AVO Modeling of Monochromatic Spherical Waves: Comparison to Band-Limited Waves

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1 AVO Modelng of Monochromatc Sphercal Waves: Comparson to Band-Lmted Waves Charles Ursenbach* Unversty of Calgary, Calgary, AB, Canada and Arnm Haase Unversty of Calgary, Calgary, AB, Canada Summary Monochromatc and band-lmted sphercal waves have dfferng reflecton coeffcent curves. To facltate comparson, a new expresson for monochromatc reflectvty s gven n terms of a weghtng functon. The weghtng functon approach, developed prevously for a specfc class of band-lmted sphercal waves (Raylegh fltered waves), shows explctly how dfferent plane waves contrbute to a sphercal-wave reflecton coeffcent. Drect comparson shows that monochromatc waves have oscllatory, non-decayng weghtng functons, and thus sample a wde range of plane waves. In contrast, typcal Raylegh wavelets produce localzed weghtng functons. These two behavors lead to reflecton coeffcent curves whch dffer beyond the crtcal angle. A brdge between these two behavors s constructed by consderng unusually narrow Raylegh wavelets. These show ntermedate propertes. Ths study shows 1) a smple and convenent method for calculatng monochromatc sphercal-wave reflecton coeffcents, and 2) a clearer understandng of how sphercal-wave reflecton coeffcents are created from consttuent plane-waves. Introducton Hstorcally, the most common approach for descrbng reflectvty of sphercal waves n sesmc exploraton has been through constructng the reflecton coeffcents for a monochromatc source. Ths approach s due to Lamb (1904) and Sommerfeld (1909) and s descrbed n Ak and Rchards (1980). Carryng out such calculatons for multple frequences allows one to obtan the reflecton coeffcent for a band-lmted wavelet va an nverse Fourer transform, as has been carred out by Haase (2004). Let t Flow 2007 CSPG CSEG Conventon 52

2 We have prevously presented a drect approach to band-lmted reflecton coeffcents (Ursenbach et al., 2005). Ths employs the Raylegh wavelet (Hubral and Tygel, 1989), whch allows the nverse Fourer transform to be carred out analytcally. Only one numercal ntegral s then requred to obtan the reflecton coeffcent for a gven geometry. The fnal result s expressed as R ( θ ) = W ( S, θθ, ) R ( θ) d(cos θ), (1) sphercal (Raylegh) PP Γ n 0 PP where θ s the angle of ncdence, θ s an ntegraton parameter, Γ s an ntegraton path n the complex plane, S0 α1 ( Dω0), α 1 s the P-wave velocty of the overburden, D s the length of the raypath from source to recever, ω 0 s the domnant frequency of the Raylegh wavelet, and W n s a normalzed weghtng functon, wth n a parameter of the Raylegh wavelet. For Raylegh wavelets, W n s an analytc functon whch can be readly programmed. In addton to provdng a speedy approach to calculatng sphercal-wave reflecton coeffcents, the Raylegh wavelet approach of equaton 1 also provdes useful nsght nto the relatonshp between plane-wave and sphercal-wave reflectvtes. The W n kernel s largest when θ and θ are smlar, and decays rapdly when θ θ s large. Thus the sphercal-wave reflecton coeffcent receves contrbutons prmarly from plane-wave coeffcents near the angle of ncdence. Indeed, as S 0 0, the sphercal-wave coeffcent approaches the plane-wave Zoepprtz result (Ursenbach et al., 2005). To obtan a smlar pcture for sngle frequences, we frst derve an expresson for the monochromatc sphercal-wave reflecton coeffcent whch s smlar n form to equaton 1. We then compare monochromatc reflecton coeffcents to band-lmted reflecton coeffcents, and monochromatc weghtng functons to band-lmted weghtng functons. We wll demonstrate that the two cases dffer sgnfcantly, but that the band-lmted results approach the monochromatc results for ncreasngly narrow bands. Theory Analogous to equaton 6.30 of Ak and Rchards (1980), the monochromatc potental for a reflected sphercal wave may be wrtten as p φω ( ) = Aωexp( ωt) R ( p) J ( ωpr)exp[ ωξ( z + h)] dp ξ PP 0 (2) 0 where φ( ω ) s the spectrum of the dsplacement potental, s frequency, A s an arbtrary scale 2 2 factor, t s tme, p and are horzontal and vertcal slownesses ( α1 = 1/ p + ξ ), R PP s the plane-wave reflecton coeffcent, J 0 s a zeroth-order Bessel functon, r s the source-recever offset, and z and h are the vertcal dstances from the nterface to the recever and source. To proceed from equaton 2 to an expresson for the sphercal-wave reflecton coeffcent, we follow steps smlar to those descrbed n detal n Ursenbach et al. (2005) for the Raylegh wavelet band-lmted case: (1) Obtan the dsplacement spectrum from the gradent of the potental parallel to the ray vector at the recever, (sn θ,0,cos θ ), where θ s the angle of ncdence. (2) Dvde ths result by the smple dsplacement spectrum obtaned usng R PP = 1. (3) Perform a change of varables for the ntegraton and defne snθ = pα1 and cosθ = ξα1. (4) Set h = z, and note that Let t Flow 2007 CSPG CSEG Conventon 53

3 2 2 z = ( D/2)cosθ and r = Dsnθ (where D= r + 4z ). (5) Note that the ntegrand now depends upon only three varables, θ, θ, and S, where D, and 1 appear only n the combnaton S α1 ( Dω ), a quantty whch provdes a measure of the mportance of curvature and sphercal effects. The fnal result can then be wrtten R ( θ ) = W( S, θθ, ) R ( θ) d(cos θ), (3) sphercal (monochromatc) PP W( S, θθ, ) = Γ PP [ J1(snθ sn θ / S)snθsn θ + J0(snθsn θ / S)cosθ cosθ] S( 1 S) exp [ (1 cosθcos θ )/ S]. (4) We have thus derved an equaton of the form of equaton 1, and can compare reflecton coeffcents and weghtng functons between monochromatc waves and band-lmted waves. Results We consder a Class 1 AVO system defned n Table 1. We assume a frequency for the monochromatc wave of 100/π Hz ( 31.8 Hz), and an nterface depth of 500 m, so S =.01cosθ. Equatons 3 and 4 may then be solved to gve the monochromatc reflecton coeffcent curve, whle, gven an approprate wavelet, a band-lmted result may be obtaned usng the method of Ursenbach et al. (2005). Recent developments have mproved ths method so that calculatons may be readly carred out for Raylegh wavelets wth large values of n, where the Raylegh n wavelet s gven by w( f) = f exp[ n( f / f0)] and f 0 s the domnant frequency. If f 0 s set equal to the frequency of the monochromatc wave, then choosng an approprate set of n values should provde a brdge between monochromatc behavor and that of a typcal sesmc wavelet. Ths s llustrated n Fgure 1a, where the Raylegh wavelet spectrum approaches a spke as n. In Fgure 1b we see a correspondng progresson n the sphercal-wave reflecton coeffcent curves. The monochromatc curve s hghly oscllatory just past the crtcal angle whle the band-lmted soluton approaches the plane-wave asymptote much more smoothly. Table 1. Two-layer, elastc nterface model employed n calculatons Densty (kg/m 3 ) P-wave velocty (m/s) S-wave velocty (m/s) Layer Layer Let t Flow 2007 CSPG CSEG Conventon 54

4 a) b) Fgure 1. (a) The spectra of three Raylegh wavelets. As n ncreases, the spectrum becomes spke-lke, so that ts reflectvty behavor should approach that of a monochromatc wave-let. Fgure 1. (b) The sphercal-wave reflecton coeffcent curves for the wavelets n (a), and the correspondng plane-wave curve and mono-chromatc sphercalwave curves. To understand these dfferences we consder the weghtng functons that gve rse to the above reflecton coeffcents. Fgure 2 dsplays weghtng functons for the three Raylegh wavelets and the monochromatc wavelet for θ = 40. The n = 15 and n = 50 wavelets do ndeed form ntermedates. They decay away from θ = θ, as does the n = 5 wavelet, but ther decay s slower and more oscllatory, approachng the behavor of the monochromatc weghtng functon. a) b) c) d) θ Fgure 2. The weghtng functons defned at = 40 for the wavelets n Fgure 1a. Note that the tals of the Raylegh wavelet weghtng functons [a)-c)] become ncreasngly oscllatory as n grows, thus approachng the appearance of the monochromatc wavelet weghtng functon n d). Thus framng calculatons n terms of weghtng functons, as n equatons 1 and 3, provdes nsght nto sphercal-wave calculatons for dfferent types of wavelets. Let t Flow 2007 CSPG CSEG Conventon 55

5 Summary and Comments Monochromatc sphercal-wave reflecton coeffcent calculatons have been re-expressed n terms of a weghtng functon. Ths weghtng functon depends explctly on only three varables: angle of ncdence, an ntegraton varable, and a sphercty parameter. The latter subsumes frequency, overburden velocty and depth. The weghtng functon s analytc and may be readly programmed n terms of these three varables. A straghtforward 1-D numercal ntegraton then yelds the normalzed reflecton coeffcent. Calculatons wth the method have shown that weghtng functons for monochromatc wavelets are non-decayng and hghly oscllatory. Comparng them to a seres of weghtng functons and reflecton coeffcent curves for ncreasngly narrow Raylegh wavelets suggests that the less smooth the wavelet spectrum s, the more oscllatory the weghtng functon wll be, and ths wll result n oscllatons n the reflecton coeffcent curve as well. Acknowledgments The authors wsh to thank the CREWES Sponsors for support of ths research. References Ak, K., and Rchards, P. G., 1980, Quanttatve Sesmology, Vol. 1: W. H. Freeman. Haase, A. B., 2004, Sphercal wave AVO modelng of converted waves n elastc sotropc meda: 2004 CSEG Natonal Conventon. Hubral, P., and Tygel, M., 1989, Analyss of the Raylegh pulse (short note): Geophyscs, 54, Lamb, H., 1904, On the propagaton of tremors over the surface of an elastc sold: Phl. Trans. Roy. Soc. (London) A, 203, Sommerfeld, A., 1909, Über de Ausbretung der Wellen n der drahtlosen Telegraphe: Ann. Physk, 28, Ursenbach, C. P., Haase, A. B., and Downton, J. E., 2005, An effcent method for calculatng sphercal-wave reflecton coeffcents: 75 th Annual Meetng, SEG, Expanded Abstracts. Let t Flow 2007 CSPG CSEG Conventon 56

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