Higher order theory for analytic saddle point approximations to the Ρ Ρ and Ρ Ś reflected arrivals at a solid/solid interface

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1 Higher order theory for analytic saddle oint aroximations to the Ρ Ρ and Ρ Ś reflected arrivals at a solid/solid interface P.F Daley ABSTACT The high frequency solution to the roblem of a Ρ P and Ρ S reflected wave at a lane interface between two isotroic homogeneous halfsaces at near grazing incidence is develoed. In reflection seismology sub-critical reflections are often all that is required in data rocessing. A distorted (comared to the zero order geometrical otics aroximation to the arrival wavelet is seen on the traces in the area of grazing incidence. For a roer velocity distribution, there may an area in the vicinity of the critical oint which also requires secial treatment. This results in the zero order geometrical otics aroximation being sulemented by a term involving Parabolic Cylinder Function of the order. Grazing incidence may occur in cross hole tomograhy or for a oint source in a thin surface layer with a large velocity contrast at the interface with the subsequent layer. INTODUCTION Ρ P reflected wave, Červený and avindra (97 and Brekhovskikh (98. In each of these works a different conformal maing is introduced so as to incororate a real arameter in the solutions by high frequency methods involving the evaluation of integrals of the Sommerfeld tye. After stating this, it should be noted that the motivation for considering the Ρ S reflected arrival is artly due to the fact that in the Ρ P reflected case the saddle oint location may be obtained analytically, while in the Ρ S case this must be done numerically. Another reason for considering this roblem tye is to introduce, in as simle a manner as ossible, software for dislaying the effects described above, which may be useful in the area of geohysical interretation. To obtain analytical exressions for wave tyes such as the reflected Ρ P and Ρ S a thorough understanding of the formalism required to obtain solutions for these roblems in the elastic case is an imortant and useful recondition. In the elastic case the saddle oints and branch oint lie on the real axis of the comlex -lane. Higher order aroximations are required when the sherical nature of the incident (P wave must be considered. This occurs for shallow (less than about /4 of a wavelength associated with the redominant frequency of the source wavelet oint sources of P wave. This may cause erroneous results at near vertical and at grazing incidence. What is considered here does not address those conditions of incidence but rather reflected arrivals that lie within a range of incident angles and as a result in an offset range that is considered in reflection seismology analysis. CEWES esearch eort Volume 5 (

2 Daley Although lane wave reflection coefficients are what are of interest here, some asects of sherically wave incidence must be introduced. Again, to kee the imlementation of the theory as simle as ossible, only additional arameters, beyond those for the comutation of lane wave reflection coefficients are required. These are the distance of both the source and receiver above the reflecting interface and some reference frequency, usually associated with the source wavelet being used in the acquisition of the seismic data being rocessed. THEOY Following the method resented in Aki and ichards (98 for the analysis of the reflected otential due to a oint source of P wave incidence at a free surface, the following resentation of the underlying theory will be followed for consistency. In the case investigated here two elastic medium are considered. The arameters describing the two media are the P wave velocities, α and α, the S wave velocities, β and β and the densities, ρ and ρ. A P wave source is located a ositive distance h above the interface and a receiver is a ositive distance z above the interface. The horizontal distance between the source and receiver is r. In terms of otentials the Ρ Ρ reflected arrival is given by e ( d r, hi, t = i H ( r ex i ( h z ( φ ω ω ω The horizontal and vertical comonents of dislacement are obtained from to yield and ( u, u ( φ, φ u = = ( r z r z e ( d r, i, = ω ( ω ex ω ( u r h t i H r i h z ( (4 e u r, z, t = ω H ωr ex iω h z d z For the Ρ S roblem, the otential is given by so that with ( e β d φ ( r, hi, t = iω H ωr ex iω h zη iωα (5 the resulting dislacement comonents are ( w, w φ, r ( r φ w = = (6 r z rz r r CEWES esearch eort Volume 5 (

3 and ( e β ηd wr ( r, z, t = ω H ωr ex iω h zη iωα (7 ( e β d wz( r, hi, t = iω H ωr ex iω h zη iωα (8 Fig.. Schematic of the incident rays due to reflection from a solid/solid interface between two elastic media in welded contact. Ρ Ś EFLECTED PATICLE DISPLACEMENT VECTO COMPONENTS Consider only the vertical comonent of Ρ Ś reflection, as all of the other three comonents of dislacement may be solved for in a similar fashion. ( e β d wz( r, hi, t = iω H ωr ex iω h zη iωα (9 e β ( (,, d wz r hi t = ω H ( ωr ex iω( h zη α ( Introducing the large order exansion for the Hankel function (Abramowitz and Stegun, 98. results in ( ω ( πω [ ω ] i r i 4 H r r 8i r e ω = π ( CEWES esearch eort Volume 5 (

4 Daley or exanding has e β iωr iπ 4 wz ( rh, i, t = ω e α 8iω r πω r ( ex iω h zη d ( which leads to iπ 4 β wz ( rh, i, t = ω e α πω r d ex i ω r h zη 8iω r r d ex i ω r h zη iπ 4 β wz( rh, i, t = ω e α πω r d ex i ω( r h zη 8iω r r d ex i ω( r h zη ( (4 Note that terms in r have been left under the integral sign in the first integral. It is taken outside in the second integral as it is already a first order term, when comared to the zero order term in the first integral (in the geometrical otics sense, as it is of the order of ( iω. Define the exonential term as η f = r h z (5 so that the saddle oint is given by the (numerical solution of 4 CEWES esearch eort Volume 5 (

5 and the second derivative, f h z h z f = r = = r (6 = η ˆ ˆ η is = = Here and h z h z f = = = η η. (7 ˆ ˆ = ( (, ( i, = α = = α = (8 i i i i i ( (, ( i,: j,4 η = β = = β = =. (9 i i j j i where a circumflex over a arameter indicates that it is evaluated at the saddle oint. It will be assumed that in the vicinity of the saddle oint K r h z = = η. ( At this oint it is convenient to introduce the change of variable in terms of the real quantity y as (Červený and avindra, 97 ( ( i 4 ye π ( y = < < ( (Figure. It might be more correct to make y dimensionless by the relacement y y. However, the resent definition is mathematically accetable. equired formulae from ( are ( d iπ 4 = e ( dy = ( iπ 4 d e dy d e d e dy dy iπ 4 iπ 4 = = (4 d d d d i dy d dy dy = = (5 CEWES esearch eort Volume 5 ( 5

6 Daley Fig.. Parameterized saddle oint contour given by equation ( in the first quadrant of the comlex -lane. The ortion of the contour Σε has been slightly modified to show that in the vicinity of the saddle oint the saddle oint contour asses through the saddle oint at an angle of π/4. The quantities ( and are the dislacement reflection coefficients secified in Aki and ichards (98 and given in Aendices A and B together with their first and second derivatives with resect to. In the integrals in equation (4, define so that with h z Q = = r η K (6 K the following may be obtained r h z = = η (7 6 CEWES esearch eort Volume 5 (

7 together with leads to and h z K = η (8 ( ( h z K = (9 η 5 5 K K = ( K or equivalently K K K K K K = ( With equation (6 it may be determined that K K K =. ( 4K K Q = K (6 and Q K = ( K K Q or equivalently K = K K ( K K K K K (4 CEWES esearch eort Volume 5 ( 7

8 Daley ( K K K K Q = K K (5 From this is follows, using equations ( and (5 with the above that and dq dy d dy = Q (6 or dq d d = Q Q dy dy dy (7 dq i = Q ( ˆ Q ( dy. (8 Exanding Q y in a Taylor series about = ( y = has dq ( ( ( (9 Q y Q Q y dy The second term in the above exansion will not contribute to the integral, leaving only the first and third terms. Thus equation (4 may be aroximated as z β iω r ( h zη d πωα dq iω r ( h zη ye dy iωt iπ 4 (,, ω w rht = e Q e A Taylor series exansion of f ( in terms of y becomes ( iω r ( h zη d Q d e 8iωr (4 8 CEWES esearch eort Volume 5 (

9 leading to iωf iωf ( f y = iωf ( y dy iω d ωa (4 and finally to h z h z f = = = ˆ ˆ η ˆ ˆ η = (4 where d h z ˆ f y = y a y ˆ = dy ˆ η iω ω ω (4 a = ˆ ˆ η h z ˆ. (44 The above exansion has taken the truncated form because f ( =. The following sequence of stes (equations (45 (49 for the determination of w( rht,, may then be taken z ω π τ β ωa y πωα i t i (,, ω wz r h t = e Q e dy dq dy ( Q 8iω r ωa y y e ωa y dy e dy (45 ω π τ β π πωα ωa i t i i (,, ω wz rht = e Q π 4 ωa ( dq dy Q π 8iωr ωa (46 CEWES esearch eort Volume 5 ( 9

10 Daley w z w w z ( rht,, ( rht,, z ( rht,, iπ iτ ωe β = a α iωe a K Q iωt iτ = iωe a K β α iωt iτ = β α ( ( Q ( dq ωa dy 8iω r dq K ωa dy 8iω r ( dq K ωa dy 8iω r (47 (48 (49 The third term is finite if K( is used to define r. Based on the above derivation, the Ρ Ś radial vector comonent of dislacement may be determined to be iωe βηˆ = iωt iωτ wr ( rzt,, K ( αa dq K ωa dy 8iω r (5 where h z r = ˆ ˆ η Using a similar derivation as was used to obtain (5, the horizontal comonent of Ρ Ś is obtained as (5 iωe βηˆ = iωt iωτ wr ( rzt,, K ( αa dq K ωa dy 8iω r (5 CEWES esearch eort Volume 5 (

11 Ρ Ρ EFLECTED PATICLE DISPLACEMENT VECTO COMPONENTS Consider now the case of Ρ Ṕ reflection at an interface between two elastic isotroic solid media. The exressions for the radial and vertical comonents of article u = u, u, are given by equations ( and (4. If the same method as dislacement, r z was used to obtain a modified saddle oint aroximation for the Ρ Ś case, the resultant exressions may be obtained as follows iωt iωτ e ur ( rzt,, = iω a K dq K ωa dy 8iω r = ˆ iωt iωτ uz ( rzt,, iωe ak ( dq K ωa dy 8iω r (5 (54 In this instance, some function values have different forms such as The derivative of f = r ( h z. (55 f with resect to is equal to zero at =, which defines the saddle oint. Also, for this roblem K and the quantities K and K has the form a r h z = = (56 follow with minimal derivation. The term ( h ˆ z ˆ =. (57 Thus all quantites required to evaluate equation (5 and (54, excet for the reflection coefficient ( and its first and second derivatives which may be found in the Aendices. a CEWES esearch eort Volume 5 (

12 Daley DISCUSSION AND CONCLUSIONS Plane wave reflection coefficients together with correction terms as a consequence of the saddle oint related to a secific reflected ray being close to grazing incidence for Ρ P and Ρ S lane wave reflections at an interface between two elastic media have been resented The addition of the correction term is similar, but not equivalent to obtaining higher order terms in the infinite asymtotic series which describes a given reflection or transmission between two elastic media. EFEENCES Abramowitz, M and Stegun, I.A., 98. Handbook of Mathematical Functions, Dover, New York. Aki, K. and ichards, P.G., 98. Quantitative Seismology, Freeman, San Francisco, CA. Brekhovskikh, L.M., 98, Waves in layered media, Academic Press. Červený, V. and avindra., 97. Theory of seismic head waves, University of Toronto Press. AENDIX A: DISPLACEMENT EFLECTION COEFFICIENTS Ρ Ṕ AND Ρ Ś AND THEI FIST AND SECOND DEIVATIVES WITH ESPECT TO HOIZONTAL SLOWNESS Two elastic media are assumed to be in welded contact. The arameters describing the two media are the wave α j =,, the S wave velocities, P velocities, j β ( j =, and the densities ( j, j lower as "". Ρ P Coefficient: ρ =. The uer medium is denoted as "" and the j 98 The Ρ Ρ reflection coefficient,, may be written as (Aki and ichards, Ρ S Coefficient: EF GH = D The lane wave article dislacement Ρ Ś reflection coefficient, interface between two elastic media has the form (Aki and ichards, 98 (A., at an ( α ( α ab cdη ab cdη = = βd β D (A. α U = β D (A. D = EF GH (A.4 CEWES esearch eort Volume 5 (

13 α U UD = β D D (A.5 ( U D UD U ( D α U = β D D D ( η (A.6 U = ab cd (A.7 ( ( η ( a b ab c dη cdη cdη U = ab cd ( ( η ( a b ab c dη cd ( η η a b a b ab c dη c d ( η η cd ( η η η U = ab cd (A.8 (A.9 D = EF GH (A. D = E F EF G H GH GH (A. D = E F E F EF ( ( G H G H GH 4 G H GH GH EF GH = D (A. (A. U = (A.4 D (A.5 U = EF GH D = EF GH (A.6 U UD = D D (A.7 ( U D UD U ( D U = D D D (A.8 U = E F EF G H GH GH (A.9 U = E F E F EF ( ( G H G H GH 4 G H GH GH (A. CEWES esearch eort Volume 5 (

14 Daley AENDIX B: EQUIED QUANTITIES FO AENDIX A The quantities requiring definition in the above reflection coefficients and their first and second derivatives with resect to are given by: (, i,. ( α = = = (B. i i i i = (, i =,. (B. i ( =, i =,. i i i i (, i,. j,4. ( β i j j j (B. η = = = = (B.4 η = ( η, i =,. i ( η η =, i =,. j =,4. i j i E = b c i E = b b c c E = b b b c c c E% = b c E% = b b c c E % = b b b c c c F = bη cη F = b η bη c η cη (B.5 (B.6 (B.7 (B.8 (B.9 (B. (B. (B. (B. (B.4 F = b η b η bη c η c η cη (B.5 G = a dη G = a dη dη (B.6 (B.7 G = a dη dη dη (B.8 G = a dη G% = a dη dη (B.9 (B. G% = a dη dη dη dη (B. H = a dη (B. 4 CEWES esearch eort Volume 5 (

15 H = a dη dη (B. H = a dη dη dη (B.4 with ( ( a= ρ β ρ β ( a = 4 ρβ ρβ ( a = 4 ρβ ρβ b= ρ β ρβ 4 ( ρβ ρβ 4( ρβ ρβ ( (B.5 (B.6 (B.7 (B.8 b = = a (B.9 b = = a (B. c= ρ β ρβ ( ρβ ρβ ( ρβ ρβ ( (B. c = 4 = a (B. c = 4 = a (B. d = ρβ ρβ (B.4 d = (B.5 d = (B.6 a = b = c (B.7 a = b = c (B.8 CEWES esearch eort Volume 5 ( 5

Conversion coefficients at a liquid/solid interface

Conversion coefficients at a liquid/solid interface P.F. aley Conversion coefficients ABSTACT When upward-propagating rays transporting seismic energy are recorded at the earth s surface, the vertical