We E Multiparameter Full-waveform Inversion for Acoustic VTI Medium with Surface Seismic Data

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1 We E16 4 Multiarameter Full-waveform Inversion for Acoustic VI Medium with Surface Seismic Data X. Cheng* (Schlumberger) K. Jiao (Schlumberger) D. Sun (Schlumberger) & D. Vigh (Schlumberger) SUMMARY In this study we develo a strategy for multiarameter FWI for acoustic VI medium with surface seismic data. hrough arameterization analysis and synthetic tests we find that it is more feasible to invert for the arameterization as vertical and horizontal velocities instead of inverting for the arameterization as vertical velocity and anisotroy fields. We develo a hierarchical aroach to inverting vertical velocity first but we kee anisotroy fields unchanged and only switch to joint inversion when vertical velocity inversion are aroaching convergence. We demonstrate the success of our strategy for VI FWI using synthetic and real data examles from the Gulf of Mexico. Our results show that incororation of multiarameter FWI imroves migration of large offset full azimuth broad band acquisition data and roduces better focused migration images. 76 th EAGE Conference & Exhibition 214 Amsterdam RAI he Netherlands June 214

2 Introduction Full-waveform inversion (FWI) as a data-driven minimization roblem aims to directly fit the observed and simulated seismic waveform in either time or frequency domain. he inversion is erformed by iteratively udating the velocity fields to reduce the difference between the two. It has been shown that the inversion is very sensitive to the starting velocity fields and data with long offsets and low frequencies is crucial for the success of FWI to overcome this sensitivity (Vigh et al. 211). Considering the imortance of data with long offsets and low frequencies in most geologic environments anisotroy is an unavoidable toic for FWI esecially at long offsets since anisotroy tends to have more ronounced effects on waves traveled for a great distance (Prieux et al. 211). In VI medium this means more horizontal velocity will be registered in middle- to long-offset data while more vertical velocity will be registered in near- to middle-offset data. o date most real-world alications of FWI still remain in isotroic medium and only a few studies have been shown to account for anisotroy. And most of those studies only account for anisotroy in waveform simulation but do not invert for those anisotroy fields (Wang et al. 212; Warner et al. 213). Multiarameter inversion for anisotroy fields even in VI medium remains a hot toic in the field (Plessix and Cao 211; Gholami et al. 213ab). In this study we develo a strategy for multiarameter FWI for acoustic VI medium with surface seismic data. hrough arameterization analysis and synthetic tests we find that it is more feasible to invert for the arameterization as vertical and horizontal velocities instead of inverting for the arameterization as vertical velocity and anisotroy fields. We develo a hierarchical aroach to inverting vertical velocity first but we kee anisotroy fields unchanged and only switch to joint inversion when vertical velocity inversion are aroaching convergence. We demonstrate the success of our strategy for VI FWI using synthetic and real data examles from the Gulf of Mexico. Method and heory By setting the shear velocity to zero we can derive a set of first-order couled VI acoustic wave equation that correctly describes the kinematics of comressional waves roagation (Alkhalifah 2; Duveneck et al. 28): ρ v t = Dσ (1) σ = CDv + f t where v and σ stand for the article velocity and stress wavefield resectively. ρ is the density f is the source term and C and D are defined as 1 + 2ε 1 + 2ε 1 + 2δ C := ρv ε 1 + 2ε 1 + 2δ D := 1 + 2δ 1 + 2δ 1 76 th EAGE Conference & Exhibition 214 Amsterdam RAI he Netherlands June 214 x (2) where VI anisotroy is described using homson arameters ε and δ and v is the P-wave velocity along symmetry axis. Our multiscale time domain imlementation of FWI iteratively udates the velocity fields (m) to reduce the misfit between the observed data (d) and the forward simulated data ( ψ (m)) which can be formulated as a standard least-squares roblem: min J(m)=1 W ( S ψ (m) d ) 2 m 2 2 (3) where S stands for the samling/rerocessing oerator W reresents the weighting oerator (to aroriately scale multicomonent/multioffset data) and ( v ) ψ = solves the wave equation system i.e. σ the VI acoustic wave equation system (1).

3 he gradient of the misfit function can be derived through adjoint-state method (Plessix 26). Vigh et al. (214) derived the gradient formula for general elastic medium. In the case of VI anisotroy the gradient regarding to anisotroy model arameters are given by: v J vti (m)= λ vx x + λ vz ( ε J vti λ vx (m)= x + ) dt2ρv 2 δ J vti (m)= λ vx x + ( ρv dt δ λ vz ( ) ( (1 + 2ε) 1 + 2δ v x dt2ρv 1 + 2δ 1 ( vx x + ) v y ) vz v x x + v y 76 th EAGE Conference & Exhibition 214 Amsterdam RAI he Netherlands June 214 x + v y v z where λ vx and λ vz stand for x y and z comonent of the back-roagating article velocity wavefields. he misfit function is minimized by iteratively udating the model arameters with line-search method: m k+1 = m k + α k g k (5) where α is the ste size resolved from line search and g is the model udating direction. In Equation (5) the model udating direction is comuted by means of conjugate gradient where only first-order information is used. Although it was widely alied in ractice interactions between different model arameters are simly ignored. A strong reconditioner is usually required to accelerate the inversion convergence. his is esecially true for multiarameter anisotroic inversion since velocity and anisotroic arameters have different hysical unit and very different strength of influence on surface seismic data. Bringing in second-order information through Newton-like method will better balance the contribution of different arameter classes and imrove the inversion. During simultaneous inversion we observe significant acceleration in the convergence when incororating second-order information and reconditioning into inversion. Here in this study all the results are obtained with Equation (5). Several studies have shown that there is an ambiguity between the deth and the anisotroy arameter δ (Alkhalifah and svankin 1995; Plessix and Cao 211). From surface seismic alone δ cannot be recovered uniquely. hus we decide to hold the δ fields unchanged during our inversion and invert only for vertical velocity and ε fields. Synthetic Examles In our first synthetic examle we did a sensitive analysis of the misfit function for an VI Marmousi model. he model is 17 km wide and km in deth. here is a total of 171 shots with shot interval of 1 m. Receivers are located 25 m aart at 1 m deth. In all of our synthetic tests the δ fields were set as zero and fixed. We first smooth both the velocity and ε model for 5 m (Figure 1). hen fractions of the difference between the smoothed model and the true model were used to generate different model variations. he misfit evaluation is conducted at a band witdth centered at 3 Hz. As we can see in Figure 1e velocity variations clearly dominate the misfit function. On the contrary ε only has very small influence on the misfit function. A joint inversion starting with velocity fields far away from the true velocity will easily cause wrong ε udates and lead the inversion into a local minimum (e.g. models indicated as λ v = and λ ε = ). hus we develoed a hierarchical strategy of inverting for velocity first for VI medium and only starts joint velocity and ε inversion when velocity model is good enough. In another words we should only start joint inversion when λ v is close to 1. Our second synthetic examle is for the VI SEAM model. he model is 4 km wide and 15 km in deth. Similar as the Marmousi test we used a total of 41 shots with shot interval of 1 m. Receivers ) (4)

4 (b) (e) v k = v sm + λvk ( vtrue v sm ) ε k = ε sm5 + λεk ( ε true ε sm5 ) 17. (c) 17. Misfit (d) λv λε Figure 1 rue velocity and esilon (b) model; 5 m smoothed velocity (c) and esilon (d) model; (e) misfit function sensitivity analysis. are located 25 m aart at 1 m deth. he starting models were the smoothed version of the true models (5 m smoothing for velocity and 2 m smoothing for ε ). In addition velocity was reduced further by 1% and ε is further reduced by 2% in the starting model fields. We start the inversion from a bandwidth centered at 3 Hz. Figure 2 shows the comarison between the inversion results derived from simultaneous inversion and from the hierarchical inversion develoed in this study. Consistent with the sensitivity analysis that was reviously discussed simultenous inversion is quickly traed into a local minimum. he working inversion strategy is to use our hierarchical aroach to start the inversion with velocity only. Only starting to invert for ε when velocity model is aroaching convergence. he comarison of the deth rofiles for the inversion results reveals reliable model udates are obtained from our hierarchical multiarameter inversion aroach. 3 (c) 3 (e) 3 (g) 4914 (i) 3 Velocity (m/s) (b) (d) (f) Deth (m) (h).16 (j) Velocity (m/s) Deth (m) Figure 2 rue velocity and esilon (b) model; starting velocity (c) and esilon (d) model; final velocity (e) and esilon (f) model derived from simultaneous inversion; final velocity (g) and esilon (h) model derived from our hierarchical multiarameter inversion. Deth rofiles for velocity (i) and esilon (j) models; green - true model blue - starting model red - simultaneous inversion and urle hierarchical aroach. Real Data Examles Our real data examle is from Gulf of Mexico. he data is acquired with a dual-coil marine acquisition technique that rovides full azimuth and long offset (u to 14 km) coverage of the survey area. he source is at 1 m deth. Receivers are slanted from 12 m to 4 m deth. he initial velocity model used 76th EAGE Conference & Exhibition 214 Amsterdam RAI he Netherlands June 214

5 in this study was built using simle velocity analysis instead of a full tomograhic inversion. Smooth anisotroic arameter fields were generated based on knowledge of regional geology. Figure 3 shows the overlay of the deth slice of the starting migration image on to of the starting ε model and the final migration image on to of the inverted ε model. Significant image imrovements are achieved after running our multiscale FWI inversion. Detailed structures revealed in ε model are in lined with the final migration image. (b).1 1 Figure 3 Deth slice of the starting migration image overlaying on to of the starting esilon model and the final migration image overlaying on to of the final esilon model (b). Conclusions We develoed a hierarchical multiarameter inversion strategy for acoustic VI medium with surface seismic data. We demonstrate our inversion strategy using synthetic and real data examles. Our results show that incororation of multiarameter FWI imroves migration of large offset full azimuth broad band acquisition data and roduces better focused migration images. References Alkhalifah. [2] An acoustic wave equation for anisotroic media. Geohysics 65(4) Alkhalifah. and svankin I. [1995] Velocity analysis for transversely isotroic media. Geohysics 6(5) Duveneck E. Milcik P. Bakker P.M. and Perkins C. [28] Acoustic VI wave equations and their alication for anisotroic reverse-time migration. 78 th SEG Annual Meeting Exanded Abstract Gholami Y. Brossier R. Oerto S. Ribodetti A. and Virieux J. [213a] Which arameterization is suitable for acoustic vertical transverse isotroic full waveform inversion? art 1: Sensitivity and trade-off analysis. Geohysics 78(2) R81 R15. Gholami Y. Brossier R. Oerto S. Prieux V. Ribodetti A. and Virieux J. [213b] Which arameterization is suitable for acoustic vertical transverse isotroic full waveform inversion? art 2: Synthetic and real data case studies from valhall. Geohysics 78(2) R17 R124. Plessix R.E. [26] A review of the adjoint-state method for comuting the gradient of a functional with geohysical alications. Geohysical Journal International 167(2) Plessix R.E. and Cao Q. [211] A arametrization study for surface seismic full waveform inversion in an acoustic vertical transversely isotroic medium. Geohysical Journal International 185(1) Prieux V. et al. [211] On the footrint of anisotroy on isotroic full waveform inversion: the valhall case study. Geohysical Journal International 187(3) Vigh D. Jiao K. Watts D. and Sun D. [214] Elastic full waveform inversion alication utilizing multicomonent measurements of seismic data collection. Geohysics acceted. Vigh D. Kaoor J. Moldoveanu N. and Li H. [211] Breakthrough acquisition and technologies for subsalt imaging. Geohysics 76(5) WB41 WB51. Wang C. Yingst D. Bloor R. and Leveille J. [212] VI waveform inversion with ractical strategies: Alication to 3D real data. 82 th SEG Annual Meeting Exanded Abstract Warner M. et al. [213] Anisotroic 3D full-waveform inversion. Geohysics 78(2) R59 R8. 76 th EAGE Conference & Exhibition 214 Amsterdam RAI he Netherlands June 214