Ray tracing equatins in transversely istrpic media Csmin Macesanu and Faruq Akbar, Seimax Technlgies, Inc. SUMMARY We discuss a simple, cmpact apprach t deriving ray tracing equatins in transversely istrpic media. The general equatins derived here are given in terms f the ray slwness vectr and the directin f the axis f symmetry f the medium, which can change rientatin depending n lcatin. The standard Thmsen parameters are used instead f stress-energy tensr cmpnents, thus allwing fr easy cnnectin with experimentally relevant quantities. The TTI results can easily be simplified t the VTI and HTI cases crrespnding t particular rientatins f the symmetry axis. Bth weak and strng anistrpy are cnsidered. The accuracy f ur frmulas is verified by cmparing traveltimes with -way wave equatin numerical results. INTRODUCTION Ray tracing is an essential tl fr many mdern imaging algrithms Kirchhff and Beam migratin and velcity update methds tmgraphy. Ray tracing equatins in istrpic media are well understd and dcumented. Hwever, taking int accunt anistrpic effects due t cracks and layers f varius strata in the gelgical mdel has becme quite imprtant fr mdern seismlgical applicatins. In this paper we present a cmpact system f equatins applicable fr anistrpic media with transverse symmetry which, depending n the directin f the symmetry axis, ges under the name f VTI fr vertical, HTI fr hrizntal, r TTI fr tilted transverse istrpy. The general thery f ray tracing in hetergenus anistrpic media is discussed in Cerveny 00. Thmsen 986 defines anistrpic parameters relevant t practical use and derives results fr phase and grup velcity fr transversely istrpic media. Sena 99 see als Dng et al. 000 present explicit results fr velcity and traveltimes in HTI media, with azimuth-riented axis f symmetry. The standard ray tracing equatins fr a medium with n particular symmetry invlve evaluating up t 8 terms in a single equatin, which is quite expensive numerically. Symmetries in the medium, r using the weak anistrpy apprximatin, will reduce the cmplexity f the equatins invlved and speed up the cmputatinal prcess. Psencik and Farra 005 als see Dehghan et al. 005 intrduce first rder ray-tracing FORT equatins fr a general anistrpic medium, applicable fr weak anistrpy values. They als present explicit FORT equatins fr Orthrhmbic, HTI riented alng the X axis, and VTI media. Here we extend these results fr transversely tilted istrpic TTI media t large values f anistrpic parameters. We derive a relatively simple and elegant system f equatins that accmmdates varying velcity, anistrpic parameters, and changes in the rientatin f the symmetry axis. We als give the simplified results applicable fr VTI and HTI media with weak anistrpy. RAY TRACING EQUATIONS Fllwing Cerveny 00, the ray tracing equatins can be written in terms f the eigenvalues Gp, x f the Christffel matrix Γ : dx i = Gp,x ; dp i = Gp,x. Here x i are the crdinates f a pint n the ray, and p i are the cmpnents f the slwness vectr p i = n i /v alng the ray at that pint. In the abve equatins, x i and p i are treated as independent variables. The Christffel matrix cmpnents are given in terms f the elastic stress tensr Γ i j x, p = c i jkl xp j p l, and the functins G are slutins f the eigenvalue equatin detγ i j Gx, pδ i j = 0. Fr a transversely istrpic medium, the nn-zer cmpnents f the stress tensr in a crdinate system with the symmetry axis in the z directin are fllwing the Vight ntatin and Thmsen 986: c = c = C, c 3333 = C 33, c = C C 66, c 33 = c 33 = C 3 c 33 = c 33 = C 44, c = C 66 3 In terms f the standard weak anistrpy Thmsen parameters: C 33 = ρv, C = ρv +ε, C 3 = ρv +δ, 4 where v is the p-wave velcity alng the symmetry axis. In the pseudacustic apprximatin we can set C 44 = C 66 = 0. The eigenvalue equatin has three slutins fr Gp, the largest ne crrespnding t the pseud p-wave prpagatin, and the ther tw crrespnding t shear waves mdes. We will discuss the p-wave prpagatin in the fllwing, since the treatment f shear waves is quite similar. VTI in the weak anistrpy limit In the limit f weak anistrpy, the eigenvalue equatin fr the p-wave then becmes: [ ] Gp i,x = v p + δx p z p r p + εx p4 r p, 5 with p r = p x + p y the radial cmpnent f the slwness. In terms f the angle θ between the vertical symmetry axis and the slwness vectr n z = csθ: Gp i,x = v p +δ cs θ sin θ + ε sin 4 θ = v p η cs θ sin θ + ε sin θ 6
Ray tracing equatins in transversely istrpic media with η = ε δ. The equatin Gp,x = will give us the magnitude f the phase velcity v as a functin f the VTI medium parameters v 0,ε,δ, and the angle θ. T cmpute the change in crdinates alng the ray left hand equatin in we have t take the derivatin f G with respect t p i variables, while the x variables are kept cnstant that is, the v 0 x,εx and δx functins will be kept cnstant. Nte that the eigenvalues G are secnd rder hmgenus functins in the p variables; this means that they can be written in the generic frm: therefre, v 0 Gp,x With sin θ = p x + p y/p : Gp i,x = v 0 p gsin θ,x ; = p i gθ,x+ p p sin θ = p z sin θ ; p z dgθ,x d sin θ sin θ. p sin θ = p x,y cs θ. 7 p x,y The equatins fr ray crdinates change in weakly anistrpic VTI media becme : dz dx, y = p z v 0 η sin4 θ = p x,y v 0 +ε η cs4 θ. 8 see, fr example, Eq. 4 in Dehghan et al. 005. Expressed in terms f the angle between the slwness vectr and the symmetry axis csθ = pk/p, equatins 6 stay unchanged, the difference in the TTI case being that the angle θ is dependent n the spatial lcatin x. T cmpute the change in crdinates alng the ray, the equivalent f Eqs. 7 are = pk p sin θ = p pk k i p i p = pcsθ cs θ = p k i csθ p i p pk p. This result reduces t Eqs. 7 if we set k = 0,0, fr the case f VTI anistrpy. By evaluating the derivatives, ne then btains: [ dx i = v pk pk 3 0 p i ηpk k i k i p + p i p 4 + εp i k i pk]. 3 The change in slwness vectr alng the ray is btained by adding t Eq. 9 terms due t changes in the rientatin f the symmetry axis: dp i p = v v 0 gθ+v sin θ ε η cs θ +v 0 ε η csθ sin θ, 4 Cnversely, t cmpute the change in slwness vectr alng the ray, we keep the p variables therefre θ cnstant in the right hand equatin. Then, we have dp i p = v v 0 gθ+v sin θ ε η cs θ, 9 r, if instead f anistrpic parameter ε ne uses the hrizntal velcity v h x = v 0 x +εx : p d p i = v v cs θ η sin θ TTI in the weak anistrpy limit v η sin θ cs θ + v h v h sin θ. 0 Transversely tilted anistrpy is esentially VTI alng an arbitrary symmetry axis, whse rientatin may vary in space. Let us assume the directin f the new symmetry axis is given by a new vectr field kx f unit magnitude k =. One can then straightfrwardly generalize the eigenvalue equatin 5 by replacing the the slwness cmpnent p z with the prjectin n the new symmetry axis pk: [ Gp i,x = v p +δx pk p pk p 4 ] +εx p pk p 4, with sin θ = pk p = p j pk p k j. 5 Nte hwever that usually the axis f symmetry changes directin rather slwly, s the derivatives k j / can be expected t be very small numerically. Therefre fr practical purpses ne can usually neglect the secnd line terms in Eq. 4 and use Eqs. 9, 0 fr the TTI case t. Nte that these terms are als typically neglected in efficient numerical implementatins f the tw-way wave equatins fr TTI media see, fr example Macesanu 0. HTI anistrpy Fr reference, we present frmulas specific t the case f HTI anistrpy where the symmetry axis lies in the xy plane. The ray equatins can then be derived frm the general TTI case by setting the vectr k = k x,k y,0 = csφ,sinφ,0, with the angle φ cnstant. One then btains dz = v 0 p z +ε η cs 4 θ dx, y = v 0[ px,y η csθpk x,y csθ+ p x,y cs 3 θ+εp x,y pk x,y csθ ]. 6 Fr the cmpnents f p, Eqs. 9, 0 still apply where θ is the angle between the slwness vectr p and the HTI symmetry axis, and v h shuld be interpreted as the vertical velcity rather than hrizntal ne.
Ray tracing equatins in transversely istrpic media STRONG ANISOTROPY EQUATIONS NUMERICAL RESULTS Fr values f the anistrpy parameters greater than sme threshld typically abut 0., the weak anistrpy apprximatin may nt be accurate enugh. In this case, ne can use the exact result fr the p-wave eigenvalue f the Christffel matrix. Fllwing the ntatins Eqs. 0 in Thmsen 986, ne can write the result as with Gθ,x = v 0 p gθ = v 0 p +ε sin θ + D θ, 7 [ D θ = v s v + 4δ sin θ cs θ ] / +4ε v s v + ε sin θ 4 8 with v s = C 44 /ρ the shear wave velcity in the directin f the symmetry axis fr bth shear mdes, and δ v = s v + +δ + v s v v s v + ε here we keep the definitin +δ = C 3 /C 33 used in the weak anistrpy case. The ray equatins can be derived straightfrwardly : dx i v = p i gθ+ p sin θ ε + D θ p 0 i sin θ 9 where the derivatives f the sin θ quantity with respect t the slwness cmpnents are given in Eq. 7 fr the VTI case and in Eq. fr HTI r the general TTI case. Als dp i p = v v 0 gθ+ v sin θ ε + D θ + v 0 ε + D θ sin sin θ 0 θ where again the secnd line terms can usually be neglected fr the TTI case and are zer fr the VTI and HTI cases. T write dwn the derivatives f the D functin, it is cnvenient t intrduce the ntatins : ρ s = v s v ; D θ = Fθ. Then D θ sin θ = D θ = [ δ csθ+ερ s + εsin θ ] Fθ [ ρs Fθ + ε sin4 θ ρ s + sin θ cs θ δ + sin 4 θρ s + ε ε ]. The abve results are applicable fr the tracing f rays assciated with pseud-pressure waves. Fr shear waves, ne can use the ther tw eigenvalue functins: G SV θ,x = v 0 p v s /v 0 + ε sin θ D θ G HV θ,x = v s p +γ sin θ. T verify the accuracy and crrectness f ur equatins, we run sme numerical tests in mdels with mderate anistrpy. The first test is run in an hmgenus VTI mdel, with a p- velcity v 0 = 3000m/s,v s = 0, and the anistrpic parameters ε = 0. and δ = 0.. We perfrm ray tracing starting frm a surface pint, and calculate the traveltimes frm the sht lcatin t pints n a vertical line situated 3Km frm the sht. T evaluate the accuracy f the equatins, we cmpare the result with traveltimes evaluated using a -way wave equatin fr anistrpic media Duveneck et al. 008. Nte that the wave equatin results are cmputed in the pseud-acusic apprximatin v s = 0, hwever, the resulting traveltimes are exact up t numerical errrs that is, there is n apprximatin related t the magnitude f the anistrpic parameters in the equatins being used. TT rati.005.004.003.00.00 0.999 0.998 500 000 500 000 500 3000 3500 4000 4500 depth - z m Figure : Traveltime ratis fr strng slid line and weak anistrpy apprximatin dashed line in a hmgenus VTI mdel. In Fig. we plt the rati R ray/wave f the traveltimes btained frm ray-tracing versus the traveltimes btained frm wave equatin mdeling. The slid line crrespnds t rays traced with the exact equatins 9, 0; nte that in this case the tw traveltimes agree with an errr margin < 0.0%. The dashed line crrespnds t the case where the rays are traced using the weak anistrpy apprximatin Eqs. 8, 9; the difference frm unit in this rati is due t apprximatin being used. We see frm this plt that the errr due t the use f the weak apprximatin in ray-tracing equatins can be quite significant in this case since relative errrs f rder 0.% in traveltime translate int abslute errrs f rder milisecnds fr the times assciated with seismic events. These errrs seem rughly cnsistent with thse fund by Dehghan et al. 005, which reprt a weak apprximatin errr f abut 0.08% fr values f the anistrpy parameters equal t half f the nes used here. One als ntes frm Fig. that the greatest errrs assciated with the weak anistrpy apprximatin appear at angles clse t 60 deg the relatin between depth z in Fig. and angle is tanθ = 3000/z nte that this is the ray angle rather than the phase angle θ emplyed in the previus sectin. The reasn
Ray tracing equatins in transversely istrpic media fr this behaviur is that anistrpic effects generally increase with angles indeed, at θ = 0 alng the symmetry axis the prpagatin is istrpic. Hwever, at right angle with respect t the symmetry axis the weak apprximatin becmes exact; ne can check this by lking at the weak apprximatin fr the functin D in Eq. 8: D weak θ = η sin θ cs θ + ε sin 4 θ D θ, in the limit where θ 90deg. VTI media. We find very gd agreement between the tw sets f traveltimes when the full exact equatins are used fr ray tracing. When the weak apprximatin is used, we find errrs f rder 0.% in traveltime evaluatin fr anistrpic parameters f rder 0%. This indicates that using the exact equatins is reccmmended if ne desires gd accuracy in mdels with mild anistrpy..005.004.003 TT rati.00.00 0.999 0.998 500 000 500 000 500 3000 3500 4000 4500 depth - z m Figure : Traveltime ratis fr strng slid line and weak anistrpy apprximatin dashed line in a vz mdel. The secnd mdel used fr testing is a vz mdel, where all parameters vary with depth nly. We take v 0 z = + 0.5z Km/s,ε = 0.z,δ = 0.05z with depth measured in Km, s that the average f the parameters ver the first 4 Km f depth are the same as in the first test. The ratis f the raytracing traveltimes and wave equatin traveltimes fr this mdel are pltted in Fig. fr the weak apprximatin and the exact frmulas. Again, we nte gd agreement between the ray-tracing traveltimes and the wave equatin nes when using the exact equatins 9, 0, and errrs f rder 0.% fr the weak apprximatin case. Nte that cmpared with the previus test, the errrs in this mdel are smaller shallw since the magnitude f the anistrpic parameters is smaller, and smewhat larger at depths greater than 4 Km. CONCLUSIONS We have describeded here a cmpact system f equatins fr ray-tracing in a transversely istrpic medium. These equatins imprve the efficiency f numerical implementatin f ray-tracing algrithms, cmpared t methds where the elements f the stress-energy tensr are evaluated individually. We give simpler frms f the equatins suitable fr weak anistrpy apprximatin in VTI and HTI media. Hwever, the general results we present are valid fr arbitrarily strng anistrpy, and extend previus published wrk valid in the weak anistrpy limit. We perfrm numerical cmparisn between traveltimes evaluated by ray-tracing, and by using a -way wave equatin in
Ray tracing equatins in transversely istrpic media REFERENCES Cerveny, V., 00, Seismic ray thery: Cambridge University Press. Cambridge Bks Online. Dehghan, K., V. Farra, and L. Nicletis, 005, Apprximate ray tracing fr qp waves in inhmgeneus layered media with weak structural anistrpy: GEOPHYSICS, 7, SM47 SM60. Dng, Y., H. Yang, and W. Sun, 000, An apprximatin f the grup velcity fr hti media: SEG Technical Prgram Expanded Abstracts, 389 39. Duveneck, E., P. Milcik, P. M. Bakker, and C. Perkins, 008, Acustic vti wave equatins and their applicatin fr anistrpic reverse-time migratin: SEG Technical Prgram Expanded Abstracts, 7, 86 90. Macesanu, C., 0, The pseudacustic apprximatin t wave prpagatin in tti media: SEG Technical Prgram Expanded Abstracts 0, 74 78. Psencik, I., and V. Farra, 005, First-rder ray tracing fr qp waves in inhmgeneus, weakly anistrpic media: GEO- PHYSICS, 70, D65 D75. Sena, A. G., 99, Seismic traveltime equatins fr azimuthally anistrpic and istrpic media: Estimatin f interval elastic prperties: GEOPHYSICS, 56, 090 0. Thmsen, L., 986, Weak elastic anistrpy: Gephysics, 5, 954 966.