The azimuth-dependent offset-midpoint traveltime pyramid in 3D HTI media

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1 The azimuth-dependent offset-midpoint traeltime pyramid in 3D HTI media Item Type Conference Paper Authors Hao Qi; Stoas Alexey; Alkhalifah Tariq Ali Eprint ersion Publisher's Version/PDF DOI.9/segam3-58. Publisher Society of Exploration Geophysicists Journal SEG Technical Program Expanded Abstracts 3 Download date 6//8 :4:5 Link to Item

2 Downloaded /4/6 to Redistribution subject to SEG license or copyright; see Terms of Use at The azimuth-dependent offset-midpoint traeltime pyramid in 3D HTI media Qi Hao* Alexey StoasNTNU Norway and Tariq Alkhalifah KAUST Saudi Arabia. Summary Analytical representation of offset-midpoint traeltime equation is ery important for pre-stack Kirchhoff migration and elocity inersion in anisotropic media. For VTI media the offset-midpoint traeltime resembles the shape of Cheop s pyramid. In this study we extend the offset-midpoint traeltime pyramid to the case of 3D HTI media. We employ the stationary phase method to derie the analytical representation of traeltime equation and then use Shanks transformation to improe the accuracy of horizontal and ertical slownesses. The traeltime pyramid is deried in both the depth- and time-domain. Numerical examples indicate that the azimuthal characteristics of both the traeltime pyramid and the migration isochrones are ery obious in HTI media due to the effect of anisotropy. Introduction The azimuthal characteristics of seismic waes in HTI media can help to identify the azimuth of fractures which is important for oil and gas exploration in fractured reseroirs. The azimuthal ariation of elocities influences amplitudes but more detectable traeltimes. Deeloping analytical prestack traeltime formulations helps in many applications including traeltime tomography and integralbased Kirchhoff imaging. For isotropic media the traeltimes analytically as a function of offset and midpoint is gien by a simple double-square-root (DSR) equation in homogeneous media. Howeer it is difficult to obtain the analytical traeltime formulations in anisotropic media since the explicit relation between group elocity and ray angle does not exist in anisotropic media. Een for transersely isotropic media with a ertical symmetry axis (VTI) the traeltime is often calculated numerically. Alkhalifah (b) deried the offset-midpoint traeltime equation or Cheop s pyramid equation for VTI media through the stationary phase method. Hao and Alexey (3) extended the offsetmidpoint traeltime equation to the case of D transersely isotropic media with a tilt symmetry axis (TTI). In this abstract we derie the analytical offset-midpoint traeltime pyramid for 3D transersely isotropic media with a horizontal symmetry axis (HTI). We specifically obtain a Taylor s series expansion of the slowness components as a function of anelliptic parameter. The accuracy of these expansions is enhanced using Shanks transformation (Bender and Orszag 978) to obtain the high-order representation. Time- and depth- domain azimuth-dependent offsetmidpoint traeltime pyramid The 3D pre-stack phase-shift migration operator defined in the offset-midpoint domain is gien by (Yilmaz ) P( x x h h z t ) dp( x x h h z ) dk dk dk dk exp( it ) h h x x where T is the traeltime shift gien by T ( q q ) z p ( x x ) s g x () px( x x ) ph h phh and P and P are the seismic data before and after prestack depth migration respectiely; ( x x ) is the lateral position ector of midpoint ( x x) is the lateral location ector of image point z is the depth of image point. ( h h ) is the half-offset ector ( q q ) are the ertical projections of the slowness ector defined at source and receier positions respectiely and ( px px) and ( p p ) are the horizontal slowness ector defined in h h midpoint-offset space. Alkhalifah (a) used the stationary phase method to calculate the horizontal slowness component at stationary point by soling equation () for source slowness p and receier slowness p g instead of the following linear relation p p p s x h p p p s x h p p p g x h p p p g x h s g p h and p according to Hence equation () can be written in terms of slowness for the source and receier gien by T( x x x x h h z) ( q q ) z p y p y p y p y () s g s s s s g g g g where ( ys y s) denotes the lateral distance ector between image point and source gien by ( ys ys) ( x x h x x h ) and ( yg y g) denotes the lateral distance between image point and receier gien by ( y y ) ( x x h x x h ). g g By setting deriaties of traletime T in equation () with respect to p s p s p g and pg to zero we obtain the equations of the type dq y (3) dp z x s 3 SEG DOI SEG Houston 3 Annual Meeting Page 3335

3 The azimuth-dependent offset-midpoint traeltime pyramid in 3D HTI media Downloaded /4/6 to Redistribution subject to SEG license or copyright; see Terms of Use at dq dp y z (4) where ( p p ) q and ( y y ) are either ( ps p s) q s and ( ys y s) for the source or ( pg p g) q g and ( y y ) for the receier respectiely. g g Equations () (3) and (4) represent the 3D ersion of depth-domain traeltime pyramid. In order to obtain the time-domain traeltime pyramid for HTI media zero-offset two-way traeltime is obtained by setting the half offset ector ( h h ) and lateral coordinate of image point equal to lateral midpoint coordinate ( x x) ( x x ) in equation () T( x x x x x x h h z) q z z (5) where q z qs qg (6) x x h x x h nmo denotes the ertical slowness for the zero-offset seismic ray. Consequently by substituting equation (5) into equation () we obtain the time-domain traeltime pyramid for the acoustic case in HTI media T( x x x x h ) nmo qs qg (7) p y p y p y p y s s s s g g g g where denotes the obseration azimuth measured from x-axis tan h h (8) h denotes the size of half-offset ector h h h. (9) Equation (7) can be employed to calculate the azimuthdependent traeltime of reflected acoustic wae in 3D HTI media. The slowness ector ( ps p s) and ( pg p g) in equation (7) can also be represented in terms of offset and azimuth. Horizontal and ertical slowness approximation In order to obtain the horizontal slowness ( p p ) from equations (3) and (4) we consider the slowness surface equation for HTI media which describes the relation between horizontal slowness ( p p ) and the ertical slowness q. According to Alkhalifah (998 a) the 3D slowness surface for an acoustic wae in VTI media is gien in the following form F VTI nmo nmo q p p () ( ) p p where ( p p ) and q are the horizontal and ertical slownesses respectiely. and are the nmo ertical and normal moeout elocities respectiely. is the anelliptic parameter where and are the anisotropic parameters (Thomsen 986). The slowness surface in TTI media is obtained by applying two separate simple rotations (Zhu and Dorman ). In Figure the p p q system denotes the slowness in VTI media; the p p q system denotes the slowness in TTI media where p is located on the p p plane. In the first rotation we perform a clockwise rotation using along the p axis. After the first rotation the q and q axes are coincide. The second rotation is a clockwise using angle about the q axis. Figure : The geometry of slowness ector rotation. The 3D slowness surface equation for HTI media can be obtained by seting p sin cos p p cos sin p () q q where corresponds to the azimuth of the horizontal symmetry axis measured from the x-axis. The perturbation of equation () can be gien in the following form p sin cos p p cos sin p () q q and the perturbation of the ertical slowness q and q can be expressed by the corresponding horizontal slownesses respectiely q q q p p (3) p p 3 SEG DOI SEG Houston 3 Annual Meeting Page 3336

4 The azimuth-dependent offset-midpoint traeltime pyramid in 3D HTI media Downloaded /4/6 to Redistribution subject to SEG license or copyright; see Terms of Use at q q q p p p p. (4) In order to obtain the partial deriatie relations between VTI and HTI media we assume that the perturbations of the slowness surface in VTI media is caused by a change in only p not p that is p. Hence we obtain the following linear algebra equation from equations () (3) and (4) q cos p p q sin p p. (5) p q q p p Obiously the existence of a non-zero solution for equation (5) should satisfy that the determinant of the coefficient matrix is zero. Consequently we obtain q p q q cos sin. (6) p p For the other case we assume that the perturbations of the slowness surface in VTI media is caused by a change in only p not p. In the similar way we finally obtain q q cos sin q p p. (7) p q q cos sin p p Equations (6) and (7) describe the deriatie relations between VTI and HTI media. Similar idea was used for mapping of moeout in TI media (Stoas and Alkhalifah ). In order to calculate the expression for p and p we consider the following relation p pcos p psin where p denotes the size of horizontal slowness gien by dq q q c (8) dp p p and the explicit expression for c can be obtained from equations (6) and (7) and denotes the azimuth of horizontal slowness ector which could be obtained from equation (6) and (7): q q tan. p p From equations () and (8) we obtain polynomials as a function of the slowness p and q independently c ( p ) ( p ( )) p (9) 3 4 nmo nmo nmo 4 3 c q ( q ) nmo ( q). () Both equations (9) and () are quartic equations with respect to ariables p and q respectiely. To derie the trial solutions we use the perturbation of respect to the anelliptic parameter ( ) _( )... p and q with p p p p () q q q q. () ( ) _( )... Substituting trial solution () and () into equation (9) and () respectiely the coefficients pi and q i i can be deried. The exact explicit expression for p i is gien in Hao and Stoas (3). Furthermore Shanks transformation can be employed to improe the accuracy of Taylor series expansion for the horizontal slowness and the ertical slowness in VTI media. The final expressions take the form p _ p p _ p p q _ q q _. q q Considering equation () we obtain the norm of horizontal slowness p and the azimuth of horizontal slowness and ertical slowness q in 3D HTI media respectiely p p p p q tan p Numerical examples sin cos sin p sinq p sin sin p cosq q p cos. To illustrate the characteristics of the traeltime pyramid as a function of offset and midpoint we select a homogeneous HTI model with a horizontal symmetry axis along x-axis. The medium parameters are km / s... Figure shows the acquisition system used for calculating the traeltime as a function of midpoint and offset for a gien azimuth in 3D HTI media. x and h are the 3 SEG DOI SEG Houston 3 Annual Meeting Page 3337

5 The azimuth-dependent offset-midpoint traeltime pyramid in 3D HTI media Downloaded /4/6 to Redistribution subject to SEG license or copyright; see Terms of Use at midpoint and the half offset along the obseration azimuth respectiely. The image point is located along the z-axis. The zero-offset two-way ertical traeltime is 3s for midpoint x. Figure 3 shows the comparison of traeltimes corresponding to different azimuths extracted from the traeltime pyramids for a common-midpoint ( x ) for a common-offset ( h ) and for ( x h). All three plots in Figure 3 depict that the traeltime decreases with an increase in azimuth from to 9. In the other words the traeltime pyramid becomes flatten with an increase in azimuth from to 9. This characteristic becomes much more pronounced when offset and midpoint are far from zero. Next we inestigate the ariation of common-offset migration isochrones with the change of the obseration azimuth. The model parameters in the first example are adopted again. To obtain the common-offset migration isochrones we assume that both the half offset h and the traeltime t are constant representing a location in our data space and we sole for potential image space contribution locations. For h.5km and t 3s Figure 4 shows the migration isochrones corresponding to azimuths Figure : The geometry of the offset-midpoint acquisition system. ranging from to 9. We clearly realize that the azimuthal ariations of migration isochrones are much more pronounced for large dip angles where the effect of anisotropy is large. For a horizontal reflector (the isochron apex) all azimuths proide the almost the same placement of energy reflecting the high accuracy of the approximation for this offset. Conclusions The azimuth-dependent offset-midpoint traeltime pyramid for 3D HTI media is deried using the stationary phase method. Perturbation in the anisotropic parameter followed by Shanks transform proides relatiely simple analytical formula to describe the traeltimes with high accuracy. The azimuthal characteristic of this traeltime pyramid equation is justifying its potential use in identifying the azimuth of the symmetry axis in 3D HTI media. Acknowledgements We would like to acknowledge the ROSE project for financial support. Figure 4: Comparison of common-offset migration isochrones along different obseration azimuth. Figure 3: Comparison of azimuth-dependent traeltimes extracted from traeltime pyramids for x =(left) h=(middle) and x =h(right) 3 SEG DOI SEG Houston 3 Annual Meeting Page 3338

6 Downloaded /4/6 to Redistribution subject to SEG license or copyright; see Terms of Use at EDITED REFERENCES Note: This reference list is a copy-edited ersion of the reference list submitted by the author. Reference lists for the 3 SEG Technical Program Expanded Abstracts hae been copy edited so that references proided with the online metadata for each paper will achiee a high degree of linking to cited sources that appear on the Web. REFERENCES Alkhalifah T. 998 Acoustic approximations for seismic processing in transersely isotropic media : Geophysics Alkhalifah T. a An acoustic wae equation for anisotropic media : Geophysics Alkhalifah T. b The offset-midpoint traeltime pyramid in transersely isotropic media : Geophysics Bender C. M. and S. A. Orszag 978. Adanced mathematical methods for scientists and engineers: McGraw-Hill. Hao Q. and A. Stoas 3 The offset-midpoint traeltime pyramid in TTI media: Presented at the 75 th Annual International Conference and Exhibition EAGE. Stoas A. and T. Alkhalifah a A tilted transersely isotropic slowness surface approximation: Geophysical Prospecting in press. Stoas A. and T. Alkhalifah b Mapping of moeout in a TTI medium: Presented at the 75 th Annual International Conference and Exhibition EAGE. Thomsen L. 986 Weak elastic anisotropy: Geophysics Yilmaz O. Seismic data analysis processing inersion and interpretation of seismic data: SEG. Zhu J. and J. Dorman Two-dimensional three-component wae propagation in a transersely isotropic medium with arbitrary-orientation-finite-element modeling: Geophysics SEG DOI SEG Houston 3 Annual Meeting Page 3339