(1977 BSSA Daley)Reflection and transmission coefficients for transversely isotropic media

Bulletin of the Seismological Society of America. Vol. 67, No. 3, pp. 661-675. June 1977 REFLECTION AND TRANSMISSION COEFFICIENTS TRANSVERSELY ISOTROPIC MEDIA BY P. F. DALEY AND F. HRON FOR ABSTRACT It has become necessary in seismology to consider more complicated models of the Earth's structure in order to obtain synthetic seismograms that are more con- sistent with actual field data. Gassmann (1964) and Postma (1955) have pre- sented results dealing with travel-time methods in anisotropic media--in par- ticular, transversely isotropic media. Kinematic properties alone, however, are not enough to conclusively interpret seismic records. Consequently, dynamic prop- erties must be considered producing a need for synthetic seismograms. One of the most efficient methods for obtaining synthetic seismograms is through the use of asymptotic ray theory (Hron and Kanasewich, 1971 ; Hron, 1973; Hron, Kanasewich and Alpaslan, 1974). A necessary step in the implementation for layered media displaying transverse isotropy is the computation of reflection and transmission coefficients at the interface between two such layers. Reflection co- efficients for a free interface and the corresponding surface conversion coeffi- cients must be computed, as well. Theoretical formulas for reflection, transmission, and surface conversion co- efficients corresponding to the zero-order approximation of asymptotic theory are presented for the above-mentioned media. INTRODUCTION Asymptotic ray theory applied to the most general case of an anisotropic medium has been presented by Babich (1961), Cerveny (1972) and Daley (1976). In these papers, it is shown that substitution of the assumed solution into the equations of particle motion yields an eigenvalue problem. The resultant eigenvalues are the nor- mal velocities of the three types of wave fronts propagating in the medium, namely, a quasi-compressional P wave front and two quasi-shear SV and SH wave fronts. The coresponding eigenvectors give the preferred direction of the displacement vectors associated with each of the three wave-front types. In an isotropic medium, the direction of the compressional P eigenvector is tangent to the wave-front normal and the shear SV and SH eigenvectors are normal and bi- normal to the P eigenvector. The three eigenvectors form an orthogonal system in an anisotropic medium, but as was shown by Musgrave (1961), the P eigenvector is not in general tangent to the wave-front normal, hence the prefix "quasi". Cerveny and Psencik (1972) present expressions for the normal velocities associ- ated with compressional and shear (SV) wave fronts in a transversely isotropic medium. Using these velocities, the normalized eigenvectors are easily calculated. For the special case of transverse isotropy only the quasi-compressional P and quasi-shear SV wave fronts are coupled, i.e., a P or SV wave front impinging on an interface produces only reflected and transmitted P- and SV-type wave fronts. Con- sequently, the problem of reflection and transmission can be treated in two dimen- sions. As this problem has been solved in great detail by Daley (1976), only the necessary basic relations will be given here. The notation throughout this paper is identical to that used in the above-mentioned work. 661 662 P .F . DALEY AND F. I-IRON I~EFLECTION AND TRANSMISSION COEFFICIENTS AT AN INTERFACE BETWEEN Two TRANSVERSELY ISOTROPIC LAYERS 1. Theoretical background. In this section a general set of linear equations using asymptotic ray theory to calculate reflection and transmission coefficients will be developed. The first case to be considered is that of a plane interface of the first order between two transversely isotropic homogeneous layers in welded contact. To simplify further, it will be assumed that the axis of anisotropy in both media are perpendicular to the interface. Only the eight coefficients corresponding to incidence from the upper layer (medium 1) will be formally derived here. The remaining eight coefficients resulting from in- cidence from the lower layer (medium 2) can be obtained using the first eight and interchanging the layers and redefining the z (vertical) axis. In our notation, the displacement vectors arriving at or leaving a boundary are denoted by a ray series e~(t_r~ ) u. (x , y, z, t) = U . . (x , y, z) • .=0 (/~)- where the subscript v specified the type of wave front and the medium. The frequency is given by ~, and r is a phase function describing the wave front. All relevant displacement vectors are assigned a suitable integral value of v so that the amplitude terms in the corresponding ray series have the following meanings. v = 0 incident wave (either P or SV) U,0 = P,0 or Sn0 v = 1 reflected P wave U.I = P.1 v --- 2 transmitted P wave U~2 = P.~ v = 3 reflected SV wave Un~ = S~a v = 4 transmitted SV wave U~ = S~4. (1) In general, for a transversely isotropic medium, the Un~ are not purely in the direc- tion of U~0 but can be expressed as a combination of two scalar amplitudes and the two unit eigenvectors (g(1) and g~(~)) (Appendix A and Figure 1). The U~ can be expressed as follows pn~ = p~.g (1) + p~g (2) (2) S~, = S~g, (1) + s~g, (~). (3) As (Or,/Ox) = (sin 8,/V,) and Shell's law is valid Also 0to _ Or, _ sin 8o _ sin 8, Ox Ox Vo V~ (4) 0r~ 1)~+1 cos G (5) Oz - ( - V, COEFFICIENTS FOR TRANSVERSELY ISOTROPIC MEDIA 663 where 0~ are the acute angles between the z axis and the v th wave-front normal, and V~ are the normal velocities. 2. Incident quasi-compressional P wave front. As shown in Figure 1, the solid lines indicate normals to P4ype wave fronts, while the dashed lines define SV-type wave- front normals. This notation will be used throughout the paper. The eigenvectors So, °~ D 2 ' , ;4 '~ ,o\;o ( ] ) S3 ..~" P, X (2) -% s~ FIG. 1 Geometry of incidence for wave-front normals and displacement vectors at the interface of two transversely isotropic layers. g(i) (Appendix A) are given in terms of components in our Cartesian system as g(1) = (l. sin 0~, (--1)v+l•, cos 0,) g(2) = (m, cos 0~, (--1)*L sin 0~). (6) v specifies wave-front type and medium J - ~ , , + ~:>>/~: o.1 + Ia~ ~ + ~, , - ~ ,~ ;, l~= [(Q(~) "<~) A(,) ,~(v);~ 1/2 \ 2Q(') f ( "(')'/e ' °'] + ""> A('> °~"'I}I~' _ [(Q('> - A~I > + As~ ) , O~.Q<,) L-',~ + ~3, - ~ my Q(,) {<~(') 2Al(') As(") 0,} 1/s = I,.,~aa -- As(~)) s + sin O, + sin s 664 P. F. DALEY AND F. HRON Al(‘) = 2(&’ + A:;‘)’ - (A:? - A:;‘)(Ai;’ + A:;’ - 2A::‘) AZ(‘) = (A;:’ + Ai;’ - 2A::‘)’ - 4(A$’ + Ad;‘)2 Ai;’ specifies elastic parameters Ci;‘, of the medium divided by the density p(‘) of the medium. The superscript ‘9.~” in this case specifies the medium only, as will become clear later. The conditions which must hold at the interface between two layers in welded contact are the continuity of the z and z components of displacement, and the con- tinuity of normal and shear stresses. Introducing Kronekers symbol (&j: = 0, i f j; = 1, i = j) , the continuity of the r component of displacement yields & ( - l)“P,,Z, sin 0, + 2 ( - 1) W~namu co8 & = Pa020 sin 00 - & ( - l)“&,Z, sin & - & ( - 1)6~o+v~.,m, COB ev. (7) Continuity of the x component of displacement has 2 4 P,,m, cos ey - f: s,Jv sin ey = P,om0 COB e. - a& S,,m, cos 8, II=3 + Vi ( - 1) *vop,vZv sin eu. (8) Continuity of shear stress requires that g$ (-l>“u.Z(u,) = uzz(uo) where ( a(%>, u*,(u,> = c:c -I$- + a(%>8 - dX > and C!” z3 = p(w),j;). The superscript on CS5 identifies the medium. At the interface C$’ = d:’ = d? and C:“,’ = C$. After some algebra the following expression results 5 &'(I, + m,) sin 2&P,, + k C%'(m, cos2 8, - I, sin” e,)s,, w=l 2-v, 0=3 v, = CF(~o cos' Oo - lo sin 20o)s.o 29o _ y] J~.',, (,,,, cos' 0. -- 1, sin' O.)p,, =1., .=,~ V. + (-1)'C~> L ~ + - - + ~ { (-1)l+a'°C~)(/'2V~ + m,)s in 20.S,. V~0 -F ( - l ) '+"°C~ > [O(So)z-"~-F O.(So~_,,,]) , (13) Continuity of normal stress lv ( J l~ ~Tlv~'38 - - ~413 ] (_1)~+1 + l~ cos' O~ p. , fdv) r~(v) ~ ~ ~(0) f~(O) ,~ -- (lo O3~ ~o,~x3 ) ~a8 m~Wl8 ) sin 20~sn~ = + (--1)" (l~ sin 20os,0 ~=3 2V~ 2~o { 0(P~),-1.~ (1, C~ ) _t,(~), } v=l OX Oz 2 V~ ~=4 f O(&)._1,, _ ~ (_1) .+8,0 C~)0(S~)._1,, + C(8~) ~=n ~ OX OZ V=O - - ~/r: b, ~,-'18 ) COS' 0~ S, (14) where ~0 denotes normal velocity of incident SV wave front. The equations (7), (8), (9), (10) and (11), (12), (13), (14) are two recursive sets of four linear equations in four unknowns. To know the solution of these two sets for a given n, the solutions for all m < n must be known. If the geometrical parameters of the layers are much greater than the wavelength, and if we are not very close to the source, taking only the first term in the ray series (n = 0) can be justified to give a reasonable approximation. This so called zero-order or plane-wave approximation will be considered. It should be noted that for an incident P-type wave front 00 - 01, and for an in- cident SV-type wave front 00 = 08. Let sin 01 = x. All other pertinent angles will be parameterized in this variable via Shell's Law. sin 0o] Vo / sin O, sin Oo l - V, v = 1, 2, 3, 4. 1 R J Using this, the following simplifying substitutions will be carried out Vt V4 k2, V-23 = kl cos 01 = P = (1- - x 2) 1/2 COEFF IC IENTS FOR TRANSVERSELY ISOTROPIC MEDIA 667 cos 0~ = Q = (1 - k12X 2) 1/2 ( x:Y cos 03 = S= 1 - n2 ] cos 04 = R = 1 - n2 ] I l l v = 0, 1, 3. C~;) = [~ v = 2, 4. l ,~(3) C(3) 3~,F33 - - ~ft3 13 ~ 61 ,~(4) ~(4) 14u~3 62 - - Tn 4L, 13 ~-~ V1 1 {m3 cos ~ 0~ - 18 sin 2 03} = ~1 V3 (l~ + ml) Vi 1 V4 (ll + m0 {m4 COS 2 04 - - 14 sin 2 04} = ~2 ~( i ) . ,--,(1) ? /~(1)-~ = /1(-"13 -~- [Tf/1U33 -- blWl3 ) COS 2 ~i el V1 ~(2) / f~(2) ~(2) - - {/2(J~13 ~ (Tf l2( J33 - - /2(.~'13 ] Cos 2 02} ~- e2 V~ g2 "-[- m2 - / ll -~- mi After implementing the substitutions in the zero-order equations, the two following sets of four linear equations in four unknowns result. Incident P-type wave front ] ~//13 klX ~"l_..2 .~ --~,4~2X J /'~03 / ____. 7 ~i xP ~i ~i ~2 l xS f32 ~2 [ L -- el 6~ xQ E2 - 62 xR J (15) Incident SV-type wave front I ~n3 x ~Q P - 1---2 kl x ml -- e ~i xQ --12 x -- m4 i:t "~ -hk~x m__2 z ml ml n ~ lxS ~2 us e2 - - ~2 xR -P01] ,b'03 P02 I L s04 d " ~ m3 -v-Q -13 ki x mi - - ~1 xQ s00. (1 6) 668 P. F. DALEY AND F. HRON Solving (15) and (16) using Cramer's method with the incident displacement set to unity, the reflection and transmission coefficient R,, tabulated below are obtained. (According to our convention, ~ = 1 corresponds to an incident P wave front and # = 3 corresponds to an incident SV wave front, while the second subscript v typifies the resultant reflected or transmitted wave front.) Rn = (--E~ + E= + Ea + E4 -- E~ -- E6)/D R~. --- (E~ + Es)/D R~8 = (E9 + Elo)/D R14 -- (En -~- EI~)/D Rs1 = ( Eu -~" E14) /D R32 = (El6 + E,6)/D Raa = ( -E l+E2- Ea + E4 + Es - E6)/D Ra4 = (E~ -t- E~s) /D D = E I+E=+Ea+E4- I -E~+E6. (:7) g0 (1) = (lo sin 00, mo cos 00) go (s) = (--rno cos 0o, lo sin 00) g(1) = (12 sin 02, --ms cos 0~.) g=(~) = (--ms cos 0,, --12 sin 02) The requirement for continuity of shear stress yields the following two equations f (l= + m=) sin 20=P~= + \ 2V~ [{ O(k~X) r,--1,4~ ,-,(4) O( &),,-1,4 + f ( l, + ~) sin 204S~4 + \ 2V4 g~(1) = (lo sin ~o, no cos ~o) g~(2) = ( _no cos ~o, lo sin ~o) g4 m = (14 sin 04, --rn4 cos 04) g4 (2) = (--m4 cos 04, --14 sin 04). ~=(u0) + ~,(u2) + ~o(u~) = 0 c°s= z= sin= (m4 cos 2 ~4--14 sin s 04) s~4~'] /J The Ei , j = 1, 18 are given explicitly in Appendix B. FREE INTERFACE The free-interface case has a wave front (either quasi-shear (SV) or quasi-com- pressional (P)) or propagating in a transversely isotropic medium and impinging on the interface of that medium and a vacuum. Reflected shear (SV) and compres- sional waves only result. The boundary requirements are the continuity of shear and normal stresses along with the previously listed conditions on the first derivatives of r, (see Figure 2). COEFFICIENTS FOR TRANSVERSELY ISOTROPIC MEDIA 669 where r~(0) F = ~55 O( P,) ~-~,o~ +{ (l°-Fm°)sin2e°P*°Vo -- (m°c°s=O°--l°sin2O°)Vo P"°}l ( is) vacuum gl ~ e~ g2 ~4- , ,~ '~ , ' o o J Oo "'--~ t / % / O ttt J S o }. X • 2 $4 FIe. 2. Geometry of incidence at a free interface. in the case of an incident quasi-compressional P wave front and F = F /(o(s,)._l,O +{ (/° + ~°) sin 20° S 'o2V~ -- (~° c°s2 0° - /° sin2 O°)V=o S.o}] (19) in the case of an incident quasi-shear SV wave front. Continuity of normal stress zz,(uo) + zz,(u2) + z~(u4) = 0 g ives rise to 670 P.F. DALEY AND F. HRON G = [_{C~]) O( P.).-z,2 ~(s) O( P.)._,,2~ Ox + ,~88 Oz ) { . . . (2 ) . .~(s ) , 02)P.2+ ,}] + (12C~) + ~msC'~ - t2c'1~ ~ cos s (12 ~(s) C (2)~ ,~a~ -- m2 ~a~ sin202p. Vs 2Vs /C 4) O(k'~x)n--l'4 J F C(4)a(S~zn-l'4 ) x~ Ox { ,m4b~, 4 z,) cos s k.t,~aa -- m4b~a)s in 204Sn4}l + (/4C~ 4) -1- " "~(*) - l C (~)~ 04)S., [7 f1(4) ~(4)'~ V~ + 2V~ where again (o) p a = L~ a-x + ~Z- ) _ ~0(~7 C (o)x3 + (m0" wz3"(°) -- ~ouzsz ~(0)~j cos s Oo)P~o V0 rz t~(o) C(0) 0}] _[_ ~o,~s3 -- mo la sin 20op. (20) 2Vo corresponds to an incident P wave front and G = Ff (S•)n--,,0 C,3 (Sz)n--l,0~ Li. ax + -~ ) t ~ (0) ,~ ,,-~(o) _ (~o e l . + ~mo t,.~ - ~o C~ °)) cos s 0o) S,o ¢o ~7 t~(0) ,,(0)~ .)7 + ~o~a -- ~ot-,z3 j sin 20os. ?J 2Vo (21) corresponds to an incident SV wave front. Equations (18), (19), and (20), (21) are two sets of two recursive linear equations in two unknowns whose solutions give the required reflection coefficients. In the zero order approximation these systems become (22) and (23) after the following simplifying substitutions. V4 x= sin02 kz=- - V2 COS 0s = P cos 04 = Q 7 ~(4) ,-~(4) V~ 1 V, (ts + ms) {m4 cos 2 04 - l, sin 2 04} 7 ~(2) C (s) 7/~(s)~ ~s = ~",,~za ~ (m2 3~ - - ,,',~za j cos 20s . Incident P wave front Poo • ~z azxQ_J L so4_] - ,1 (22) COEFFICIENTS FOR TRANSVERSELY ISOTROPIC MEDIA 671 Incident SV wave front -~- 800 • ~1 ~lx s4 - ~lxQ (23) Solving (22) and (23) yields the following coefficients of reflection. Ro2 = (~ lx2PQ- ~1~I)/H Ro4 = - 2e lxP /H R~4 = ( -~x2PQ + el~s)/H R52 = - 2~I~lxQ/ H H = ~lx~PQ ~- e1¢ol (24) SURFACE CONVERSION COEFFICIENTS A receiver situated on the Earth's surface, i.e., at the interface of a transversely isotropic medium and a vacuum, records not only the disturbance caused by an incident wave front at that point, but also the disturbances resulting from the two reflected wave fronts. Let ~ denote the interface, G the location of the receiver at the interface and u~ ~ (G) the displacement registered by the receiver. (v = 0 or 0 depending on whether the incident wave front is of the P or SV type.) Thus u~(G) = u~(G) ~- u~2(G) T u~4(G) (25) where the first term on the right-hand side of (25) is the contribution from the in- cident wave front and the 'second and third terms are contributions from the reflected P and SV wave fronts, respectively. In the zero-order approximation in asymptotic ray theory an arbitrary displace- ment vector can be defined by u~(G) = A~(G) expi~(t - f,(G))n~(G) n. (G) being a unit vector. As to(G) = r~(G) = T4(G) at the interface, for an incident P-wave front (25) becomes u0Z(G) = Poo(G) exp i~(t - ro(G)){n0(G) + Ro~n2(G) + R04n4(G)} (26) and similarly since r~(G) = r2(G) = T4(G), the resultant expression for an incident SV wave front is u0Z(G) = Soo(G) exp i~(t - rs(G))n~(G) ~- R52n~(G) ~- R~4n4(G)}. (27) The R~ are given by (24). 672 P. F. DALEY AND F. HRON Let g,(G) = n~(G) + R~2ns(G) + R~4n4(G). g~(G) is called the surface conversion vector and the x and z zomponents of g,(s), g,~, and g~, are called the surface conversion coefficients. From (24) it follows 2x(el"m4 -{- ~lx21~)PQ g0~ = H 2e1(~o1~2 -~- klx214)P g0~ = H 2o~1( ~1 m4 ~- ~1 X212) Q g~ = H 2~1x( ~1~2 + kl x214) PQ g~ = H (28) CONCLUSION The formulas for reflection and transmission coefficients have been tabulated for elastic waves impinging on an interface between two transversely isotropic layers and on an interface between a transversely isotropic layer and a vacuum. Surface conver- sion coefficients related to the free interface case have also been presented. Numerical results using the formulas presented will be given in a forthcoming paper which is currently under preparation. ACKNOWLEDGMENTS During the course of this work the authors have received financial support in the form of re- search grants provided by the Research Center of Amoco Production Company and the National Research Council of Canada. REFERENCES Babich, V. M. (1961). Ray method for the computation of the intensity of wave fronts in elastic inhomogeneous medium in Problems of the Dynamic Theory of Propagation of Seismic Waves, Vol. 5, Leningrad University Press, Leningrad (in Russian). Cerveny, V. (1972). Seismic rays and ray intensities in inhomogeneous anisotropic media, Geophys. J. 29, 1-13. Cerveny, V. and I. Pseneik (1972). Rays and travel time curves in inhomogeneous anisotropic media, Z. Geophysik 38,565-577. Daley, P. F. (1976). Reflection and transmission coefficients for transversely isotropic media (Masters Thesis), Department of Physics, University of Alberta, Edmonton. Gassmann, F. (1964). Introduction to seismic travel time methods in anisotropic media, Pure Appl. Geophys. (Milan), 58, 63-112. Hron, F. (1973). A numerical ray generation and its application to the computation of synthetic seismograms for complex layered media, Geophys. J. 35,345-349. Hron, F. and E. R. Kanasewich (1971). Synthetic seismograms for deep seismic sounding studies using asymptotic ray theory, Bull. Seism. Soc. Am. 61, 1169-1200. Hron, F., E. R. Kanasewich, and T. Alpaslan (1974). Partial ray expansion required to suitably approximate the exact wave solution, Geophys. J. 36,607-625. Musgrave, M. J. P. (1961). Elastic waves in anisotropic media, Progr. Solid. Mech. 2, 64-85. Postma, G. W. (195