesion will be able to povide beneficial esults. In the fist stage of expeiment planning, it is easonable to estict ouselves to linea invese poblems to analyze the basic elations between the model paametes and paametes of the obsevation system. A technique based on the singula value decomposition (SVD) analysis of the lineaized fowad map is a typical choice fo such a poblem (Menke, 1994; Lebun et al. 2001; Osypov et al. 2008). We adopt this appoach to compae some pomising offset VSP acquisition scenaios fom the FWI point of view. Acquiing VSP data using cuently available 3C tools is a costly opeation due to extended time equied fo pope geophone clamping to the boehole wall. An altenative method is to use hydophones that can be easily deployed in the well, and which do not equie clamping. Mazetta et al. (1988), Geenwood et al. (2011) showed that pocessing the pessue wavefields povides the images that can be intepeted. The main focus of the cuent study is to ensue theoetically that the infomation equied fo successful elastic invesion exists in the pessue wavefields, and that futhemoe, it is possible to extact such infomation in the pesence of a given level of uncoelated noise. Also, we ae inteested in simultaneous invesion of the typical VSP multicomponent ecodings and the pessue measuements. An example of using such data was shown by Diks et al. (2002) whee the combined pessue and 3C data wee used fo sepaating the upgoing and downgoing events egisteed in hoizontal wells. In the pesent wok, we answe the following questions: 1) Is thee the possibility fo ecoveing elastic paametes in a boehole vicinity using only VSP pessue measuements; 2) Does the simultaneous invesion of pessue and displacement measuements impove the quality of esults. In the study being epoted, we ignoe the pesence of fluid-filled boehole and conside the pessue ecodings as the pessue in the elastic media. This assumption ignoes impotant issues elated to ecoding hydophone data, one of which is the tube waves, which ae the souce of stong coheent noise. Method The theoetical basis of this study elies on analyzing the following linea equation: Lm d, (1) whee L is a lineaized fowad map, d epesents the esidual data vecto (in ou case, displacements, pessue, o both), and m is the unknown model paametes petubation within the taget aea. In this pape, we ae dealing with a composite linea system: Lˆ( 1) d( 1) m, (2) Lˆ( N ) d( ) f N f whee an opeato Lˆ( ) is a lineaized fowad map coesponding to a single fequency. We conside thee cases to compose the system shown in equation (2): elastic lineaized fowad maps, lineaized fowad maps fo pessue, and the oint nomalized displacement pessue fowad maps. Pemultiplying equation (2) by the conugate matix L leads to the nomal system of linea equations: SEG Las Vegas 2012 Annual Meeting Page 1

2 Lˆ( ) Lˆ( ) m Lˆ( ) d ( ), (3) whee 1,...,. The system (3) plays the cental ole N f in ou study. Afte equation (3) is constucted fo the fixed acquisition scenaio, SVD fo the self-adoint matix of equation (3) is caied out; i.e., [ L L] VV, whee the columns of the matix V ae the eigenvectos, and the diagonal enties of the matix diag ( i ) ae eigenvalues of equation (3) aanged in noninceasing ode Estimated model paametes m ae obtained by applying the -pseudoinvese matix (Kostin and Tcheveda, 1995) on the data contaminated by the noise; i.e., m [ L L] 1 d [ V ] d, whee the diagonal V enties of the matix 1 (1 1 diag i ) ae set equal to zeo stating fom the index. The index detemine the condition numbe of the tuncated system, cond 1, which may be consideed as the egulaization paamete. Cond. numbe contols the fundamental tade-off between the esolution and vaiance (o stability) of the invesion esults. The optimal choice of the egulaization paamete is a difficult poblem in an actual situation. A numbe of techniques exist fo selecting this paamete (Vogel, 2002). In ou case, the situation is much simple because the tue model is known as well as the noise distibution. As a esult, we can compae the best achievable invesion esults with the tue model. In the cuent wok, we use the meansquaed eo (MSE) minimum citeia. Fo each acquisition scenaio when the noise type and level ae known and fixed, we scan the egulaization paamete within some ange. We select the value that povides the minimum of MSE with espect to the tue model. Invesion with such egulaization will povide the best achievable esults in a given situation; i.e., the esults will have the best tade-off between esolution and stability fo a given noise level. It is inteesting to undestand the contibution each facto makes into the final esidual. The MSE is defined as MSE Em 2 m, whee E means the math expectation. MSE can be epesented as: 2 MSE tace[cov m ] bias( m, m ) = 2 2 V B (4) The fist tem is a tace of the model covaiance matix which is esponsible fo vaiance of the ecoveed model. The model covaiance matix may be obtained by having SVD of matix (3) and the noise covaiance matix (Menke, 1994). The second tem in equation (4) defines the bias of the estimate. Bias is diectly connected with esolution accoding to bias m Rm, whee R is a model esolution matix coesponding to a given egulaization paamete, i.e. R [ V V ]. It is obvious that the bette is esolution the smalle the bias tem is. To compae esults of invesion fo diffeent scenaios we used a value of elative mean-squaed eo RMSE MSE m V m B / m / / The smalle the minimum RMSE value, the moe pefeable is the acquisition scenaio fom the point of view of the possibility of eliable infomation extaction. The SVD analysis capabilities in compaing diffeent acquisition scenaios ae best illustated by an example. Numeical example A fagment of the model used in this study is shown in Figue 1. The model has a vetically-inhomogeneous paamete distibution and a small petubation is located in the boehole vicinity. The elative petubation in P- and S- impedances ( Ip, Is ) does not exceed 1 to 1.5 %. The petubation in density is in the ange of 4 to 6%. We assume that the elastic paametes along the well ae known exactly, and we take them as a backgound model fo invesion. The pofiles fo the initial model and the tue models at the distance of 200 m fom the well ae shown in Figue 2. Analysis is pefomed within the taget aea X = [15, 550] m and Z = [4130, 4190] m. The total numbe of gid points in the taget aea is N =120 and X N =20; Z hx 4.5m, hz 3 m. The eceives ae placed along a vetical boehole fom the suface to a 4350-m total depth in fixed incements. We conside two cases, when the eceive spacing is equal to 6 and 12 m. The pessue souce is located at the suface with the offset 2130 m fom the wellhead. Ricke wavelet with dominant fequency f 0 60 Hz was used as a souce function. A key element of the pesent study is calculation of the matix in equation (3) by using the Bon appoximation. The pocedue is divided into seveal steps. In the fist step the explicit computation of Geen matices is pefomed. Fo an abitay backgound model this step is a computationally expensive pocess because a lage numbe of simulations should be pefomed (equal to the numbe of gid points in the taget aea). In this example, 1D backgound medium is consideed; theefoe, the numbe of fowad modeling uns is essentially educed and is equal to the numbe of the depth levels in the taget aea. Computations ae pefomed in the fequency domain using an iteative solve fo elasticity with a semi-analytical peconditione (Neklyudov et al., 2011). The second step is constuction of the matix L fo a given acquisition scenaio. As soon as the Geen matices ae calculated, the coesponding matices can be calculated almost on fly. In SEG Las Vegas 2012 Annual Meeting Page 2

3 this pesent study, we used N = 25 unifomly spaced f fequencies within the ange [5,125] Hz with a constant incement f 5Hz. To constuct the data vecto d, a wavefield scatteed by the petubation is geneated at each fequency. The wavefield is defined as the action of coesponding lineaized fowad map on the vecto of petubed paametes. An example of a 2C seismogam is pesented in Figue 3. The scatteed wavefield looks quite complicated because it is geneated not only by the downgoing P-wave excited by the souce but also by the conveted, eflected and multiple events oiginating in the unsmoothed efeence model. To calculate the pessue esponse, we use the definition of pessue though the stess tenso components; i.e. P 0.5 ( XX ZZ ). We assume that the data ae contaminated with additive white Gaussian noise, which povides a given signal-tonoise atio (SNR) with espect to the stongest aival in the calculated seismogams fo the tue model. An example of MSE vesus egulaization paamete cuves fo a fixed noise level 40dB is shown in Figue 4. The cuves wee used to detemine optimal esolutionstability tadeoff. The minimum achievable elative meansquaed eos (RMSE) fo diffeent acquisition scenaios ae shown in Figue 5. We epesent RMSE fo each model paamete independently as the coloed bas: the blue pat of a ba coesponds to the esolution tem of RMSE, and the ed pat coesponds to the vaiance tem. The bas demonstate the contibution of esolution and vaiance tems fo a paticula case. Note that the minimal RMSE is sought fo the composite dimensionless model vecto consisting of Ip, Is, pats. So we assume the invesion pefoms fo all elastic paametes simultaneously. RMSE is minimal when the displacements and pessue data with 6-m eceives spacings ae inveted simultaneously (2C+P case). The best achievable model fo this scenaio is shown in Figue 6B in compaison with the best achievable model fo moe taditional scenaio of 2C data invesion with 12-m eceive spacing (Figue 6C). These figues demonstate the achievable esolution of the ecoveed model paametes with a pedefined noise level in the data. One may conclude that the P impedance is esolved much bette than othe paametes (Figue 5). S- impedance, and especially density, may be ecoveed with much less esolution fo any consideed acquisition scenaio. With inceasing noise level, this behavio of ecoveed elastic paametes becomes moe and moe visible. Pessue data invesion (P case) gives the lagest RMSE: howeve, P-impedance is ecoveed quite well. P- case is not vey suitable fo IS and density econstuction, but it should povides quite easonable P impedance ecovey. Conclusions We have pesented a methodology fo peliminay quantitative evaluation of FWI esults as a function of the acquisition scenaio and data noise. The method is based on SVD analysis of the lineaized invese poblem. We explicitly constuct the data kenel matix using the Bon appoximation. The elastic Geen matices used fo that constuction ae calculated in the fequency domain. This appoach may be vey attactive because it enables one to compae the diffeent acquisition scenaios in a shot time. We applied the method to compae the esolving ability of some offset VSP acquisition scenaios. Paticulaly, we exploed the possibility of using the hydophone ecods to ecove the elastic paametes in the boehole vicinity. By means of a numeical example, it was shown that these data may povide acceptable P-impedance ecovey, which is compaable with that fo typical multicomponent VSP measuements. Recovey of S impedance and density ae not as good as in the P-case as in the 2C-case. Simultaneous invesion of hydophone and geophone data acquied in the boehole enables one to achieve almost the same esolution as fo 2C data invesion but with slightly impoved stability of the esults. This fact is especially visible fo Is, econstuction. Deceasing of the eceive spacing up to 6 m does not t impove the esolution of the invesion damatically; howeve, the invesion pocess becomes moe stable. Acknowledgements This study was pefomed in coopeation with Schlumbege Moscow Reseach and patially suppoted by RFBR gants , , Figue 1: A fagment of the model showing the taget aea and acquisition geomety (onshoe scenaio). SEG Las Vegas 2012 Annual Meeting Page 3

4 Figue 2: Density and impedances pofiles fo tue (ed) and backgound (blue) models Figue 3: 2C- scatteed wavefield caused by the petubation (Xcomponent) Figue 4: MSE of ecoveed model vesus condition numbe fo diffeent acquisition scenaios. Receive spacing is 6m. Data ae assumed to be contaminated with 40dB white Gaussian noise. Figue 5: Sensitivity chats fo diffeent acquisition scenaios. Length of each ba coesponds to elative mean-squaed eo of the ecoveed paametes Ip, Is, with espect to the tue model. Receive spacing is 6 and 12 m; SNR=40dB. Figue 6: A) model petubation; B) Best achievable esult of invesion 2С+P data with 6-m eceive spacing C) Best achievable esult of invesion 2С data with 12-m eceive spacing; Data ae assumed to be contaminated with 40 db white Gaussian noise SEG Las Vegas 2012 Annual Meeting Page 4

5 EDITED REFERENCES Note: This efeence list is a copy-edited vesion of the efeence list submitted by the autho. Refeence lists fo the 2012 SEG Technical Pogam Expanded Abstacts have been copy edited so that efeences povided with the online metadata fo each pape will achieve a high degee of linking to cited souces that appea on the Web. REFERENCES Diks, V., M. Xin-Quan, N. Randall, J. Blanco, M. Ebetta, J.-L. Gomes, and P. Dillon, 2002, Highesolution images fom 4-C hoizontal-well VSP data: 72nd Annual Intenational Meeting, SEG, Expanded Abstacts, Geenwood, A. J., J. C. Dupuis, A. W. Kepic, and M. Uosevic, 2011, Boehole hydophone acquisition: Some pitfalls and solutions: Boehole Geophysics Wokshop, Expanded Abstacts, BGP14. Kostin, V., and V. Tcheveda, 1995, R-pseudoinvese fo compact opeato in Hilbet spaces: existence and stability: Jounal of Invese and Ill-Posed Poblems, 3, Lebun, D., V. Richad, D. Mace, and M. Cue, 2001, SVD fo multioffset lineaized invesion: Resolution analysis in multicomponent acquisition: Geophysics, 66, Mazetta, T., M. Oton, A. Kampe, L. Johnson, and P. Wuenschel, 1988, A hydophone vetical seismic pofiling expeiment: Geophysics, 53, Menke, W., 1994, Geophysical data analysis: Discete invese theoy: Academic. Neklyudov, D., I. Silvestov, and V. Tcheveda, 2011, Fequency domain iteative solve fo elasticity with semi-analytical peconditione: 81st Annual Intenational Meeting, SEG, Expanded Abstacts, Osypov, K., D. Nichols, M. Woodwat, O. Zdaveva, and C. E. Yaman, 2008, Uncetainty and esolution analysis fo anisotopic tomogaphy using iteative eigendecomposition: 78th Annual Inte national Meeting, SEG, Expanded Abstacts, Vogel, C., 2002, Computational methods fo invese poblems: SIAM. SEG Las Vegas 2012 Annual Meeting Page 5