Anelatic media Reflected PP arrival in anelatic media P.F. Daley and E.S. Krebe ABSTRACT A homogeneou wave incident on an interface between two anelatic halfpace in welded contact i conidered. In the anelatic ene, a homogeneou wave i defined by the condition that the propagation and attenuation vector are collinear. It ha been indicated in a number of paper over the pat everal decade that the proper definition of the real and imaginary part of the vertical component of the lowne vector in the reflection coefficient are not obviou for ome ditribution of the quality factor, Q. Thi can reult in anomalou behaviour of both or either of the amplitude and phae of the PP reflection coefficient when diplayed veru the incident propagation angle or equivalently the real part of the horizontal component of the incident lowne vector. In an earlier work (Krebe and Daley, 007) the quetion of anomalie in the amplitude and phae of the PP plane wave reflection coefficient for thee ditribution of the quality factor Q in adjacent anelatic halfpace wa dicued in coniderable detail. In what follow, the above paper (Paper ) will be referred to often to minimize repetition of previou dicuion. The problem of the PP reflection coefficient i addreed again here. Thi i done within the context of two elected approximate method, of varying complexity, which produce acceptable behaviour for the anomalou quantitie, from a numerical viewpoint. What caue thi behaviour in the PP reflection coefficient may be attributed, at leat in part, to improper ign being aigned to the real and imaginary part of the radical defining the tranmitted P wave vertical lowne vector component. However, thi may be looked upon a a ymptom rather than the actual caue of the problem. Conideration of the PP plane wave reflection coefficient i the firt matter dealt with and the dicuion i then extended to the high frequency geometrical optic olution of a Sommerfeld type integral, uing zero order addle point method for determining the particle diplacement vector of the reflected PP diturbance due to a P wave point ource incident at an interface eparating two anelatic media. One approximation of the addle point method wa preented in detail in Paper and another approximate approach wa uggeted and i expanded on here. The accuracy of approximation to the addle point method are etablihed through comparion with an "exact" (numerical integration) olution. INTRODUCTION A recent paper in the geophyical literature (Krebe and Daley, 007, (Paper )) contain a fairly detailed dicuion related to the problem of the anomalou behaviour of the PP particle diplacement reflection coefficient at an interface eparating two anelatic media, together with a urvey of the relevant literature publihed over the pat everal decade by a number of author. A thi wa preented in Paper intereted reader are referred there for a fairly thorough introduction to thi topic. One of the In what follow, the PP particle diplacement reflection coefficient will be referred to imply a the PP reflection coefficient a it i thi quantity that i ued throughout thi work. CREWES Reearch Report Volume (009)
Daley and Krebe objective of thi work i to determine what i required, from a high frequency (geometrical optic) olution perpective, to produce acceptable accuracy in numerical modeling technique. A the addle point method i computationally fat, when compared to more exact olution, it i the leading candidate for ue in obtaining inight into problem related to wave propagation in geological tructure diplaying anelaticity. It ha been etablihed that for PP reflection the quantity cauing the anomalou behaviour i the vertical component of the lowne vector aociated with the tranmitted P wavefront. Thi parameter appear a a radical in the PP reflection coefficient but doe not appear in an exponential term in the Sommerfeld type integral, o it i not included in the computation of the addle point The problem mentioned above occur in an expected region, where the addle point i near a branch point. (However, there i no indication that the total range of incidence i not affected in ome manner.) The zero order addle point method i not valid in thi region a the reflected PP geometrical optic olution here hould be in term of a higher order addle point method approximation. For that reaon, a modification of the addle point approximation will be purued, whoe major effect i (een mot clearly but not necearily) limited to thi region. The intent i to obtain numerical reult that are in reaonable agreement with a numerical integration olution, at leat in the pre-critical region. In the pot-critical region the olution i required to be conitent with what would be een in a imilar type of problem for the elatic cae. A numerically correct remedy i ought for the anomalou behaviour, which may not be theoretically rigorou. In Paper, the topic of the plane wave SH reflection and tranmiion coefficient at an interface eparating two anelatic media wa dicued, a well a the PP reflection coefficient at a imilar boundary type. Only the PP reflection problem will be conidered here. It wa determined that the anomalou cae occurred where QP < Q P ( upper or incident halfpace, lower or halfpace of tranmiion) and that adjutment to the formal mathematical addle point olution were required to be introduced to enure phyically realitic reult. The plane wave PP reflection coefficient will be given a curory review and, apart from plane wave diimilaritie between the two approximate method conidered, addle point olution will be compared with an exact numerical integration approach. One of thee, a major topic of Paper, will be commented on here together with the tandard addle point method, for which it ha been hown in Paper that an additional aumption mut be made at the reflection coefficient tage to avoid erroneou reult. Thi reult in the introduction of a modification, which force the reflection coefficient to diplay a behaviour that could be conidered conitent with the elatic cae, and will be dicued in more detail in a later ection. Each of the anelatic halfpace ( upper, lower) are parametrically defined by a real P wave velocity, V P, a real S wave velocity, V S, a denity, ρ, and two quality factor related to P and S mode of wave propagation, Q P and Q S. The complex P and S velocitie may be defined in term of Q, where Q may or may not be a function of frequency, a v ( V )( iq) v = α : [ P Compreional] or v β : [ S Shear] = +. Here v are the complex velocitie, =. In lowne pace, the following CREWES Reearch Report Volume (009)
Anelatic media quantitie are required: p α =, p α =, p3 β = and p4 = β, with "" referring to the upper halfpace and "", the lower. It hould be made clear that the dicuion preented here i under the aumption that QP > Q P with Q P and Q P. The quantitie Q S and Q S are both choen to be greater than Q P. The method of implementing a frequency dependent Q i taken from the paper of Futterman (96) and Azimi et al. (968). Thi variation will be ued in the computation of ynthetic trace, and will be formally introduced a later. SADDLE POINT METHOD THEORY Conider an interface between two anelatic halfpace where z = 0 define the interface with the z axi choen to be poitive downward. A P wave point ource i located at r = 0 at a ditance z0 above the interface in a cylindrical coordinate ytem and a receiver i imilarly poitioned at z above the interface at an offet of r. Thu, both the ource and receiver are located in the upper () halfpace, eparated by a horizontal ditance of r. The lower halfpace i deignated a (). The reflected PP potential from the interface may be written a iω p φ( r, z, ω) = F ( ) PP ( p) J0 ( pr) exp i Tˆ ω ω ω ( p) dp 4πρα ξ () 0 Aki and Richard (980, 00) or equivalently a ( ω) iω F ( ( ) ( ) ) p φ r, z, ω = PP p H 0 ( pr) exp i Tˆ ω ω ( p) dp 4πρα ξ () (Abramowitz and Stegun, 980), where F ( ω ) i the Fourier time tranform of what will aumed to be a band limited ource wavelet, indicated a f ( t ) in the time domain. ( J0 ( ζ ) i the Beel function of zero order, H ) 0 ( ζ ) i the Hankel function of type one and order zero, PP ( p) i the PP reflection coefficient at an interface between two anelatic olid (Aki and Richard, 980, 00) and ( ) ξ 0 Tˆ p = z+ z. (3) The radical ξ j and η j are the vertical component of the lowne vector, and may be defined in term of the complex velocitie α j and β j (or alternatively in term of the related point, p j, in the complex p plane ) and the integration variable p, the generally complex horizontal component of the lowne vector, a and / / ( α p ) ( p p ) ( j, ) ξ = = = (4) j j j CREWES Reearch Report Volume (009) 3
Daley and Krebe ( / / p ) ( p p ) ( j ) η = β = + =,. (5) j j j The radical η ( j =, ) appear only in the reflection coefficient ( ) j PP p, a doe ξ. The horizontal and vertical component of the particle diplacement vector may be obtained from the reflected PP potential, equation (), uing the formula (Aki and Richard, 980, 00) φ φ u = ( ur, uz) =, r z which lead to the integral expreion (6) (,, ) ( ) φ rzω iω Fω (,, ) ( ) ( ) exp ˆ p ur r z ω = = PP p J ωpr iωt ( p) dp r 4πρ α (7) ( rz,, ) F( ) ξ 0 φ ω ω ω (,, ) u ( ) ( ) exp ˆ z r z ω PP p J ωpr iωt ( p) pdp. (8) = = 0 z 4πρα 0 Equivalent expreion in term of Hankel function of type one and order zero and one follow from equation (). Only the vertical component of reflected PP particle diplacement will be dealt with here a the horizontal component i imilar. After introducing the zero order Hankel function H ω pr, a in equation (), and retaining jut the firt term in it aymptotic ( ) ( ) 0 expanion for large argument, the high frequency or geometrical optic formula for the vertical component of reflected PP particle diplacement (equation (8)) may be written a where F ( ω) e 3 iπ 4 ω uz ( r, z, ω) = PP ( p) exp iωt ( p) p dp 4πρα π r (9) The addle point, ( ) ξ 0 T p = pr + z + z. (0) p i given by the olution of ( p) p z+ z0 p z+ z0 dt = r = r = 0, () dp ξ ξ p= p p= p where a tilde above any quantity indicate that it i to be evaluated at the addle point, p = p. The notation ued in Paper ha been retained, o that a addle point in the complex p plane will be denoted a p, except if it lie on the real p axi, where the 4 CREWES Reearch Report Volume (009)
deignation p 0 will be ued. Expanding ( ) ha ( ) ( ) dt Anelatic media T p in a Taylor erie in the vicinity of p = p ( p) p= p ( ) T p T p + p p + () dp a d T ( p) dp 0 p = = define the location of the addle point. Further, p dt dp ( p) 0 = T ( p ) = ξ p= p z+ z p The definition τ ( p) = Re T( p) and ( p ) Im ( ) T p τ ( p ) and ( p ) repectively. Thu, i T( p). (3) κ = are introduced, where κ are the real travel time term and exponential attenuation term, ω ha a Taylor erie expanion of the form ( ) ( ) ( ) ( ) ( ) i ω T p i ωτ p ωκ p + i ω T p p p + (4) p= p In general, the addle point olution to the integral in equation (9) may be written a (Brekhovkikh, 980, Marcuvitz and Felen, 973) u z ( rz,, ω) ( ω) ( ) ( rt ( p )) ω F PP p p e = 4πρα iωτ ( p ) ωκ ( p ). (5) The alternate addle point approximation ued in Paper for thi pecific problem differ from the above a it derivation yield a real-valued addle point, p 0. It i dealt with quite comprehenively in that work and will not be preented here. The approximation to the addle point olution of the above tated problem, which will be ued in the numerical experiment mentioned previouly, will now be dicued in more detail. The bai for a numerical integration procedure ued to etablih the numerical accuracy of thee approximation i Equation (8). CREWES Reearch Report Volume (009) 5
Daley and Krebe FIG.. The PP reflection coefficient at the interface between the two anelatic halfpace decribed in Table. Approximation ( A ) (grey curve) i compared to the elatic cae (black curve). Anelaticity i introduced uing frequency independent complex velocitie. 6 CREWES Reearch Report Volume (009)
PP REFLECTION COEFFICIENT Anelatic media The anomalou addle point olution dealt with here i for a pecific ditribution of quality factor in the two anelatic halfpace: Q P and Q P with Q P > Q P (Figure ). The reflected PP particle diplacement in the upper halfpace, where the ource and receiver are located, ha been a longtanding problem. The motivation here i not to proceed with a rigorou mathematical analyi, but rather to obtain formulae of the geometrical optic type that may be ued for the numerical modeling of thi anomalou behaviour uch that the reult are in reaonable agreement with more accurate method of olution, uch a numerical integration. Thi may require taking ome mathematical libertie. However, thi will be deemed acceptable if the reult i the production of realitic numerical reult. The addle point approximation dicued in Paper will firt be briefly conidered. T p, with repect to p, when A i tandard convention, the derivative of the function ( ) et equal to zero pecifie the addle point, p. T( ) p, in thi firt cae, i eparated into real and imaginary part. The value of p 0 i obtained from the olution of { ( ) } p= p0 d Re T p dp = 0. A thi olution involve only real value, the addle point i a real valued quantity, located on the real poitive axi in the complex p plane. Subtituting thi value into the imaginary part of the exponential function T( p ) produce the attenuation factor expected for wave propagation in an anelatic medium. Thi modification of the typical addle point method reult in an improved behaviour of the ξ = p p = α p and a a conequence, of the PP reflection radical ( 0) ( 0) coefficient. A indicated earlier, a more detailed analyi of thi pecific procedure may be found in Paper. In what follow thi will be referred to a Approximation. The plot of the amplitude and phae veru the incident P wave angle for the PP reflection coefficient approximation are hown in Figure, together with the elatic cae. The parameter of the media are given in Table and are the ame a thoe ued in Paper. The definition of the manner in which frequency independent (hyteretic) value of Q j = P, P, S, S are introduced i given in the Introduction. The inert in the phae j ( ) panel give a clearer picture of the behaviour of the phae in the vicinity of the elatic cae critical point. From Aki and Richard (980, 00) it may be een that the numerical integration path for thi particular problem i along the real axi 0 < p <. Aume a cloed contour integral in the complex p ( ) p plane, and neglect the contribution from any pole contained within thi cloed contour. For a function of p, ay ( ) p ζ, which i analytic at all interior point (except the poible aforementioned pole), within thi imple cloed contour C, may be written a a conequence of Cauchy Theorem (Churchill and Brown, 003) a CREWES Reearch Report Volume (009) 7
Daley and Krebe FIG.. A chematic of the addle point and/or numerical integration path pecific to A. The addle point, p, lie on the line connecting the origin with the point Approximation ( ) p. The imaginary axi i caled ignificantly with repect to the real axi (black curve). The two numerical integration contour employed are indicated in the figure a C and C. Rr ( ) ( ) ( ) ( ) ζ p dp = ζ p dp + ζ p dp + ζ p dp = 0. (6) C 0 0 Rc Auming that certain radiation condition are atified for ζ ( p), uch that it tend to zero a R, ζ ( p) dp 0, leaving (Figure ) Rr Rc ( p) dp = ζ ( ) ζ p dp. (7) 0 0 The above equation how that the numerical integration along the two path produce identical reult. That part of the contour related to the integration along the real p axi will be denoted a C, while the other contour in equation (7) will be labelled C. It i known from earlier work that the addle point olution for thi problem i contrained to lie on the line joining the origin and the point p α = = p, where α and p were 8 CREWES Reearch Report Volume (009)
Anelatic media previouly defined. That the correct addle point path lie along thi path cannot be aumed from equation (7) becaue the addle point i in the vicinity of a branch point along at leat part of thi path. However, a dicrepancy of the type oberved in earlier numerical reult related to thi problem would indicate ome additional problem. FIG. 3. The PP reflection coefficient at the interface between the two anelatic halfpace decribed in Table. Approximation ( A ) (grey curve) i compared to the elatic cae (black curve). Anelaticity i introduced uing frequency independent complex velocitie. CREWES Reearch Report Volume (009) 9
Daley and Krebe It wa uggeted and demontrated by graphing the PP reflection coefficient in Paper that forcing the radical ξ ( ) = p p, in the PP reflection coefficient, to take on it complex conjugate value for all value of p along the addle point line in the complex p plane produce good numerical reult.. Thi will be referred to a Approximation. The plot of the amplitude and phae veru the P wave incident angle of the PP reflection coefficient for thi approximation a well a the elatic cae are preented in Figure 3. An inert in the phae panel provide a clearer picture of the behaviour of the phae in the vicinity of the elatic critical point. The value for the media parameter ued in the computation are the ame ued in the Approximation cae above. Before conidering the addle point ynthetic trace reult related to thee two approximation the equality of the two numerical integration contour hould be addreed. The model ued ha both the ource, and receiver line located at a ditance z = 000m above the interface between the two anelatic halfpace. If the elatic cae i taken a an guide, the critical ditance i at an offet of about 55m. The total offet range conidered i from r = 0m to r = 00m with the horizontal ditance between two adjacent receiver being 40m. In theory, integration along the real axi C i equal to the integration along the line from the origin of the p plane to the point p = p ( C ). However in practice, the p ( ) two integration are equal only if the complex conjugate of the quantity ξ i ued along p = to p = p. If thi i not done, the critically the econd integration path ( C ) from 0 refracted (head) wave diplay unphyical propertie in that it propagate backward in pace and time with apparently incorrect amplitude. Thi behaviour i hown in panel (a) of Figure 4. Panel (b) how the correct reult if it aumed that the integration along the real p axi ( C ) i the indicator of correct reult. It may be inferred from thi that forcing the radical ξ to diplay the ame behaviour in the PP reflection coefficient and in any ubequent zero addle point computation i jutified, if only from a numerical perpective. In computing the ynthetic trace by the numerical integration method the velocitie are aumed to be frequency dependent. Thi dependence i etablihed in the fahion decribed by Futterman (96) and i dicued in detail in Aki and Richard (980, 00). The introduction of an additional parameter for each halfpace, a reference 0 CREWES Reearch Report Volume (009)
Anelatic media FIG. 4. Integration along C in the complex p plane. Panel (a) how the erroneou reult obtained if the contraint that ξ i not replaced by it complex conjugate in the integration proce. In panel (b), the correct ynthetic numerical integration trace are diplayed. If the reult from integrating along C were overlaid in panel (b), the match would be almot exact. CREWES Reearch Report Volume (009)
Daley and Krebe frequency, f 0, of the Gabor ource wavelet - fr = f0 = 30Hz. The relevant formulae at ome other circular frequency, ω, in term of ω π f Q ω and V ( ω ) and complex value velocity ( ) o that which reult in R = of the real value ( ) R v ω are determined by the equence of relation ω Q( ω) = Q( ωr ) ln πq( ωr) ωr V ( ω) V ( ω ) ( ωr ) ( ω) (8) Q = R. (9) Q i = + = or v( ω) V( ω) Q( ω) ( v α β). (0) Equation (8)-(0) correpond to thoe which appear in Zahradník et al. (00). In the next ection zero order addle point ynthetic trace will be compared with thoe obtained from numerical integration for the anelatic model dicued above. ZERO ORDER SADDLE POINT APPROXIMATIONS Synthetic trace computed uing the two approximate zero order addle point method decribed above are hown next. The model ued i a decribed for the numerical integration ynthetic trace. The firt offet i at r = 0 with 30 further receiver placed at 40m interval o that the maximum offet i 00m. If the elatic parameter are ued the critical ditance i 55m which lie at about trace 9 in the common hot gather of 3 trace. The pot-critical offet were included to ee how the addle point approximation behave without the incluion of the critically refracted PPP (head) wave (Červený and Ravindra, 97) or any modification of the addle point approach to accommodate the range of offet where a higher order approximation hould be ued to compenate for a addle point in the vicinity of a branch point (Marcuvitz and Felen, 973, and Červený and Ravindra, 97, a example). A mentioned earlier, a Gabor wavelet i ued with a predominant frequency f0 = 30Hz and a dimenionle damping factor γ = 4. The time ampling rate i m in all trace. The manner of introducing a frequency dependent Q into the addle point computation i the ame a that ued for the numerical integration trace decribed in the previou ection. Gabor wavelet: f( t) = in ( π ft) exp ( π ft γ) 0 0 where γ i a dimenionle damping that control the amplitude pectrum width in the frequency domain and the ide lobe of the wavelet in the time domain. CREWES Reearch Report Volume (009)
Anelatic media FIG. 5. The two addle point approximation, A and A, dicued in the text plotted together the numerical integration, NI,olution of the vertical component of reflected PP particle diplacement due to P wave incidence on an interface eparating two anelatic halfpace. Each of the gather conit of 3 receiver paced at 40m interval for an offet range of 0m r 00m.A 30Hz Gabor wavelet wa ued and the time ampling rate i m. ( ) diplacement for P wave incidence at a plane interface between two anelatic halfpace. Zero time on the trace correpond to an actual time of 60m. Each of the trace are 500m CREWES Reearch Report Volume (009) 3
Daley and Krebe The three panel in Figure (5) how the addle point approximation together with the numerical integration reult for the model decribed above and in Table. To preerve pace, only 500 time point of the trace were plotted, tarting at 60m which wa aigned a the zero tie point in all of the trace in thi figure. The two addle point approximation are indicated in Figure (5) by A and A and numerical integration trace by NI. In the three panel in Figure (6) the two approximation and the numerical integration trace are compared by plotting all three on a ingle axi for the offet r = 0,600, and 00m. The line type ued in the plotting of the three different trace are defined in the figure caption. The fit between all approximate trace with the numerical integration trace i quite evident, except at r = 00m, which i in the range of offet for which the reflected and critically refracted arrival interfere. Thi i known a the interference zone. More on thi topic may be found in Červený and Ravindra (97). After viewing the panel in Figure (6) it become evident, that at leat in the high frequency or geometrical optic olution, the two addle point approximation produce conitent reult, with approximation producing the bet fit with the numerical integration trace. Thi i to be expected, a A incorporate more approximation to produce ynthetic reult. CONCLUSIONS A comparion of addle point approximation for the cae of PP reflection due to incidence of a P wave, emanating from a point ource, at an interface eparating two anelatic media ha been preented. The two zero order addle point approximation conidered provide numerically proper approache of dealing with the anomalou amplitude behaviour, oberved in problem of thi type due to a pecific ditribution of Q P value in the two halfpace. Acceptable reult, when compared to the exact olution, obtained from a numerical integration algorithm were realized. Thi comparion of reult wa done within a numerical context and it may be concluded that if the PP reflection coefficient diplay appropriate behaviour, o too will the aociated addle point approximation. Further invetigation i till required to determine a proper theoretical explanation for the anomalou behaviour oberved. The firt of thee approximation, becaue of it minimized memory requirement and computational peed i in the proce of being implemented in a plane layered tructure with an arbitrary number of layer and receiver. Thi i being done for both amplitude VSP, modeling application. veru offet, ( AVO ), and vertical eimic profile, ( ) 4 CREWES Reearch Report Volume (009)
Anelatic media FIG. 6. A comparion of trace at elected offet of r = 0, 600 and 00m. Synthetic trace obtained uing the two addle point approximation, A and A, and the numerical integration, NI, approach are plotted on a ingle time axi. The code identifying each trace type i given in the inert in panel. CREWES Reearch Report Volume (009) 5
Daley and Krebe REFERENCES Abramowitz M. and Stegun I.A., 980. Handbook of Mathematical Function, Dover, New York, NY. Azimi Sh. A., Kalinin A.V., Kalinin V.V. and Pivovarov B.L., 968. Impule and tranient characteritic of media with linear and quadratic aborption law, Izv. Phy. Solid Earth,, 88-93. Aki K. and Richard P.G., 980. Quantitative Seimology, W.H. Freeman and Company, San Francico, CA. Aki K. and Richard P.G., 00. Quantitative Seimology, nd edition, Univerity Science Book, Saulalito, CA. Červený V. and Ravindra, R., 97. Theory of eimic head wave, Univerity of Toronto Pre, Toronto. Churchill, R.V. and Brown, J.W., 003. Complex variable and application, McGraw-Hill, New York, NY. Futterman W.I., 96. Diperive body wave, J. Geophy. Re., 67, 579-59. Krebe E.S. and Daley, P.F., 007. Difficultie with computing anelatic plane wave reflection and tranmiion coefficient, Geophy. J. Int., 6, 4-55. Marcuvitz N. and Felen L.B., 973. Radiation and cattering of wave, Prentice Hall, New Jerey. Rudd, B.O., 005. Ambiguou reflection coefficient for anelatic media, Sixth Workhop Meeting on Seimic Wave in Laterally Inhomogeneou Media, Geophyical Intitute, Academy of Science of the Czech Republic and Department of Geophyic, Charle Univerity.(available online at http:/www.ig.ca.cz/activitie/poter/poter.php). Zahradník J., Jech J. and Moczo P., 00. Approximate aborption correction for complete SH eimogram, Stud. Geophy. Geod., Special Iue 00, 33-46. ( ) ρ g cm V ( m ) ( ) HP V km Q P HS Q S fr ( Hz ). 500 000 5 5 30. 5000 000 40 0 30 Table. Parameter of the anelatic two halfpace model ued in producing the numerical reult preented. Thee real quantitie are ued to pecify either a contant Q type tructure or with the additional of the halfpace reference frequencie, frequency dependent. f R, an anelatic model that i 6 CREWES Reearch Report Volume (009)