Velocity Modeling in a Vertical Transversely Isotropic Medium Using Zelt Method

Transcription

1 Velocy Modelng n a Vercal Transversely Isoropc Medum Usng Zel Mehod ABSTRACT Maryam Sadr *, H.R.Ramaz and M. Al Rah Receved 05 July 008; receved n revsed 1 January 009; acceped 1 February 009 In he presen paper, he Zel algorhm has been exended for ray racng hrough an ansoropc model. In ansoropc meda, he drecon of he propagaed energy generally dffers from ha of he plane-wave propagaon. Ths makes velocy values o be vared n dfferen drecons. Therefore, velocy modelng n such meda s compleely dfferen from ha n an soropc meda. The velocy model for ray racng s parameerzed n erms of blocky rapezod cells where he velocy changes nsde he cells lnearly. Thomsen s approxmaons n weakly ansoropc meda were used o esmae ansoropc velocy vecors. Rays were raced n drecon of group vecor n he vercal ransversely soropc (VTI) meda, whereas, he ansoropc Snell s law mus be sasfed by he phase angle and phase veloces across he nerface. The synhec examples are gven o demonsrae and verfy he ray racng algorhm. Refleced and urnng waves were raced hrough he soropc and ansoropc velocy models. Laeral and vercal velocy varaon caused devaon on rajecory of he ravelme curve. The resuls show ha he dfference beween soropc and ansoropc ravelmes ncreases wh offse, especally when he rao offse/deph exceeds 1.5. KEYWORDS Ray racng, Vercal Transversely Isoropy (VTI), Zel mehod, sesmc velocy modelng. 1. INTRODUCTION In a homogenous ansoropc medum, he wave behavor s que dfferen from ha n soropc meda (Helbg, 199) and he velocy of a sesmc wave changes wh he drecon of propagaon. Parallel cracks, srafed layerng, orenaon of grans and rock folaon cause velocy ansoropy (Posma, 1955; Krey and Helbg, 1956; Levn, 1978; Berryman, 1979; Helbg, 1981; Schoenberg, 1983; Thomsen, 1986). There are dfferen ray racng mehods o model he velocy dsrbuon. Shoong or bendng algorhms are generally used o race he ray pah from a source pon o a recever (Julan and Gubbns, 1977). Cerveny (1985) appled paraxal exrapolaon o esmae he ravelmes. Fne-dfference mehods were used by Reshef and Kosloff (1986), Vdale (1988), Podvn and Lecome (1991), and Van Trer and Symes (1991) o esmae he frs arrval ravelmes. Moser (1991) presened a graph heory o calculae he shores ray pah beween source and recever. Vnje e al. (1993) and Lomax (199) smulaed he wave frons nsead of ray pahs. Zel and Ells (1988) presened a ray racng algorhm n a -D soropc velocy model. The model s dvded by a seres of rapezodal blocks n each layer. The velocy whn he rapezodal cell s defned lnearly. Zel and Smh (199) nroduced a mehod of sesmc ravelme nverson for deermnaon of -D velocy model and nerface srucure. The ray racng algorhm was a modfcaon of ha presened by Zel and Ells (1988). They defned velocy nodes for he corners of he rapezod. Zel (1999) appled sesmc refracon/wdeangle reflecon ravelmes o oban -D velocy model and nerface srucure. In hs paper, we presen a ray racng mehod whch enables o compue he ravelme hrough each ray n an soropc and vercal ransversely soropc (VTI) meda. We develop he Zel and Ells s (199) algorhm based * Correspondng Auhor, Maryam Sadr, Ph.D. Suden of Mnng Engneerng, Dep. of Mnng, Meallurgy and Peroleum Engneerng, Amrkabr Unversy of Technology, Tehran, Iran (msadr@au.ac.r). H.R Ramaz, Asssan Professor, Dep. of Mnng, Meallurgy and Peroleum Engneerng, Amrkabr Unversy of Technology, Tehran, Iran. M. Al Rah, Assocae Professor, Insue of Geophyscs, Unversy of Tehran, PO Box , Tehran, Iran. Amrkabr / MISC / Vol. 3 / No.1 / Sprng

2 on ray racng n an ansoropc model. Thomsen s (1986) approxmaons for weakly ansoropc meda were used o esmae he ansoropc velocy vecors. Slawnsk e al. s (000) equaons for horzonal ansoropc meda were exended n a dppng ansoropc layerng model o compue he ray drecon. The man advanages of hs mehod are smplcy and mnmum number of parameers needs o defne he model. Ths mehod s suable for mul-sho recordng o defne opmal geomery of recevers n sesmc operaon plannng.. VELOCITY MODEL In ansoropc meda, he ravelme funcon s governed by an advanced form of he ekonal equaon. Soluon of hs equaon by he mehod of ray racng s dscussed by Cerveny e al. (1977). The knemac ray racng calculaons were performed by solvng he ray racng sysem (Cerveny e al., 1977). In hs sudy, he earh model was supposed o be homogenous and soropc/ansoropc composed of a sequence of layers. Each layer s dvded no rapezodal cells (Fg.1). The nerface of layers s specfed by an arbrary number of boundary nodes conneced by lnear nerpolaon. The number and poson of he nodes may dffer for each nerface. The P-wave velocy whn he rapezodal cell s descrbed as follows (Zel and Smh, 199): V ( c1x + c x + c3z + c xz + c5 ) ( x, z) ( c x + c ) 6 7 where, c s lnear combnaon of he corner veloces ( V 1, V, V3 and V ). In he case of P-wave propagaon n VTI (vercally ransverse soropy) meda, velocy deermnaon requres he esmaon of hree parameers; V p (P-wave 0 velocy n he drecon of he symmery axs), ε (fraconal dfference beween veloces perpendcular and parallel o beddng), and δ (he varaon n P-wave velocy close o he symmery axs). Under he assumpon of weak ansoropy ( ε << 1, δ << 1 ), he Thomsen (1986) s approxmaon for P-wave phase velocy s: ( θ ) ( 1 + δ sn θ cos θ + ε sn θ ) V p V p 0 where, ( θ ) p (1) () V s he phase velocy n he phase angle θ measured from he vercal axs of symmery. The Thomsen ansoropc parameers mus be defned for each velocy nodes (Fg.1). Fgure 1: The corner parameers of he cell n an ansoropc model: ε and δ are Thomsen ansoropc parameers; V s he vercal velocy. 3. RAY TRACING In ansoropc meda, he drecon of he propagaed energy (group vecor) generally dffers from ha of he plane-wave (phase vecor). Therefore, he ray angle Φ s dfferen from he phase angle θ excep a φ θ 0 and Π (Fg.). Fgure : Relaonshp beween velocy vecors n an ansoropc model. V (θ): phase velocy; V (Φ): group velocy; θ: phase angle; Φ: group angle. Shoong mehod s appled o race rays hrough he - D velocy model. Ray racng s carred ou by numercal solvng of he wo dmensonal ray racng equaons by Cerveny e al., (1977): dx anθ dz dθ z dz ( V anθ V ) V x where, θ s he angle beween velocy vecor and z-axs, V s he velocy of a cell (equaon 1), and V x and V z are paral dervaves of velocy wh respec o x and z drecon. The rays are raced wh P-wave group vecor hrough he model, whereas he Snell s law should be sasfed across he nerface wh phase angle and phase veloces: snθ1 snθ () V p ( θ1) V p ( θ) The phase angle n each layer can be calculaed by solvng he equaon 5 (Thomsen, 1986) usng he Newon-Raphson algorhm. (3) 36 Amrkabr / MISC / Vol. 3 / No.1 / Sprng 011

3 ( V ) ( dv d ) ( V ) ( dv d ) anθ + 1 ( θ). θ anϕ 1 an θ. 1 ( θ). θ where, φ s he ake-off angle. In hs paper, he subscrps and refer o he medum of ncdence and ransmsson, respecvely. Once he phase angle, θ, s found, he ray parameer, x 0, can be calculaed as: snθ x0 V (1 + δ sn θ.cos θ + ε sn θ ) 0 Equaon (7) can be solved for he ransmed phase angle ( θ ): ( V0 x0( ε )) sn + ( V0 x0. ) sn sn + ( V0. x0 ) 0 δ θ δ θ θ (7). The ransmed group angle, φ, whch he group velocy vecor makes wh he normal o he nerface, can be calculaed based on he ransmed phase angle, θ, from equaon (8), (Slawnsk e al., 000): V 1 φ cos 0 cosξ snξ 1 ( r( ξ )) +.( δ cosξ ε cos ξ ) snξ r ( ξ ) [ ( )] V.snξ δ cosξ ε cos ξ 0 where, ξ s he phase laude ( ξ 90 θ ), and r ( ξ ) V 0 1 ( 1+ δ sn.cos cos ξ ξ + ε ξ) The ravelme a he endpon of he ray s esmaed by numercal negraon along he ray pah (Eq. 10). By lnear nerpolaon across he endpons of he wo closes rays ha surround he pon of neres, he ravelme assocaed wh a specfc recever locaon s deermned. n d V 1 (5) (6) (8) (9) (10) where, s ravelme, d s ray sep lengh, and V s he ray velocy. reverse faul whch has nerruped he horzonal layers. The faul s ermnaed n nd layer and connued o he 5 h layer. A. A homogenous-soropc model To sar he ray racng, we assumed a homogeneous soropc condon for D velocy model. In hs model, he velocy n each layer was supposed o be consan. The 5 h layer has he lower velocy han he upper layers. Therefore, he srong reflecon surface s made a he op of hs layer. Table 1 shows he velocy values for he model. TABLE 1 PARAMETERS FOR THE HOMOGENOUS-ISOTROPIC VELOCITY MODEL. layer V p 0 (m/s) Rays were raced hrough he model by he new mehod. The ray pahs and correspondng synhec secons are ploed n Fg.3. The synhec sesmograms were produced by convoluon of ravelmes wh 60 Hz Rcker wavele. The amplude s he same for all races. Fgure 3a shows he refleced rays from he second layer (he sho s locaed on x8 km). The sho locaon moved o x1.5 km and he ray racng carred ou on he 5 h layer (Fg. 3b). Because of he velocy conras beween 3 rd and h layers, he crcal angel occurs a 3 degree on hs nerface. Therefore, he long offses were mssed on he 5 h layer durng he ray racng. Ray racng hrough he 6 h reflecor has been shown n Fg.3c. The sho s locaed a x1.5 km. Presence of he faul n hs model was caused laeral velocy varaon durng he ray racng on 6 h layer. Velocy changes and dppng reflecors were caused he ravelme dsorons on he reflecon curve.. SYNTHETIC EXAMPLES Mehodology examnaon accomplshed by he compuer program was wren for ray racng n -D soropc and ansoropc meda. To es he algorhm, a velocy model conssng of 7 layers was defned. The velocy model wh soropy/ansoropy parameers and velocy graden was consdered. Ths model conans a Amrkabr / MISC / Vol. 3 / No.1 / Sprng

4 B. A heerogeneous-soropc model Vercal velocy graden s consdered for he prevous velocy model. Ray bendng occurs when he vercal velocy changes whn a rapezodal cell. Table shows he velocy gradens n each layer. (a) TABLE PARAMETERS FOR THE HETEROGENOUS-ISOTROPIC VELOCITY MODEL. layer V p 0 (m/s) vercal graden (b) Due o vercal velocy graden, urnng waves were generaed durng ray racng. Fgure a shows he urnng waves whch ravel n he curvaure pahs. The lnear evens n sesmogram are relaed o he urnng waves produced n he frs layer (Fg. b). The unconformy locaed n he mddle par of he secon causes me dsorons on he hyperbolc ravelme curve corresponded o he 3 rd reflecor. (a) (c) (b) Fgure 3: Refleced P-wave rays were raced rough he homogenous-soropc meda. Velocy model and refleced rays wh relaed synhec sesmograms are shown for (a) second layer, (b) fourh layer, and (c) sxh layer. Fgure : (a) Refleced rays were raced on he boom of he 3 rd layer n he heerogeneous-soropc meda. (b) Lnear evens n sesmogram are relaed o he urnng waves generaed n he frs layer. The hyperbola s reflecon curve of he 3 rd layer. 38 Amrkabr / MISC / Vol. 3 / No.1 / Sprng 011

5 C. A homogenous-ansoropc model The D-VTI model seleced from Grechka and Tsvankn (00) comprses he hree homogenous layers. The parameers for he weak ansoropc model have been shown n Table 3, where ε and δ are Thomsen parameers for a VTI medum. Vp 0 s he vercal p-wave velocy. The veloces n each layer are assumed o be consan wh dfferen ansoropc parameers. TABLE 3 PARAMETERS FOR THE WEAKLY ANISOTROPIC VELOCITY MODEL. (a) layer V p 0 (m/s) ε δ We raced he rays correspondng o he P-wave reflecons from he source pon locaed a 1850 m wh dfferen propagaon angles o he boom of he 3rd layer (Fg.5). (b) Fgure 6: Comparson of soropc and ansoropc ray racng resuls (source pon s locaed a x1850 m). (a) Ray drecons change n ansoropc layers, (b) he dfference beween ravelmes compued by he ray racng hrough he soropc and ansoropc models. Fgure 5: Refleced P-wave rays were raced rough he ansoropc meda. The source pon s locaed a x1850 m. All he rays should sasfy he ansoropc Snell s low. The crcal angle for he refleced rays s checked n each nerface by solvng he equaon (7). Imagnary answers for hs equaon mean ha he ray s unable o penerae o he lower layers and should be refleced. The boundares should be smoohed before rayracng because he feaures wh abrup dp could make shadow zones. For esng he ansoropc effec, he ray racng was performed hrough he model wh soropc assumpon ( ε δ 0 ). Fg.6a shows he comparson of ray pahs under soropy and ansoropy condons. I seems ha he dfference beween he wo ray pahs s relaed o he ansoropc parameers. The dfference beween ravelmes compued from he soropc and ansoropc models ncreases wh offse. Ansoropc parameers cause he sgnfcan effecs on he long offses especally when he rao offse/deph exceeds 1.5 (Fg.6b). 5. CONCLUSION A new mehod of ray racng based on Zel and Ells s (199) algorhm for vercal ransversely soropc meda has been developed. The new mehod of ray racng has been examned n dfferen velocy models (soropc/ansoropc). The man advanages of hs mehod are smplcy and a mnmum number of parameers needs o defne he model. Ths mehod s suable for mul-sho recordng o defne opmal geomery of recevers n sesmc operaon plannng. The synhec daa presened o es he mehod show ha he dfference beween ravelmes compued from he soropc and ansoropc model ncreases wh offse. Ansoropc parameers cause sgnfcan effecs on he long offses especally when he rao offse/deph exceeds ACKNOWLEDGMENT I s necessary o hank Dr. C. Zel from Rce Unversy for sendng hs paper. Amrkabr / MISC / Vol. 3 / No.1 / Sprng

6 7. REFERENCES [1] J.G. Berryman, Long-wave elasc ansoropy n ransversely soropc meda, Geophyscs, vol., pp , [] V. Cerveny, I.A. Molokov, and I. Psenck, Ray mehod n sesmology, Unversa Karlova, [3] V. Cerveny, The applcaon of rayracng o he numercal modelng of sesmc wavefelds n complex srucures, Handbook of geophyscal exploraon, Geophyscal Press, London, [] V. Grechka, I. Tsvankn, PP+PSSS, Geophyscs, vol. 67, pp , 00. [5] K. Helbg, Sysemac classfcaon of layer-nduced ransverse soropy, Geophy. Prosp., vol. 9, pp , [6] K. Helbg, Foundaons of ansoropy for exploraon sesmc. Handbook of Geophyscal Exploraon. Oxford, Pergamon, 199. [7] B.R. Julan, and D. Gubbns, 3-D sesmc ray racng, Geophyscs, vol. 3, pp , [8] Th. Krey, K. Helbg, A heorem concernng ansoropc srafed meda and s sgnfcance for reflecon sesmc, Geophy. Prosp., vol., pp. 9-30, [9] F.K. Levn, The reflecon, refracon of waves n meda wh ellpcal velocy dependence, Geophyscs, vol. 3, pp , [10] A. Lomax, Wavelengh-smoohng mehod for approxmang broad-band wave propagaon hrough complcaed velocy srucures, Geophyscs. J. In., vol. 117, pp , 199. [11] T. J. Moser, Shores pah calculaon of sesmc rays, Geophyscs, vol. 59, pp , [1] P. Podvn, and I. Lecome, Fne dfference compuaon of ravelmes n very conrased velocy models: a massvely parallel approach and s assocaed ools, Geophys. J. In., vol. 105, pp. 71-8, [13] G.W. Posma, Wave propagaon n a srafed medum, Geophyscs, vol. 0, pp , [1] M.Reshef, and D. Kosloff, Mgraon of common-sho gahers, Geophyscs, vol. 51, pp , [15] M. Schoenberg, Reflecon of elasc waves from perodcally srafed meda wh nerfacal slp, Geophys. Prosp., vol. 3, pp. 65-9, [16] M.A Slawnsk, A. Raphael, R. Slawnsk, and B. James, A generalzed form of Snell s law n ansoropc meda, Geophyscs, vol. 65, pp , 000. [17] L. Thomsen, Weak elasc ansoropy, Geophyscs, vol. 51, pp , [18] J. Van Trer, and W.W. Symes, Upwnd fne dfference calculaon of ravelmes, Geophyscs, vol. 56, pp , [19] J. Vdale, Fne-dfference calculaon of ravelmes, Bullen of he Sesmologcal Socey of Amerca, vol. 78, pp , [0] V. Vnje, E. Iversen, and H. Gjoysdal, Travelme and amplude esmaon usng wavefron consrucon, Geophyscs, vol. 58, pp , [1] C.A. Zel, and R.M. Ells, Praccal and effcen ray racng n wo-dmensonal meda for rapd ravelme and amplude forward modelng, Canadan Journal of Exploraon Geophyscs, vol., pp , [] C.A. Zel, and R.B. Smh, Sesmc ravelme nverson for -D crusal velocy srucure, Geophys. J. In., vol. 108, pp. 16-3, 199. [3] C.A. Zel, Modelng sraeges and model assessmen for wdeangle sesmc ravelme daa, Geophys. J. In., vol. 139, pp , Amrkabr / MISC / Vol. 3 / No.1 / Sprng 011