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Reflected wave propagation in a wedge Ishill, Hiroshi 1969

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REFLECTED WAVE PROPAGATION IN A WEDGE by " HIROSHI I SHII B.Sc, Tohoku University, 1963 M.Sc, Tohoku University, 1965 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in the Department of GEOPHYSICS We accept this thesis as conforming to the THE UNIVERSITY OF BRITISH COLUMBIA September, 1969 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and Study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the Head of my Department or by his representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department The University of British Columbia Vancouver 8, Canada Date 2-UU Sept . \°.t> °l ii ABSTRACT The behavior of elastic body waves in a dipping layer overlying an elastic medium has been theoretically investigated by a multiple reflection formulation. Although the diffracted wave is not included in this formulation, its importance is studied by investigation of the amplitude discontinuities within "the wedge. For a plane SH wave incident at the base of the dipping layer perpendicular to strike, a series solution has been obtained. Numerical values of the amplitude, phase and phase velocity are calculated on the surface. For waves propagating in the up-dip direction the amplitude versus frequency curves for a constant depth to the interface change slowly with increasing dip for dip angles less than 20°. However for waves propagating in the down-dip direction the character of the amplitude curves change rapidly. In these cases, it is found that the diffracted wave plays an impor tant role. In addition to satisfying the boundary conditions at the surface and the lower boundary of. the wedge, the dif fracted wave must also satisfy additional conditions along a dipping interface between the wedge boundaries due to the geometrical nature of the reflected wave solution. It is found that the phase velocities vary rapidly with both period of the wave and depth to the interface. For incident plane P and SV waves, the complexity of the problem due to the converted waves does not allow the solution to be expressed in series form. However, a com putational scheme has been developed which allows the iii calculation of the disturbance due to the multiply reflected waves. For both incident P and SV waves, numerical values of displacements and displacement ratios are calculated on the surface. It is found that the displacement ratios for incident SV waves are much more sensitive to dip than are there for incident P waves. For incident P and SV waves propagating in the down-dip direction with a propa gation direction oL, (3 = 120°, the amplitude ratio versus frequency curves for constant depth to interface do not have significant peaks for dip angles greater than 15°. The maximum discontinuities caused by the outgoing wave are also calculated to determine the role of the diffracted wave. As subsidiary problems the energy relations between waves at an interface between elastic media are determined in terms of propagation direction in a cylindrical system and the complex propagation direction is interpreted using the Rayleigh wave. The final study is to determine by a reflected wave formulation the displacements due to periodic and impulsive line sources of SH waves in the wedge overlying an elastic medium. A formal solution is found by which the contributions due to head and reflected waves are determined by evaluation of the integrals by the method of steepest descent. Using ray paths, the contributions iv of the integrals have been interpreted. The range of existence of head waves has been examined and the discon tinuities associated with diffracted waves studied. In the case of a free or rigid lower boundary of the wedge, the dispersion relation has been determined. V TABLE OF CONTENTS Page ABSTRACT ii LIST OF FIGURES viiLIST OF TABLES xACKNOWLEDGEMENTS xii CHAPTER 1 GENERAL INTRODUCTION ' i 1.1 Preliminary Remarks 1 1.2 Summary of Previous Studies 2 1.3 Scope of This Thesis 4 CHAPTER 2 MULTIPLE REFLECTION OF PLANE SH WAVES BY A DIPPING LAYER 7 2.1 Introduction2.2 Wave Equation and Fundamental Solution 8 2.3 Reflection and Refraction Coefficients 9 2.4 Multiple Reflection Solution for a Wedge 14 2.5 Numerical Computations and Discussion 20 2.5.1 Amplitude Disctoninuity at Q-<^>^~J\Z 20 2.5.2 Surface Amplitude Characteristics 21cL 2.5.3 Phase Velocity at the Free Surface 30 CHAPTER 3 MULTIPLE REFLECTION OF PLANE P AND SV WAVES BY A DIPPING LAYER 32 3.1 Introduction3.2 Equations of Motion and Boundary Conditions 32 3.3 Reflection and Refraction Coefficients 36 3.4 Computation of Displacement in the Case of a Dipping Layer 46 vi 3.5 Displacement Discontinuities 50 3.6 Surface Displacements and Displacements Ratios 53 3.6.1 Incident P 53.6.2 Incident SV 6 CHAPTER 4 HEAD AND REFLECTED WAVES FROM AN SH LINE SOURCE IN A DIPPING LAYER OVERLYING AN ELASTIC MEDIUM 61 4.1 Introduction4.2 Equation of Motion and Boundary Conditions 62 4.3 Steady State Plane Wave Solution - 65 4.4 Formal Steady State Solution for a Line Source 72 4.5 Evaluation of the First Series Term of the Integral - 73 4.5.1 Contribution from the Saddle Point (Reflected Waves) 76 4.5.2 Contribution from the Branch Point (Head Waves) 78 4.6 Aperiodic Solution 80 4.7 Interpretation of the Travel Time, 82 4.8 Range of Existence of Head Waves 84 4.9 Discontinuities 88 4.10 Dispersion Equation for the Lower Boundary Free and Rigid 94 4.11 The Horizontal Layer Solution 96 4.12 Computation of Displacement Seismograms 99 vii CHAPTER 5 SUMMARY, CONCLUSIONS AND FURTHER STUDIES 105 5.1 Summary and Conclusions 105.2 Suggestions for Further Studies 109 BIBLIOGRAPHY 111 APPENDIX I ENERGY RELATIONS 113 APPENDIX II EXPRESSION OF A FREE.RAYLEIGH WAVE USING COMPLEX ANGLES 116 APPENDIX III EVALUATION OF THE SECOND SERIES TERMS OF THE INTEGRAL 120 viii LIST OF FIGURES FIGURE PAGE 2-1 Cylindrical coordinate system ( T 0 2/) used in this problem. ' 10 2-2 Reflection and refraction at a boundary inclined at an arbitrary angle 6tk - H . 2-3 Multiple reflection and refraction for a wedge-shaped medium with a wave incident with propagation direction oL • 15 2-4 Displacement discontinuity along the edge of outgoing reflected wave for unit ampli tude incident waves with propagation direc tion cL . " 22 2-5a Amplitude surface for the parameters dip angle and (r=£/3jLH/CbiT for an inci dent wave with propagation direction d = 60°. 24 2-5b Amplitude surface for the parameters dip angle and <T~2sf37ZH/CbtT for an inci dent wave with propagation direction Oi = 120° . 25 2-6a Amplitude surface for the parameters propa gation direction cL and fr=j2/?JJGH/CbiT for a horizontal boundary. 27 2-6b Amplitude surface for the parameters propa gation direction oL and <S*-Zt/JJZ H/CbiT for a dip angle 10°. 28 2-7 Amplitude surface for the parameters dip angle and T—JZ,JLT/CMT for an inci dent wave with propagation direction oL = 60°. 2- 8 Phase velocity (Cu/Cbi ) curves versus CT = ^^3 70H/6biT for a dip angle of 10° and propagation direction cL . The thin horizontal lines are the phase velocities for the horizontally layered case. 31 3- 1 Reflection and refraction of waves at a boundary inclined at an arbitrary angle 8dL with the nomenclature for angles between rays and the horizontal and boundary surfaces indicated. 37 ix 3-2 Reflection of waves at a free surface with nomenclature for angles between rays and the free surface indicated. 44 3-3 Flow diagram showing the computational scheme used to calculate the amplitudes and propagation directions of the reflected waves in the wedge and thus the displacement and displacement ratio at any point. 49 3-4 Maximum displacement discontinuity of the radial component from the exiting P waves and tangential component from the exiting SV waves for an incident P wave with propa gation directions oi - 60° and oi = 120°. 52 3-5 Maximum displacement discontinuity of the radial component from the exiting P waves and tangential component from the exiting SV waves for an incident SV wave with propa gation directions = 60° and ^ = 120°. 54 3-6 Horizontal and vertical displacements versus the parameter (T = Z/TjZH/CaiT for inci dent P waves witli propagation directions cL = 60° and — 120 for the range of dip angles 3°^6<K^30°. 55 3-7 Displacement ratios V/H versus the parameter §- = zJKlLH/CatT f°r incident P waves with propagation directions oC = 60° and oC « 120° for the range of dip angles B°— 6^.= 30° 57 3-8 Horizontal and vertical displacements versus the parameter c? ~Zf3 7Z\A/C(x\T for inci dent SV waves with propagation directions (S = 60° and ($,= 120° for the range of dip angles $°£9<i&300. 58 3- 9 Displacement ratios H/V versus the parameter 6~" -2,J~3K,H/Ca.iT for incident SV waves with propagation directions (3 = 60° and .|S = 120y for the range of dip angles S%d^k30°. 60 4- 1 Geometry of the problem: the line source ( S ) is located at (d, 0 ) and the receiver ( R. ) at (T, 0 ) in the wedge bounded by the free surface ( Q=—S\ ) and the boundary (0 = 0ji, ) between the two media. 64 X 4-2 The oii-plane (0Cr = X+l^) on which Re(A-s)7>0 and the regions of positive and negative Im(As) , separated by the curves bg and L^g , indicated. Nota tion: 5 - saddle-point; B , B' - branch points; L - original path of integration; Lis - path of steepest descent through the saddle point; LM , Viz. - paths of branch line integral; U8 - branch cut Re(As)=0; and bg - curve along which lm(/ls) = 0 • ^5 4-3 Basic ray paths used in physical inter pretation of contributions from branch and saddle points. 83 4-4 Ray paths of the head and reflected waves expressed by the first series term of the integrals. 85 4-5 Maximum value of Qx for which the head waves shown in Figure 4-4 exist versus the ratio of source to observation distances. The observation and source points at 5° from the free surface. 87 4-6 Maximum value of the wedge angle ( 6i+ Qz ) for which the head waves of the types shown in Fig. 4-4 exist for an observation point at 5° from the free surface and d/T=lO.O. 89 4-7 Discontinuities in medium (1) due to inter action of the wave with the vertex. The lined areas indicate the regions for which the geo metric wave from thelast reflection exists with the term from which it arises indicated in brackets. 90 4-8 Relative amplitudes of the displacement dis continuities due to a plane initial wave close to the x-axis for propagation upward ( 771 = -) and downward ( 771 = +) . 92 4-9 Coordinate system for the horizontal layer case with the source ( S ) at (dl,0 ) and , the receiver ( R. ) at . 97 4-10 Three cases for which theoretical seismograms were calculated. The parameters used were: Hi = 9.59 km, H^= 3.00 km, 0 - 99.6 km, ck - 10.0 km, and the displacement parameter C = 0.05 sec. 100 xi 4-11 Ray paths which contribute to.the theo retical seismograms. 101 4-12 Displacements of the component waves for the geometries given in Figures 4-10a, 4-10b and 4-10c. 102 4-13 Synthesized seismograms resulting from the displacements of Figure 4-12. 104 A-l Coordinate system used to calculate the complex angle of a free Rayleigh wave. 117 A-2 The c^-i-plane showing branch cuts and integral paths for evaluation of the second series term of the integrals. Notation: B , C - branch points; £ -saddle point; L - original path of integration; Ls - path of steepest descent through saddle point; and LL (I— 1 , Z • - -) - paths of branch line integral. 123 A-3 Ray paths of the head waves expressed by the second series term of the integrals with the four combinations of 77L(+, — ) and £(\,Z) corresponding to the four second series terms of the integral in (A-3.1). 130 LIST OF TABLES TABLE 1 Notation used in Figure 3-3. PAGE 50 xii ACKNOWLEDGEMENTS I wish to express my sincere thanks to Dr. R. M. Ellis for his guidance and encouragement and for many hours of discussion during the course of the entire investigation. Thanks are due to Drs. R. D. Russell and D. E. Smylie for reading the manuscript and Dr. G. K. C. Clarke for his comments on Chapter 4. I appreciate the constant interest and encouragement of Dr. Russell during my studies at the University of British Columbia. Helpful discussions are acknowledged with my colleague Mr. 0. G. Jensen, who also provided me with his program for plane waves incident on a horizontally layered system. I would like to express my appreciation to Professor Akio Takagi, Chief, Akita Observatory, Tohoku University for granting educational leave and to Professor Ziro Suzuki who suggested studies at the University of British Columbia. This manuscript was typed by Miss Judi Kalmakoff. This study was supported by the National Research Council (Grant A-2617) to Dr. R. M. Ellis and the Defence Research Board of Canada (Grant 9511-76) to Drs. R. M. Ellis and R. D. Russell. A University of British Columbia Graduate Fellowship during the second year of this study is gratefully acknowledged. CHAPTER 1 GENERAL INTRODUCTION 1.1 Preliminary Remarks Elastic waves play very important roles in the determination of'the crustal structure and the internal constitution of the earth. In the past decade new ana lysis techniques coupled with advances in instrumentation have lead to rapid expansion of our knowledge concerning the seismic properties of the earth. However, the ana lyses are restricted by the limited number of models which are available - mainly for horizontally layered structures. In earthquake seismology, surface waves have been parti cularly useful for interpretation as they yield an average structure over the propagation path and hence the hori zontally layered formulation has proved to be adequate in most cases. However, body wave applications (e.g., Phinney (1964), Ellis and Basham (1968), Ibrahim (1969)) have only been moderately successful as body wave amplitudes depend on a localized region beneath the observation point which may be geologically complex as indicated by the reflection studies of Clowes et al (1968) . Hence it is necessary and important to investigate the behavior of waves in a dipping layer to obtain an understanding of the more complex models. 2 1.2 Summary of Previous Studies The interpretation of horizontally layered struc tures has been dominated by the theoretical studies of Haskell (1953, 1960, 1962). He considered an input wave at the base of a horizontally layered system and by apply ing the boundary conditions obtains propagator matrices which carry the displacements and stresses from one boundary to the next eventually obtaining a relation between the input wave at the lower boundary and the surface motion. For incident P and SV waves, the frequency domain input function may be eliminated by taking the ratio of the vertical and horizontal displacements. For incident P, the experimental V/H ratio versus frequency, and for inci dent SV, the experimental H/V ratio, can then be compared with theoretical Haskell ratios to determine crustal struc ture. Haskell's formulation is also applicable to surface wave studies. Several studies for non-parallel boundaries have been done, mainly relating to surface waves. Hudson (1963), Nagumo (1961) and Sato (1963) dealt with SH waves in a wedge-shaped medium. Hudson studied SH waves from a line source in a wedge-shaped medium with a rigid lower surface. He obtained a solution composed of multiply reflected 3 and diffracted waves. Using this solution he investi gated the effect of diffraction at the apex of the wedge by means of an approximate form of the diffracted pulse and found that the diffracted wave amplitude decreases as (wnere T and T, are the distances of the source and observation point, from the vertex). Nagumo considered two dimensional elastic wave propagation in a liquid layer overlying a rigid bottom. He found that mode solutions exist. From the solution he investigated dispersion relations of the wave. Sato studied the dif fraction problem of SIT waves at an obtuse-angled corner due to incident plane SH pulse parallel to one of the free boundaries and calculated diffracted wave forms which he found diminished rapidly away from the vertex. Lapwood (1961) , Kane and Spence (1963) , Hudson and Knopoff (1964) , McGarr and Alsop (1967) and others have studied Rayleigh wave transmission in a wedge-shaped medium. Lapwood investigated wave forms from a line pulse source on one of the free boundaries of a right angle, using integral transformation and approximation procedures. Kane and Spence (1963) and Hudson and Knopoff (1964) considered Rayleigh wave transmission on elastic wedges with free boundaries. The first authors employed an itera tion procedure and the latter employed a Green's function technique in order to calculate transmission coefficients 4 of the Rayleigh wave. Using an approximate variational method, McGarr and Alsop (1967) computed the reflection and transmission coefficients for Rayleigh waves normally inci dent on vertical discontinuities. Conversely, there have only been a few studies (Fuchs (1966)) and Kane (1966)) on the effect of non-parallel boundaries for body waves; nevertheless body waves constitute an initial section of a seismogram which is very often used in analyses. Fuchs synthesized seismograms due to a primary P signal propa gating along the median plane in a solid wedge with free boundaries, by taking a summation of reflected waves. He determined the dispersion of the body waves and particle motion. Kane employed a tree diagram which is obtained by reflecting the wedge rather than the rays and a vector which carries nine pieces of data. Thus, he calculated theoretical seismograms due to an input plane P pulse for the teleseismic response of an array of stations located on a uniformly dipping crust. In this way he demonstrated the signal distortion effects of the geometry. However the amplitude characteristics which are used for interpretation of crustal structure were not investigated nor was the diffracted wave. 1.2 Scope of This Thesis The objective of this thesis is to extend the theory of body wave propagation in a dipping structure using a re flected wave formulation. Although the forms of the dif fracted waves are not investigated, determination of the amplitude discontinuities due to the reflected wave within the wedge indicates its importance. 5 First, in Chapter 2, a plane SH wave incident at the base of a dipping layer is considered as in this case no converted waves are present. A solution by multiple reflection is obtained and the amplitude charac teristics and phase velocity calculated on the surface in terms of depth to the interface, period of the wave, and dip angle. The discontinuities which result from the last reflection and which are related to diffracted waves are determined. This development in Chapter. 2 serves as a guide for solving the more difficult problems of Chapters 3 and 4. In Chapter 3, the corresponding problem is studied for incident P and SV waves. The complexity does not allow a series solution to be obtained; however, a com putational scheme is developed which allows the calcula tion of the displacement and phase velocities. As sub sidiary problems , the energy relations between waves at a boundary are given in terms of the propagation direction and the complex propagation direction interpreted. In Chapter 4, propagation of SH waves from a periodic and impulsive line source in a dipping layer overlying an elastic medium is investigated using a reflected wave formulation. The contributions due to head and reflected waves are determined by evaluating the integrals by the method of steepest descent and a 6 comparison made with a horizontally layered case through the case of numerical examples. The range of existence of head waves is determined and the discontinuities associated with diffracted waves studied. The study is summarized and suggestions made for further investigations in the final chapter. The theory for multiply reflected waves as developed in this thesis could serve as a useful starting point for the study of diffraction. Techniques such as the geometrical theory of diffraction as developed by Keller (1962) appear to be applicable; however, they may not be practical due to the complexity introduced. In this theory for small wavelengths, Keller uses diffraction laws similar to laws of reflection and refraction which are derived from Fermat's principle. Away from the diffracting surfaces, he is able to use dif fracted rays just like ordinary rays. By the use of the reflected wave solution and such a diffracted wave procedure, it may be possible to obtain a more satisfactory description of elastic waves in a wedge. 7 CHAPTER 2 MULTIPLE REFLECTION OF PLANE SH WAVES BY A DIPPING LAYER 2.1 Introduction The calculation of the amplitude characteristics of waves propagating in horizontally layered media has been greatly simplified by the matrix formulation of Haskell (1953, 1960, 1962). The application of this formulation has proved to be a powerful method for deter mining the crust and upper mantle structure using surface waves. However, body wave applications have only been moderately successful. Even though for surface waves the regional structure may conform closely enough to the layered theory to allow a successful interpretation, the body wave amplitudes may not be useful for interpretation as they depend only on a localized area beneath the station which may be geologically complex. It is, therefore, important to study the effect of dipping boundaries on the charac teristics observed at the surface. Fernandez and Careaga (1968) have suggested that a model of this type may be required to explain body wave observations at La Paz. As an initial study of body waves interacting with a wedge overlying an elastic medium, all waves internally reflected between the free surface and the dipping layer due to a plane SH wave incident on the wedge perpendicular to the direction of strike will be considered. The objective 8 is to calculate the amplitude characteristics in terms of distance from the vertex, depth from the surface, and the period of the wave. On the basis of the results of pre vious workers, it is expected that the multiply reflected waves will play the most important role in a seismogram at observation points distant from the vertex and will be explicitly investigated in this study. For the diffracted wave, the boundary conditions are expressed and calcula tions made to indicate its importance in particular situa tions. This simple case in which there is no coupling between wave types serves as a guide for solving the more difficult problems of incident P and SV waves as well as being of interest in its own right. Further, as surface wave, refracted wave and reflected wave components are obtained by evaluating the contribution of poles, branch points and saddle points respectively in terms of multiple reflection, the solution of the present problem is an impor tant step leading to the solution of these more complex problems. 2.2 Wave Equation and Fundamental Solution In this problem with a dipping boundary it is found convenient to choose a cylindrical coordinate system ( T , 0 H, ) related to a cartesian system ( % } -"fr ^ 2L ) as 9 shown in Figure 2-1. For a plane SH wave propagating in the x-y plane, the motion is independent of Z and the displacement has only a z-component. Assuming a time variation of the form > the equation of motion ~ct~9±F (2,1) becomes in cylindrical coordinates fr* + ^fr+^fer+ K)u="0 (2.2) where £b = o;/Cb We choose as the fundamental solution of this equation A • (3 which is a plane wave of ampli tude A propagating in the cL s direction. The only non-zero component of stress is 2.3 Reflection and Refraction Coefficients We now consider two elastic media divided by 0= 0^ with waves from medium (2) incident on the interface (Figure 2-2). The solutions in media (1) and (2) can be written as 12 . L-feb,rcos(0-|s; itbzrcos(e-oL). L%irco$(e-r) .IXz~ AL -o -4- Afji'S The boundary conditions at the interface require continuity in displacement and stress. At O — Bd. wo can there fore write (2.5) ffao)|~( Pae)z The condition of equality of phase at G — OeL leads to £blcos(ed-p) = ^cos(e^-^)=^coS(ea-T) (2.6) Using (2.3) and substituting (2.4) into (2.5), we obtain Arji. = S S i n(6di- op - A S (n(9^- g> AL ASin(0d- {$)- S sin(Od-T) Arf--, 5sin(Gd-^)-gsin(edi-r) Ai Asin('0A-p)-SS/n(0d-T) where & = C^Z/CM and % — //(z//M-\ Using (2.6) and the geometric relationships between the angles cL, $ t T and the angles ; [,p ; lT (Figure 2-2), we have (2.7) 13 70 (2.8) (2.9) is = ed + -^- - sin'^cosCea-oO) and Sin(ed-p) = .-7 i-(i/^)cosa(^-oc) with ed .<oc< 6a -v- .^c Finally, substituting (2.9) into (2.7) we obtain An s I n (0d- oC) + CI / S) cosa ("^"oQ" AL Sin(0d-o/o-(|/g)yA--cos-(0d-5y Arf • _ • asin(eA-oO  Ac Sin(0d-oC)- (1/ S)^AA-cos^Oci-.oc) Thus, we have been able to denote the reflection and refraction coefficients in terms of the initial propa gation direction, the dip angle, and the elastic constants. In the case where the waves are incident on the boundary from medium (1), the same process yields the fol lowing equations: (2.10) AL A$m(0d-oO-r §/7^A*C0S*( 0d.-op (2.11) Aff=' 2&S\n(6<k~oc)  Ai ~ ASin(0d-ol)i- \SJ \- A^cos^-oO 14 and ' (2.12) P = -ir^+Sirr^ACosce^-oc)) with 0^H- Tt < cL <.0* + £7G If AACOSa(0d-oC)> I then J|-AaCOS*(eA~oO~ must be replaced by - I J A^COS^CO^ — Oi) ~ T for the solution to remain finite at infinity. For waves incident on the free surface, we have (2.13) T = Z7L - cL with 0 < OL < Jb 2.4 Multiple Reflection Solution for a Wedge A ^u"^b^TCOS(e- oi) Consider a wave A^'O incident on the boundary Q= 0^ from medium (2) (Figure 2-3) and assume a resulting reflected wave ^'' and refracted wave of the forms V, » '/ A oltb^TCOS(0-T,) n«Al-A-k-ei*b'rco8(e-^ IM* 2-3. Multiple reflection and refraction for a wedge--. Eig. 2 Siped medium witk a Wave incident with propaga tion direction cC 16 Using equations (2.8) and (2.10), the boundary conditions are satisfied for A"_ sin(e^-oO-r (i/S)v/Aa"~cos2-(6d-oC) A, -ssn(ed-^)~(!/s)/A"-cos-(aa"oC) Tj = ZdcK-i-Z'JZ-oL (2.14) and /\ = 2$\n(6rj-cL) Si n(ed-oC) - ( i / §) /^cos^ce^-oc) Pi = Od-T - Sin'(•S'COS(edL-oC)) (2.15) To satisfy the boundary condition at 9=0 , we assume a reflected wave i>b,rcos(e- r,o By equation (2.13), the boundary condition is satisfied provided that To satisfy the boundary conditions at 0 = 0^ , we must assume the reflected and refracted waves l-feb|TCOS(0-Ti) A*/A',- AL- 6 ^//= Al - A(• AL' SL^B^TGOS^0~ ^ 17 which satisfy the boundary conditions for •* Asin(0d-VH 8/1 - Aacos*(aL- r/) (2.16) and (2.17) using equations (2.11) and (2.12). These steps are then repeated. However, it is not an infinite process for it terminates whenever-^ < V*, < X + 6JL or IL < T^< JO + 0dL in these cases the wave propagates down the wedge without further collision with the boundaries. The last term of the series in medium 1 is of the form A,(7cA.)e^rc°S(e-^ where L - Tl max 18 This gives rise to a discontinuity in the displacement, at 0 = <£> — JZ . Therefore, the solution for the dif-'71 fracted waves must be of a form that will give continuity of displacement and stress at 0 = (h — JG as well as satisfy-ing the boundary conditions at Q — 0 and 0 = 0^ . As we shall see in the next section, the discontinuity at ®~ $ri~ ~"S ""n most cases SmaH indicating that the multiple reflection solution usually dominates the seismo-gram. Further, Sato (1963) has shown that the amplitude of the diffracted wave decreases rapidly a\\Tay from the region of the ray theory discontinuity and hence will be small at surface points distant from the vertex. We then write M^A^fTtAje^05^^ K=A,f:(JcAw)e^TC0S(9_T-) and (2.18) N =A,i(^Aje^ircos(e'T-} (2.19) 19 where A ~ , : _ (2.20) (2.21) \ sin (e*- ol)-(cos^eI=^ ft- eA+f--sin^i-cosceA-oc)) L TTt max |_j — TTt max ] (2.22) (2.23) (2.24) The solution can then be written in one of two forms depend ing on whether the last reflection is from 0=0 or Q _= 0^ . In the first case we have Ui = -N,+ Ni' for oie4<-}c A/ , (2.25) '^i=Ni+M,. ' ^or rL~^^e^eA with 20 and in the second case we have WITH TO < TJ, < The amplitude A, the phase (H) , and the phase velocity Cv (in the direction 0 = constant) may be written as: (2.26) A- yHrCeCUO** Im(U(f (2.27) (2.28) v-/ Re(u,)-rx;— ~ lm(^i)— x / (2-29) 2.5 Numerical Computation and Discussion For the numerical computations, the values chosen for the parameters were JJLX = (t 8 $Z and Ct>z/Cb\-\ which correspond to the crust - upper mantle model used by Haskell (1960) . 2.5.1 Amplitude Discontinuity at As discussed in the previous section, the last reflection, which does not collide with a boundary, gives rise to a displacement discontinuity and corresponding to 21 "tKis~a ^''-function in the stress at @ ~ 3^ • The magnitude of the displacement discontinuity versus dip angle is shown in Figure 2-4 for various angles of incidence as applicable to teleseismic waves. In the case where the magnitude is small the reflected wave solution adequately describes the physical problem. However, if the discon tinuity is large, then a diffracted wave with a large ampli tude in the region of d — '^—TC is required to provide continuity in displacement and stress. We see that for the incident wave propagating in the up-dip direction (oL<\£]00 ), the discontinuity is small for dip angles less than 15°. However, for incident waves propagating in the down-dip direction (OC>^0 ) the dis placement discontinuity is large for some ranges of small dip angles. In these cases the diffracted wave is impor tant because the internally reflected wave propagates out of the wedge after a small number of reflections. How ever, for surface points distant from the vertex, it is expected that the reflected wave amplitude will give a good approximation in most regions to the true amplitude as the discontinuity surface becomes distant from the free surface. It should also be pointed out that in addition to the discontinuity within the wedge, discontinuities are 21a generated by the vertex on reflection of the incoming wave and each refraction into medium (2). In the reflected wave theory these appear as displacement and stress dis continuities radiating from the vertex. The effect of these will not normally be large on the surface of the wedge except close to the vertex as the amplitude decreases rather rapidly with distance and the wave will be partially reflected at the lower boundary of the wedge. However in any study of diffracted waves their relative impor tance should be investigated. 2.5.2 Surface Amplitude Characteristics One effect of interest is the effect of a varia tion of dip angle on the amplitude characteristics at the surface for a constant depth to the boundary and fixed 22 O o CO Q LLJ Q DIP ANGLE (DEGREES) Fig» 2-4 0 Displacement discontinuity along the edge of out going reflected wave for unit amplitude incident waves with propagation direction cLo 23 propagation direction. In Figures 2-Sa and 2-5b the ampli tude surfaces are plotted for the parameters dip angle and JO H/Cbt T (His depth to the boundary) for oL = 60° and 120°. For fixed H and Cbl these are then amplitude surfaces for varying dip angle and period. For an incident wave with propagation direction o(, = 60°, we see that the amplitude characteristics change slowly with increasing dip angle except in the range near 25°. However, for an incident wave with cL = 120° the amplitude characteristics change rather rapidly with fre quency over the range of dip angles considered. The curves 2-5a and 2-5b are typical of the characteristics for inci dent waves propagating in the up-dip and down-dip direction respectively. One of the reasons for this is indicated in Figure 2-4. For a propagation direction cL = 60°, the amplitude discontinuity is small at 0=^—70 for Q^<^Z\° but for a propagation direction oL = 120°, the discontinuity is significant over most of the range of dip angles considered. This indicates that in these regions of rapid changes of the amplitude characteristics, the multiple reflection solution as presented here does not fully describe the physical situation but that the dif fracted wave is likely to play an important role. From a physical viewpoint, this larger diffracted wave for waves propagating in the down-dip direction arises as the wave 24 PROPAGATEON DIRECTION » SO° 2 JlwH Cb, T 2-5aa Amplitude surface for the parameters dip angle and =JZ/3"7GH/CbiT f°r an incident wave with propagation direction oC=60°<> 25 o ro BamndiAiv Pig. 2-5b. Amplitude surface for the parameters dip angle and 6^=£/3 70H/CbiT for an incident wave with propagation direction oi=[ZQ° • 26 only collides with the boundary a very few times before it propagates out of the wedge and hence this wave, which leads to the diffracted wave, is still of significant ampli tude. Hence we see that except for large dip angles and waves with large incident angles propagating in the down-dip direction the reflected wave solution adequately des cribes the physical problem. Also of interest are the amplitude surfaces for the parameters cT~ and propagation direction for a constant dip angle. These are shown in Figures 2-6a and 2-6b for dip angles of 0° and 10°. These two graphs are similar in the range of propagation direction 45° to 90° in that for in creasing propagation direction this amplitude versus curves oscillate more rapidly. It should be noted that the curve for a dip angle of 10° and propagation direction oL = 45° has one additional oscillation between = 0 and cT" = 49 as compared to the curve with no dip. For propagation directions greater than 110° the amplitude curves change rapidly with increasing angle. Again it should be noted from Figure 2-4 that we expect the diffracted wave to play a significant role for this range of propaga tion directions. Finally, the amplitude surfaces for the parameters -r; = ^JSL£- and dip angle (Figure 2-7) are considered. The DIP ANGLE = 0° Pig0 2-6&0 Amplitude surface for the parameters propagation direction oL and 6"=Zj3 7tH/C\,\T for a horizontal "boundary. DIP ANGLE • 10° 4-i UJ Q • 2 0. < 135 / / / / 45 60 0 7 14 21 28 35 42 49 2 73 TTH Cb,T ... _ Pig. 2-6b. Amplitude surface for the parameters propagation direction cL and (T = £/3~X.H/GbiT for a dip angle 0^=10°. PROPAGATION DIRECTION - 60° 14 21 28 35 42 49 56 63 70 77 2 IT r Pig* 2-7. Amplitude surface for the parameters dip angle and T=2JDr/CbiT an incident wave with propagation direction ©1=60° * 30 major feature of these curves is the increase in the rate of oscillation for the amplitude versus . T curves as the dip angle increases. Hence for a constant distance from the vertex the spectral character of a seismogram is expected to change rapidly with, a changing dip angle. 2.5.3 Phase Velocity at the Free Surface In Figure 2-8, the phase velocity curves are plotted for a dip angle of 10° for various propagation directions . These differ markedly from the phase velocity for the horizontally layered case (thin lines) as dispersion is present which depends oh both the period of the wave and depth to boundary. The amplitude of the phase velocity oscillations increases with increasing propagation angle until an angle is reached which corresponds to vertical incidence for the horizontally layered case. Beyond this angle the oscillations then decrease. Clearly, measure ments of phase velocity on a wedge-shaped medium will deviate markedly from the horizontally layered case due to variations with both period of the wave and depth to the interface. 30a An explanation of the large variations in the phase velocity which occur for incident waves in the up-dip direction is as follows. In this case waves propa gate toward the vertex and then have their direction reversed and propagate out of the wedge. For small inci dent angles ( oC = 75°) , the amplitudes upon reversal of direction will still be large resulting in a significant contribution to the phase velocity with little change in the transfer function compared to the horizontally layered situation. 31 A1I0013A 3SVHd Pigo 2-8o Phase velocity GV/C-bi curves versus 6"=£>/3 70H/CbiT for a dip angle of 10° and propagation direction cC o The thin horizontal lines are the phase velocities for the horizontally layered case* 32 CHAPTER 3 MULTIPLE REFLECTION OF PLANE P AND SV WAVES BY A DIPPING LAYER 3.1 Introduction , In the previous chapter the problem of a plane SH waves incident at the base of a dipping layer was con sidered. A solution by multiple reflection for waves incident at any angle perpendicular to strike has been obtained for the amplitude, phase and phase velocity. In this chapter the case of incident P and SV waves is considered. Experimental investigators who use displace ment characteristics to interpret crustal structure (e.g., Phinney, 1964) have been limited in their calculations to horizontally layered structures. Consequently, the present analysis will expand the number of models available for interpretation purposes. 3•2 Equations of Motion and Boundary Conditions In this problem, it is again convenient to choose a cylindrical co-ordinate system ( T Q ) % ) related to a cartesian system ( % , ^, % ) as shown in Figure 2-1. For plane P and SV waves propagating in the x-y plane, the motion is independent of & and the displacement has only T and 0 components. The equations of motion in cylindri cal coordinates are: 33 ^ at* ~ U+War T ae-. C3"1} where: 2COa = -™(T.lAe) 7p "90^ (3.4) and is density; and y/. , Lame's constants; r\Xr and , displacements in the T and 9 directions. The stress components are expressed by = » (Mi _ Jig. , J_9U^ Using equations (3.3) and (3.4) in the equations of motion, we obtain r9*® • .-a i r/.V3*0 i ' ^ I 1 • a-^\ „ 71 at* ar1 T r f ae* ' 1 J 34 and in the stress relations, we obtain 9T (3.9) (3.10) where" J-^L and Ci , the P and S wave velocities respectively. Assuming a time variation ~ c *() of the form ^ , equations (3.1), (3.2), (3.7) and (3.8) become 1 a® u T where ^ .2. b ^& CO a. Q CT. Gb and V (3.11) (3.12) (3.13) (3.14) 9 + I 2 ar2 1 rar ' r^sd* Substituting (3.11) and (3.12) into (3.9) and (3.10), we have (3.16) 35 We choose as the fundamental solution of equations (3.13) and (3.14) , and F'6 plane waves propagating in the cyC and |3 directions respectively. Substituting the fundamental solutions into (3.11), (3.12), (3.15) and (3.16) we have the following expressions for displacements and stresses: 2i •, i'kbTcosfe-6) ZL Fsiri(e-0)e P (3.i7) e- s'\n(_e-oL)Ql . Kb (3.18) -^c:Fcos(e-p)sm(e-p)ei*brcos(0'p) fe = -z f c%{ E cos(e-OL)S i n (ee^TC0S(0 -+ (l~zcos^e-p))FellbrcoS(e^H (3.20) 36 3.3 Reflection and Refraction Coefficients In this section the reflection and refraction coefficients in terms of the initial propagation direction and the elastic constants will be considered. First con sider two elastic media separated by 0= 0^ with waves from medium (2) incident on the interface (Figure 3-1). The solutions of the equations of motion, (3.13) and (3.14), in media (1) and (2) can be written as lto^TCOS(e~oLrJi) (3.21) o3a,= Dine^TCOS(9^+[)rte where 60 The boundary conditions at are (3.22) re, = re 37 CD CO b cT CD ti CD Reflection and refraction of waves at a boundary inclined at an arbitrary angle 0* with the nomen clature for angles between rays and the horizontal and boundary surfaces.indicated. 38 For a solution of the form (3.21), the displace ments and stresses in medium (1) and medium (2) are from expressions (3.17) to (3.21) n - [-Larncffl ^ .L%a\rcos(e-dr£ ^-o^CinCQSCe-oOe^00306-^ -^pinSin(e-P)e^TC0S(e-^ -t DrAsin(e- prJt)&1 WcosCe-r<-CM -t-l^— Crx sin(e~c^rx)6 Khz. 39 -^P.CblcosCe-p^sinCe-MB^e^1™^"^ (3.24) + {I-*cos*Ce - M 6^r C°S(e _ cosCe-^sinO-ooCine^005^"^ + {i-^e-piDI„el*wrC0S(e-p) + (i^-M}^t,'tt05(''.'rt) Application of the boundary conditions (3.22) leads imme diately to the equality of phase which for incident P or SV waves yields respectively ia.cos(e*-oi) j = ^fl2C0S(eA_^)= ^cosceu-Pm) .cs-25) ft b4 cos (e*- p) J = ta,cos(0A- ibl c os (©A - pr5) which is Snell's Law expressed in a cosine form. The boundary conditions (3.22) then yield the fol lowing equations 40 CD. crT S ^ CO. V-ccx. I CD • f— i I 8 5 ' 00 o _ o , i I 8^ o CCu I CO O O -6 CD 8 X CD I CD cO 8 .E Q_| CO c X CO. CD CO CO o o < cn. I CD o OQ. CQ. CO o O CD c\> ,5 CA5 ^ l CD CO cA> y I CD CO O o t 'co 9 5 co o o y o ll cS 3 o o O CN( Q4 CAi II •Q_i CD Sr CD 41 To solve equations (3.26) in the case of either an incident P or SV wave, the angles oiRSL » $TJL ' oCr-J- and §r£ "lust De determined in terms of the incident and boundary angles. For a P wave incident from medium (2), the following geometric relationships are evi dent from Figure 3-1. I 06= 0^ + "^7 70 - boL oCT:c = 0^ + -Jr JO — LoCr^. for 0^<^oC^ 0^4" ' lJs^nS (3.25) we then obtain ztr^ e^^jr-7ltS\n[{cOS(3A-ct)}^Zed + 27l~oC $rSL= %+Xx"t5fn~1{tc^/Ca2)COS(0^-o6)} _( (3.28) ^r**5x70™sin l(u{/Ca2)CoS(0d-cC)} where the principal value of the inverse sine is taken in each case. It is easily verified that (3.28) also holds for A.-^.-25 <c (X,<^ • For incident SY waves, we find (3.29) 42 (3.29) Hence, by setting Din—0 in (3.26) and substitution of (3.28), the reflection and refraction coefficients for an incident P wave can be determined. Similarly, by setting Q-ft = 0 and the use of (3.29) allows us to determine the reflection and refraction coefficients for an incident SV wave. For P waves incident on the boundary from medium (1) we determine the following expressions for the angles. OLTJL= dd -f \ jc - s i n'{COS(0A- oi)}^ze^-i-zjc-oi PrX^^+ir^-Sin'l{(Gb,/Ca,)COS(0^~o6)} ^ J (3.30) cLr$ = 6A + -jr^* Sin ^{{Uz/c^COSidcK-oL)} Pr± = 6* + jr^ +5mH{(Cb^/Ca|)COS(0<A-o(.)} and for S waves ^rx^^^i:^--SIn ,{(Ga(/cbl)C0S(e^-(S)} o (3 ^ = 6* H- JG + sy n'{(GWCb,)cosf^-- jS;} Equations (3.26) may then be used to determine the reflec tion and refraction coefficients for waves incident from medium (1) by the use of (3.30) and (3.31) and the follow ing substitutions: and Cb2~^ Obi ' snould ^e noted that Ar^- and 3r$-are in this case amplitudes in medium (2) and C-rJL an^ DTJ)_ amplitudes in medium (1) . Finally, we consider waves incident on the free surface 0—0 (Figure 3-2). The solutions of (3.13) and (3.14) in medium (1) can be written as ©,= Aine -f- Arx6 - itb\rcos(e-§) '• • r,tbircos(e-prjL) (3.32) Following the same procedure as before we find the equa tions between the amplitude coefficients to be ~~COSoLrjzS\noLrJL ^ cos (3TJL si niV •\-ZC0S*pTJL •-4-vb*Cospsfng For P wave incidence the following relations are seen to hold £COSI co§oLS\r\cL (3.33) and the free surface indicated. 4 5 COScLrsL— COSdL £0S ^rJL— (Cbi/Ca\) COS oi Sinod= ^i-cosvT odrJL= -§-J"c + 5\r\]{cosoi}-=Z7io-oL and for an incident SV wave C0So^=(CAi/Cb»)C0S(3 COSprji= COS £ Sin p - /"Pcos2^" cCrx = x X.+ 51 n"1 {(GA>/Cbi) cOS £> Using (3.34) and (3.35) in (3.33), we obtain for incident P waves ^ •__ 4^bfcos^sino(,/i-i/^cos:cxl ~ (\-z v£ cos* oiT ^ V . (3.36) p ;<lcoSct Si not (I - zVbi COS*oQ ^ r ^ cos2-^s inotyi -• v£cos^ -t (1 ~zvbTcos VJ* in 46 and for incident S waves A -zv^cosps\r\$(\-2coszp)  rjL cosa p s i n (3 / - cos^ + (i - z cosAp " ^1 n (3.37) P^O —~ L K . In expressions (3.36) and (3.37), it should be noted that when the argument A in a square root is negative, then \f~/\ must be replaced by -~L\J~/\ for the solu tion to remain finite at infinity. As a check on the ampli tude coefficients in this form, the energy flux equations have been derived in Appendix I. <> 3.4 Computation of Displacement in the Case of a Dipping Layer To deteirnine the amplitude at any point in a wedge we must sum the complex amplitudes of all waves which arrive. From (3.23) we see that this requires the calculation of the amplitude coefficient and propagation direction for each wave. In the process of computation, complex angles have been employed in order that the cases for total reflection and incident angles greater than the critical angle are auto matically involved in results. In Appendix II, investiga tion of a Rayleigh wave written in terms of complex angles 4 7 shows that the real part of the angle indicates the propa gation direction and the imaginary part gives the decrease of amplitude. As a P and S wave arise from the initial refrac tion and from each reflection from the free surface and the boundary between media, the rays increase in number 71+ I as Z where fl is the order of reflection. The reflection process is terminated whenever the propagation direction is between 70 and 70+'9^ for in that case the wave propagates out of the wedge. For computation pur poses a further artificial termination was introduced by neglecting all waves whose amplitude was less than (o—{CT^ (the displacement amplitudes are normalized by the displace ment amplitudes which the incident wave would have on the free surface in the absence of the boundary). For computational purposes, it is important to note that the amplitudes and propagation direction for a wavefront is the same at all points. Hence, if the ampli tudes ( K ) and propagation directions ( TTl ) are determined for all waves reverberating in the wedge, then the total amplitude of motion at any point may easily be calculated. Further, as the Kf and rffl of reflected waves depend directly on M and TTL of the input wave, it is important at the initial refraction and each reflection to store Kl and for waves which may generate further waves. The rather 48 complex computation scheme used, can best be understood by examination of the flow chart (Figure 3-3). M and TVb for the refracted waves are first calculated and stored in vectors in order that they later can be used to cal culate the amplitude and phase at any point in the wedge. As the P wave is to be followed through the wedge, the amplitude and propagation direction of the S wave ( H$ and OTls ) are also temporarily stored in a vector which will later be used to investigate waves due to S \</ave conversion, f^l and T)1 are then calculated and stored for the ray which propagates through the system as P (at the same time storing f^g. and TTLs in the subsidiary vector) until P either propagates out of the system or the amplitude is less than £ . The waves generated by the S wave of this order and higher order waves they may generate are then examined (with attention again first focussed on the P), then those generated by the next lowest order until finally the refracted S and its resulting waves are examined. Upon completion of this calculation the vertical and horizontal displacement and the vertical-horizontal displacement ratio may then be calculated for different In these computation the values chosen for the parameters were Cbi/Cai= 0. £78<t , Caz/C&\~ I .^£>7 Cbz./Ga\= 0.73 II and |. I'7 £ • which cor respond to crust-upper mantle model employed by Haskell (1962). No Yes llnout parameters LN=i Determine M and m for : refracted P . and S and store in STP , STS. Store Ms in RECS(N). Determine whether reflection from free surface or boundary. Determine M and m for reflect ed P and 8 for incident P and store in STP,STS. Store Ms,ms in RECS(N). Mo Determine wether reflection from free surface or boundary. Determine M and m for reflec-ed P and S for incident S (using Ms and m5 from RECS(N)) and store in STP and STS. Store Ms,ms in RECS(N) .  Yes i) < b VO No Yes N = N-No Yes M = 0 Caluculation of total displace ment and vertical-horizontal displacement ratio Fig. 3-3. Flow diagram showing the computational scheme used to caluculate the amplitudes and propa gation directions of- the reflected waves in the wedge and thus the displacement and dis placement ratio at any point. Notation is given in Table 1. IE Print [End] 50 Table 1.' Notation used in Figure 3-3. STP(K,2) - complex storage matrix for p amplitudes and propagation directions STS(K,2) - complex storage matrix for S amplitudes and propagation directions RECS(L,2) - complex matrix to temporarily, retain S amplitudes and propagation directions of S rays which may generate further significant amplitudes N-1 - no. of reflections a wave has undergone M - amplitude m - complex propagation direction Subscripts p and s indicate P and S wave types 3.5 Displacement Discontinuities As discussed in the previous chapter, the last reflection which does not collide with boundaries gives rise to a diffracted wave which in the reflected wave solution appears as a displacement discontinuity. When the displacement discontinuity is small, only a small diffracted wave is required to provide continuity in dis placement and stress and hence the reflected wave solution adequately describes the physical problem. Large discon tinuities will require large diffracted waves; however, at large distances from the vertex the solution is still expected to be adequate as diffracted waves decrease rapidly with distance. 51 For incident P waves the magnitude of the maximum discontinuity of both the radial component from the last P wave and the tangential component from the exiting S wave is shown in Figure 3-4 for incident waves with propa gation directions cL - 60° and 120°. Several points should be noted. The discontinuity in the case of down-dip propagation is much larger than for the up-dip direc tion. This is expected since fewer reverberations occur before the wave propagates out of the wedge. The discon tinuity from P waves is relatively large in comparison to that for S waves and rapid changes in the amplitude result as the maximum discontinuity is associated with different exiting waves for different dip angles. The particularly rapid decreases observed result when an SV wave generating an exiting reflected P wave reaches the critical angle. The maximum P wave discontinuity at the next calculated point (calculation interval = 0.25°) is then due to another wave which may be of much lower amplitude. 52 30nindlAIV 3AI1V~I__ Pig., 5-4o Maximum displacement discontinuity of the radial component from the exiting P waves and tangential component from the exiting S.Y waves for an incident P wave with propagation directions <x>60° and cC=12.0° » 53 an incIdent SV wave (Figure 3-5), the, dis placement discontinuities are negligible for dip angles less than 21° for an incident wave with (9= Q0° indi cating that this solution very closely approximates the complete solution. Again the discontinuity is larger for the incident wave propagating in the down-dip direction. However in the case for j3 = 120° the large discontinuity for the outgoing P wave is for large dip angles rather than the smaller dip angles ( O^^JZO" ) as found for the incident P wave case. The physical reason for this is not clear. As discussed in Chapter 2, discontinuities exist in medium (2) due to the reflection of the incident wave and refraction of waves back into the lower medium. Except close to the vertex, the amplitudes of the resulting dif fracted waves are expected to be small. 3.6 Surface Displacements and Displacement Ratios 3.6.1 Incident P Horizontal and vertical displacements are plotted versus the parameters (T=^/3~JGH/CaiT * C H is depth to the interface) and illustrated in Figure 3-6. For an 53a initial propagation direction of 60° the vertical component changes very slowly with increasing dip angle. The hori zontal displacement changes rather more rapidly; however the major changes in character occur for 0^ JZ>0° where from Figure 3-4, we see that the role of the diffracted ray becomes important. For QC = 120°, a more rapid change in character of both the vertical and horizontal surfaces 54 3Cin±ndlAIV 3AI1V13H Pig,* 3-5c Maximum displacement discontinuity of the radial component from the exiting P waves and tangential component from the exiting ST waves, for an incident S¥" wave with propagation directions @=60°and p=120°0 55 in u o \~ O <_ U> ce O 1-o IU ce a z o to o cc CL o < CC < X o I-z UJ s UJ o < _l CL </> a < o I-Ui > o <0 o H o LU CC Q Z o < a. o cc CL to O co cc Ui I-o < CC <. X (J UJ £ UJ o < _J 0. CO o z o N CC o X o z o t-o 00 cc UJ I-o < ce < X o a UJ 5 Z g B ^ Y- CL < co O a § 3 CL H Q: > oo o . & O 5 CM £ < cc < X o 1-z UJ s UJ o < _l CL oo Q z o I-o UJ cc a z g CO s o cc CL z o N ce o X «> * M O 3anindwv Pig. 3-6. Horizontal and vertical displacements versus the parameter <y=;i/3JtH/Ca,T for incident P waves with propagation directions oC=60° and oC~120 for the range of dip angles 5°^©d^ 30°• is evident with increasing dip angle. This is particularly-true in the horizontal where for 0dL/M7° tne surface is rather featureless. However it should be noted that the discontinuity curves in this case indicated that the diffracted wave may be important over most of the range of dip angles. The displacement ratios V/H which are of interest in practical analysis are shown in Figure 3-7. For the initial propagation direction oL - 60° , the ratios are very similar for dip angles less than 10°. For dip angles greater than 10°, the peaks move to larger values of (5s and increase markedly in amplitude. For oL = 120°, the ratios change much more rapidly even at small dip angles with the peaks moving to increasing 6^ again. However in this case the amplitude decreases until for 9<K~=£Q° the V/H ratio is almost constant for variable 6^ 3.6.2 Incident SV From Figure 3-8, it is seen that the displacement surfaces- exhibit significant character. One particular feature is that for dip angles greater than 18°, the period of the variation of horizontal displacement becomes short and the corresponding amplitude small for |3 = 60°. It should be noted that this change occurs before diffracted waves become INCIDENT P WAV Lb PROPAGATION DIRECTION • 60 ° PROPAGATION DIRECTION =120° -COIT caiT Pig. 3-7« Displacement ratios ?/Bversus the parameter 6"=A^3JcH/CaiT for incident P waves with propagation directions oC=60° and o(=1200 for the range of dip angles 5% 0^ 30°. 58 59 significant. For (B = 120° the horizontal displacement oscillations become very small for dip angles greater than 10° while the vertical displacement oscillations more slowly decrease in amplitude with increasing dip angle and at the same time the period of the variation lengthens. The above features are most evident in the dis placement ratio curves H/V (Figure 3-9). For even small dip angles marked differences from the horizontally layered curves are evident. At the larger dip angles for propaga tion direction |3 = 60° we see the rapid oscillations due to the horizontal component and for JQ — 1200 the ratio becomes featureless. INCIDENT SV WAVES PROPAGATION DIRECTION = 60 ° PROPAGATION DIRECTION = 120° 60 DIP =30° 25° 20" 10' 'ai T 5 5 0 DIP =30° 25° 20« 10' 'at T Pig... 3-9. Displacement ratios H/V versus the parameter fr--£/3JoH/C*fT for incident ST waves with propagation directions (3 =60° o and ^=120° for the range of dip angles 5%QA^30° » 61 CHAPTER 4 HEAD AND REFLECTED WAVES FROM AN SH LINE SOURCE IN A DIPPING LAYER OVERLYING AN ELASTIC MEDIUM 4.1 Introduction A number of workers (e.g., Lapwood (1961), Hudson (1963), and Hudson and Knopoff (1964)) have investigated the propagation of surface and diffracted waves in wedge-shaped media. Hudson (1963) pointed out that the solution in the case of a rigid lower boundary could be divided into two parts - the multiply reflected and diffracted wave solutions. However, the early part of the seismogram con sisting of head and multiply reflected waves which are often used in interpretation has not been well studied for a dipping layer overlying an elastic medium. In the present chapter the author will establish one method of solution and will theoretically investigate the problem for an SH line source in an elastic wedge overlying an elastic medium. In Chapter 2, a solution for the problem of multiple reflection of plane SH waves by a dipping layer has been found. By integration of a solution of this type, the dis turbance due to a line source is sought which does not in clude the diffracted wave term. However, for a transient input, an observation point distant from the vertex receives the reflected and refracted waves earlier than the diffracted 62 waves which result from collisions with the vertex. There fore the present solution should apply to the composition of the initial section of the seismogram. The formal solution is evaluated by the steepest descent technique as recommended by Honda and Nakamura (1954) for evaluation of branch line integrals and as applied by Emura (1960) and others. In this way the wave forms and the ranges of existence of the head waves are determined for various dip angles for comparison with the case of a horizontal layer. For all computations the following elastic parameters cor responding to those of Haskell (1960) are used in the S wave velocity in the upper layer Cbl = 3.64 km/sec, the velocity ratio ^-Cbz/c^— 1.2-7 > anc* the rigidity ratio tions of poles, the surface wave problem reduces to find ing poles of the finite series expression of our solution. Further, the discontinuities in displacement associated with the diffracted waves have been found and hence this problem is separated from the determination of the solu tion due to other waves. The solution of the present prob lem is therefore an important step for the consideration of surface and diffracted waves. 4.2 Equation of Motion and Boundary Conditions S =/<*//<i= 1.88 As surface waves are obtained from the contribu te propagation of SH waves through a system con-63 sisting of an elastic medium of rigidity jX\ > density f , and a dip angle 6, •+• d& > overlying an elastic medium of rigidity JJLZ and density j\ (Figure 4-1) will be considered. .The free surface is Q — —Q\ and the boundary between the elastic media is Q — . A car tesian system ( % } % ) is related to the cylindrical coordinates ( T '0 , E ) by the standard relationships %~TCO$Q » ^=TSln0 ' and 2,= 2 . The motion is generated by a line source ( S ) °f SH waves located, at ( ck. 7 0 ) in the cylindrical coordinate system. For the above problem, the motion is independent of Ei , and the displacement has only a z-component. Then, assuming a time variation of the form Q, , the equation of motion V U,= 7^-r-T l/=l,£ (4.1) Obi <?"D becomes in cylindrical coordinates where C^i= sjj^lffl is tne velocity of the S waves and Free surface S(d5o) (2) Pigo 4-lo Geometry of the problems.the line source (s) is located at (d,0) the receiver (R) at (r,0) in the wedge bounded by the free surface (0=~6,) and the boundary (0=Sb) between the two media0 65 The only non-zero component of stress is The boundary conditions then become (Pae),= 0 o.t 0=~el (4.4) and (kt d = Qz (4.5) To solve the line source problem, a plane wave solution satisfying the boundary conditions will first be obtained. The line source solution can then be obtained by integration of this solution with respect to the cylindri cal angle. 4.3 Steady State Plane Wave Solution The initial displacement due to a plane wave is expressed in the form -^b,{(d-^)CoSoi.i, + |^-|5Jno6Lj-= A.: • e 66 where c(.i is the angle between the x-axis and. wave normal of the SH waves and may take on complex values in the eva luation of the effect due to a line source. For 07 0 IXQ represents waves downgoing from the x-axis, while for 0 "x. 0 7 riiQ represents waves upgoing from the x-axis. In those cases where the waves interact with the boundaries, the former collides with the dipping boundary first and the latter with the free surface first. In Chapter 2, the reflection and refraction of SH waves in a dipping layer has been investigated in detail. The solution by multiple reflection is obtained in the same manner. For any observation point in the wedge, four expressions are required to express the motion depending on whether the initial direction of the wave is positive or negative and whether the final reflection is from the boundary (1) or the free surface (2). Using the same pro cedure as the derivation of (2.18) and (2.19), these expres sions are ^(N)=A^lSNe (4.7) 4(N)=A,C(70A^)e S2(H')=A;E:(£A)e tt 67 where the maximum numbers of reflections from the boundary |\j and ^v]/ are seen by examination of the phase in (4.7) to (4.10) to be determined by K, + e*Z zfNHM + ^NG^ Jt-Q\ (4.ii) or (4.12) (4.13) and [\j/ by or 70+^ ^(M'+Oe.+^N'e*^ (4.14) The expressions for the direct wave and the wave once reflected from the free surface are from (4.6) C_A J>~^b\^CO$(cii+\Q\)~itb\(kCO$cll 7)0-r\'v,& (4.15) and A)0— r\i& (4.16) 68 The reflection coefficients for a wave which started as a downgoing wave with respect to the x-axis are similar to expression (2.20); namely tf ASin(&+-*(1HX (4.17) and for a wave which started as an upgoing wave with respect to the x-axis is * As In{ e^lle^^ (4.18) z\ and § are defined by A=Cb^/Cbl and &~Mz/Ml The solution on and close to the surface due to the initial disturbance and satisfying the boundary conditions (4.4) and (4.5) is, (using the same method as for the derivation of 2.25 and 2.26) 14 = So + So + S* (N -1) + 5^ (N - 0 + -ST(N- 0 + Sl( M - o (4.19) for conditions (4.11) and (4.13) <= 5o+^ + S!(N-O + S1(N-O+S~(N')+SI(N/; (4.20) 69 for conditions (4.11) and (4.14) u;= 50+ s> s>)+ SI(N)+s;v-o+ si (4.21) for conditions (4.12) and (4.13) <= S0+ S~ + S*(N) + Sl(N)+ S~(N')+ Sl(lM') (4.22) for conditions (4.12) and (4.14). It should be pointed out that these formal solutions are not a physical solu tion for an incident plane wave ©.£ a particular real angle. oCl but are the plane wave forms satisfying the boundary conditions from which the line source solution will be obtained. The last terms of the series expressions (4.7)-(4.10) give rise to discontinuities in displacement and stress which serve as boundary conditions for the dif fracted waves. These will be investigated in detail in a later section. Neglecting these last terms of the series, u;=50+So+s>-i)+jSi(N-o + sr(N'-o+sioi'-o is valid everywhere in medium 1. For evaluation of the displacement due to a line source, it is convenient to express formulae (4.7)-(4.10) and (4.15) and (4.16) as 70 5>)=A±(7tA;)e^,R;|COSK_e;i) 71=1 H=i k/ ^=1 \fc=i 7 where p+ = (4.26) b> - Al6 (4.27) b^A-G (4.28) (4.29) 71 n2 . • (4.30) t&n6ui = • (4.31) tan 8^.2 = (4.32) (4.33) tane0 = rsin|e|/{dL- rco$\e\] tan6j= TSinf^i-e)/{A-rcos(^a,+e?} . 72 4,4 Formal Steady State Solution for a Line Source In order to generalize the results to the case of a line source, the operator TC + I oo ~kb\j c^cxl ^ (4.35) -loo is applied to the plane wave solution. In particular, the displacement IXo due to the initial disturbance can be written using (4.6) as 7t+loo -Loo ~t0O Equation (4.36) can easily be derived from the results of Nakamura (1960). When {^,0 i-s large, (4.36) can be approxi mated by the asymptotic formula, Uo = /\ /^ly_e~^b|P^-tf1 (4.37) l' (to which are the outgoing waves from the line source. 73 Using (4.19) to (4.22) and (4.23) to (4.28), the solution in the wedge corresponding to a line source (4.36) can be obtained as 4.5. Evaluation of the First Series Term of the Integral In this section the integration of the terms which are produced by waves which are reflected once by the boun dary between the media are evaluated. (As a guide to the evaluation of the higher order series terms, the contribu tions due to waves twice reflected from the boundary bet-wreen the media are calculated in the Appendix III.) From (4.23)-(4.26) and (4.38), we see that they have the fol lowing forms: f (4.38) (4.39) - loo where (4.40) 74 and Jc.= 1 Z an(i 7Yl~ H" ., — • From the euqations (4.17) and (4.18), 0^0^ and $ = 201+0.* and R,^ and are given by the equations (4.29) to (4.32). For this integration, the original path L is taken in the plane for which R.6(A-s)/>0 . The integrand of (4.39) contains the two-valued function \$~\J\— A^COS^^^-rol*) and its branch points are given by the relation C0S(^1TU-+-c/_r,) " and are therefore located on the real axis of the oil -plane at the points g(6B= 60~ 0,^ ), B'(©B'= ^~60~ ^'°')1" (Figure 4-2) where COS6o= ~ TL , tne refractive index. To facilitate evaluation, it is assumed that the medium is very slightly absorptive by setting -yi— Tlo~i<0 where £, is a very small positive quantity. (This assumption does not affect the final results which correspond to £—^ Q but is only a technique to facilitate evaluation of the integrals.) The branch point B is then displaced by [3/^1—parallel to the positive imaginary axis on the ot^-plane. We choose the branch cut given by R*Q[/\-S)—0 which is defined by: cos (x+^) s (n (x+$ m) cos h u- s i n h y = n01 cost* + 0*) co s h"^ - s i nA(#+<f>™) s! n rf ^> n0 C4 •41} y 76 In Figure 4-2, LB is 0 , ^B is ln\(Xs)=0 and the signs of Im(As)in the O^t-plane are indicated by plus and minus. When for large distances from the origin, £ b^b{ R-IACOS(O(.I- eJJ) vanishes along the path L , the path, of integration can therefore be shifted on the Riemann surfaces. The region where I m (COS (OCL — 8|j2.)J<^ 0 OT P, vanishes at a large distance from the origin is shown by hatching. The original path L , along which the relations Im(Sl)ldl)<0 anci Im(s)^ 0 hold, can be replaced by L$ and ( L i , L>z. ) >' where passes through the saddle point and ( L i , L.2.) goes around the branch point B • The dotted lines denote that they are on the second Riemann sheet where RLs^A-s")^ 0 . Each of them is drawn along the path of steepest descent, cosOfc-Ocosh^.l <4'42) and co5(x-0^)cosh^=cos(0B- e^) (4.43) respectively. 4.5.1 Contribution from the Saddle Point (Reflected Waves) From (4.42), L,s makes an angle ^jp- with the x-axis. In the neighbourhood of the saddle point the 77 following approximations are valid: <*l-e£= ftf* (4.44) and C0StdLi-da)=\-l?7z (4.45) The contour integral along Ls for (4.39) is then (4.46) Expanding the term A, i^l) near the saddle point and using Watson's lemma (Jeffreys, 1956) we obtain ^S-Ai / A, (0(JJ6 (4.47) where, if 0(JL > 6 B Sin(0,w+Q- S^/l/A^CoS2(0,w+ 9£) and, if Q,^ < 6g (4.49) 78 where 4.5.2 Contribution from the Branch Point (Head Waves) Next the contribution from the integral along L| is considered. L| and L,^ are taken along the path of steepest descent around B an^ tend to ^jp- + 0(^ 4~ L<2° . Setting Sin(^VoU)' we have from (4-39) , uLlLa= tb,Ai x / 0m , Tn. (4.5 0) It should be noted that along near B that and Im(A-S)>0. (4'51) From (4.43) along Li^, we can write cosC^r~0^)-cos(eB-0a)-iT: x>o C4.52) therefore CUE,= L^/Sin(oCc-e^) (4.53) 79 along {JZ near B . Putting ot I = 0 B + W-+ i,V (4.54) for very small X{ and V" , we have approximately (u+lu)sin(e8-eu)=L'"c (4#55) On the other hand, along L^. we have Sm(cxU+ (f^)- S\n(0o+ VL+ IV") Zl - I/A* -jriU+LU) (4.56) A and from (4.55) Hence, in the limit when along , we have using (4.51) M •V5in(0B-61^) ' (4.58) As CO 80 we have | -IK R£ COS (66-6,7)+ i-(i-i/Air{5in(eB-e-)p'e L ^ (4.60) 4.6 Aperiodic Solution For computation of synthetic seismograms, it is convenient to choose a displacement of the form 0(t)=- ^ A>0 , OO Performing the operation 0-00 on (4.37), (4.47) and (4.60) we obtain the following solu tions (1) Direct Waves y --/y ^ . A \ 81 where tD= do/C Uc=-A'i (2) Waves reflected once from the interface X JO A AT(C) 7TV Cos {|- tan ll^i^L-t- 70 c A. for 9|i > $B Uc=-At A -r-^jpj* x COS)4-tan'-^^-i-^-tit \ £Li&~ y^[SL/C'b\ . (3) Head Waves (4.62) (4.63) U = A- ZJZJC-S /Cbi l"2 /A(I~I/A^ (R-)^ {sin(e6-e-)}3/^ x A | m ^ {iTp^yr cosl-Ltan <^&L + f jcj (4.64) where fL „ R * CO S(6 8 - 6^) u «i« — - • . Cbi " H -| A. 82 4.7 Interpretation of the Travel Time In a study of wave propagation from a point source in a horizontal layer, Honda and Nakamura (1954) evaluated the contributions from branch points and a saddle point and associated the time factors of these contributions with head waves and reflected waves. Similarly, in this section we will determine the reflected and head wave travel times and find that they are given by ^p\jL and H^lil ' the time factors of the saddle point and branch point con tributions . Consider first the path SAR. i-n Figure 4-3. Then ' SA+ AR. S'R. SAfl Cb, Cbi __ JJTcosdz-rcos(ea^ejj^-f {AsV/^Ws^iel^e)}" c b| + Also, the travel time along the path S8CR, is given by /SBCR Cbt ' Cb, eisin eA (4.65) . r- , + ^kcose*-rcos(&-e; 6 ^sine^rsinCe^e)] + rsin(0^-e) tan(eB+e*) J cbiSm(0B+02) 4-3o Basic ray paths used in physical interpretation of contributions from branch and saddle pointso 84 G (cicos0^-rcos(e^e))cos(eB+ e*) +(dsiK\0A+r5in(ea-e))5in^B-f &0} (4.66) -bl since cos(eB+6;0== CM/CM. , ra Hence we have verified the interpretation of $y,\SL and tn tj,.0 as the reflected and head wave respectively for /)TL= 4~ a^d X— I . In a similar manner, D"fc,0 and H"^JJI can ^e interPreted for different values of 0TL and Jl as shown in Figure 4-4. Obviously, for the observation point distant from the vertex, these waves arrive earlier than the diffracted waves which are pro duced by collisions of waves with the vertex. Hence this solution should adequately describe the early section of the seismogram. 4.8 Range of Existence of Head Waves The range of existence of head waves can be deter mined by considering the process by which the integral (4.39) is evaluated. In Figure 4-2, when ©B/'^IJZ. » we can f°rm a closed contour which connects with the original path and which includes contributions of the saddle point and the branch point on application of Cauchy's theorem. On the 17V other hand when 0g<^ Q\j> > a closed contour cannot be (a) (b) (c) (d) T?i£ 4-4• Ray paths of the head and reflected waves expressed by the first series term of the integrals. CO 86 made without excluding the integral around the branch point. Therefore in the first case we have contributions from both the branch point (head waves) and saddle point (reflected waves) while in the second case there is only a saddle point contribution (reflected waves). It is clear that the critical condition to decide the existence of head waves is DB - 0|JL • Obtaining UB ANCL Djj2 by the use of (4.17), (4.IS) and (4.29), (4.30), (4.31), (4.32) respectively and using the critical condition we obtain the following transcendental equations: tan1- rsinJ^gj+gggH^_ ros~U„ ft d-rcosUdi+zo^-T-e) ~ A ^ C4.es) (k-rcostze^^^J"c ~K-dz-ze{ (4.69) _^ Q T4.70) In Figure 4-5, the range of existence of head waves whose paths are illustrated in Figure 4-4 is shown for 0(=^° and the observation point on the line 8~Q. The abscissa is the ratio of source to observation point distances from the apex while the ordinate is the maximum for which head waves exist. As expected from the small Pig. 4-5. Maximum value of 6Z for which-the head waves shown in Pig. 4-4. exist versus the ratio of source to observation distances. The observation and source points are 5° from the free surface. 88 path difference, the range of existence of (a) and (b) are close as are (c) and (d). The range of existence also decreases with increasing number of reflections and decreasing ratio cL/T For an observation point at 5° from the free surface and a source-vertex to observation-vertex distance ratio of 10.0, Figure 4-6 shows the range of existence of head waves with changing S\ , which corresponds to a change of depth of the line source. It is noted that with in creasing 0| , the dip angle for which the head waves (a) and (b) exist linearly increases while for head waves'of type (c) and (d) it linearly decreases. 4.9 Discont inuities Discontinuities in displacement and stress in medium (1) arise because the first collision of the wave with a boundary changes from the free surface to the boundary between the media as oi\, passes through zero. This colli sion with the vertex results in a diffracted wave which is not considered in this solution. When both initially up-goihg and down-going waves and the interface of the last reflection are considered, we have four cases of the com bination of the discontinuity as shown in Figure 4-7. The cross-hatched areas indicate the regions for which the 0 5 SO 15 9. (DEGREES) FIgo 4-6. Maximum value of the wedge angle (6|+6A) for which the head waves of the types shown in Pig* 4-4» exist for an observation point at 5° from the free surface and d/r=10o0» (a) (b) o (c) (d) Pig, 4_7. Discontinuities in medium (l)due to interaction of the wave with the vertex. The lined areas indicate the regions for which the geometric wave from the last reflec exists with the term from which it arises indicated in brankets. 91 geometric wave from the last reflection exists. From equa tions (4 .11) - (4 .14) the equations of these lines of dis continuity are • + (OS, ® Si o) s; (ft) si e = 2(N-i)6i-i-xNe^-7c (4.7i) .6= 7C-zN0i-2Ne^ (4.72e = j^-ziu'+oe^-zti'e^ (4.74, The discontinuities indicate a discrepancy of my solution from the complete solution of the physical prob lem. In order to obtain a quantitative estimation of the displacement discontinuities, plane waves incident toward the vertex and propagating at very small angles upward (m = -) and downward (m=+) have been examined. The result ing discontinuities are shown in Figure 4-8. For this geometry the reflected wave solution is a good approximation to the complete solution for 02,<C!3°. However for 0^,^2,1° the diffracted wave plays an impor tant role. However, this formulation should adequately describe the early part of the seismogram as the diffracted waves from the vertex will arrive later than the initial phases. It is seen that coincidence of the discontinuities in Figures 4-7b and 4-7c leads to at least partial cancel lation of the discontinuities. Two special cases are of Fig* 4-8„ Relative amplitudes of the displacement disconti nuities due to a plane initial wave close to the x-axis for propagation upward (m=-) and downward (m=+)o 93 interest as total cancellation results. (1) Lower boundary free or rigid When the lower boundary of the wedge is either free or rigid then A-fo and A-fe, are +1 or -1 respectively, The condition 6> + 0*= N + N' leads to (4.71) = (4.74) or (4.72) = (4.73) so that the two lines of the discontinuities are coincident and no discontinuities exist. Hence the solution is complete and no diffracted waves exist. (2) Surface Source If the line source is placed in the surface ( 6,= 0 ) then from (4.17) and (4.18) A+t = A-fe. For the particular situation N + N' • we have (4.71) = (4.74) or (4.72) = (4.73) and the two discontinuities coincide, hence in this case no discon tinuities exist in medium (1). In this discussion the discontinuities in medium (2) are again expected to be less important than those in medium (1). 94 4.10 Dispersion Equation for the Lower Boundary Free and Rigid In this section the dispersion equation is derived for a dipping structure in the simple case where medium (2) is either air or rigid. When medium (2) is air or rigid, A^> and-A-^ become +1 or -1 respectively. For 2Tn.(9\-Td&)<^ | we can write cosC^i+^m(0,+e4> = cosoCL-^7n(e,+eA)sinoCu In this case S*(N)=AL-e •K±0 6 (4.75) <VKIWA P '"^ 6  (4.76) 95 Similarly the expressions for S,(N) anc! S^CH) can °e obtained. Operating with -loo poles appear from the relation I \y — \J which yields Sin(-feb,rce,+ejsmoCL)=o cos(feiT(0, + 6^)sincx:L) =o For real oLi , we can then write (niz (4 where C^i is the phase velocity as COSoil ~ Cb\/C>n ( 0<Coci<CJC ). This expression is the same as that obtained by Nagumo (1961) for a sloping rigid bottom. Further, if we put T(Gi-t©^)~H ( H is the depth in the case of a horizontal layer) , the dispersion relation (4.77) coincides with that of the horizontally layered case. Nagumo (1961) has called Cyi and %L = -^-s— the formal phase and group velocity to differentiate from the observed velocities. 96 4.11 The Horizontal Layer Solution It is interesting to derive the solution for a horizontal layer using my method as the transition of the solution to the horizontal layer case may suggest a method for obtaining the surface wave solutions for the dipping layer. For the diffracted wave problem, it is useful to study this transition as the quantitative and qualitative behaviour of the discontinuities as they approach zero for zero dip angle may indicate the nature of the diffracted solution. The same ( X-, ^ ) coordinate system is used with the x-axis now being horizontal (Figure 4-9). The source is placed at ( cL , 0 ) in the layer of thickness H-H1 + H2. Employing the same procedure as for the dipping layer, we obtain the displacement for the time variations 70 + loo e -f-e , <~7 Auf ~i1lb|{(4-;<)cos Wat . -L^bi{(^^cosoCu+(^(nH,+ nHz)+^)sinoCL} "T C •T O - it b! {(<L-PQ COSoLl + (Z( (71.+ 0 H, + UH_)+U)5' Hoci} I + e. J J -i&blRoCOS(cLi~60) -HLb\RoCOS {d-i-6~) 6 +6 bl I ni -loo (4.78) Free surface S (d9o) x < X-x R ( x8y) (!) 7TTT (2) y Pigo 4-9» Coordinate system for the horizontal layer case -with the source (S) at (d,0) and the receiver (R) at (xBy)9 98 (4.78) \ O "I- g j- &0C where A,= ASinoCo-Wi-^cosV.t RXLI=/W^M^Hr HO -lj (4.79) tan d~^=(z H,+^) / (ck- PC) t(\ndnr{z(w~HO~^/(ck-x) tan dtz^ (znH + VrVfa-pc) (4.80) As equation (4.78) is o£ the same form as (4.38), formulae (4.62) or (4.63) and (4.64) can be applied for the reflected and head waves respectively. Therefore the variations of D7Tl <m. the waveforms depend only on the values rtfljj. , 6^ and QQ . Equation (4.78) can also be derived from (4.38), the formal dipping layer solution, if we take the limit as rsin6i=Hi <xr\A rs\n9^H (4.81) 99 In the case of a horizontal layer, surface waves appear from contributions of poles. As our solution trans forms to the horizontal layer solution, we could investigate surface waves in the case of a dipping layer if the finite series expression of our solution can be changed into a compact form which corresponds to a normal mode expression. 4.12 Computation of Displacement Seismograms For the direct wave, head waves, and waves once reflected from the boundary, displacements have been cal culated for the three cases shown in Figure 4-10. Elastic constants are again those employed by Haskell (1960) . Ray paths of waves which Contribute to the seismogram are shown in Figure 4-11. The component waves are shown on the time-displacement plot of Figure 4-12 and the arrival times corresponding to the ray paths indicated by lettered arrows. A detailed feature is the small amplitude of the head waves compared to the direct and reflected waves. This is expected from inspection of equations (65) and (66) which show that the head waves decrease as [ /( and the reflected waves as I/("R^A)' Although the travel times changed significantly, the wave forms of the refracted and reflected waves do not undergo large changes for the three cases illustrated. 100 Fig. 4-10. Three cases for which theoretical seismograms were caluculated. The parameters used were: H,= 9.59 km, -HA= 3.00 km, D= 99-6 km, d= 10.0 km, and the displacement parameter c= 0.05 sec. 101 102 Pig„ 4-12. Displacements of the component waves for the geometries given in Pigs. 4-10a„. 4-10b9 and 4-10co 103 Figure 4-13 represents the seismograms synthesized from the components of Figure 4-12. The seismograms look very different.. However, the different arrivals are all recognizable except for the head wave (b) is embedded in the wave forms of the direct wave (c) and the reflected wave (d) in the case of the horizontal layer. A very noticeable feature is the late arrival of the reflected wave (e) in the case of the horizontal layer. More multiply reflected waves will appear as later phases. As the dis tance between the observation point and the vertex is 10.0 km and the velocities of medium (1) and medium (2) are 3.64 km/sec and 4.62 km/sec respectively, the diffracted waves hardly contribute to the section of the seismogram shown here as the diffracted wave arriving 4 to 5 sec after the first arrival is due to a head wave of small amplitude interacting with the vertex. 104 (a) 23 * 25 a Travel time (sec) 29-(b) d 23 t a 25 29 Fig* 4-13» Synthesized seismograms resulting from the dis placements of Flgo 4-12o 105 CHAPTER 5 SUMMARY, CONCLUSIONS AND FURTHER STUDIES 5.1 Summary and Conclusions In this paper, the behavior of elastic waves in a dipping layer overlying an elastic medium has been investigated in terms of body waves in order to expand the models available for the interpretation of crustal structure. In Chapter 2, the reflected wave solution for a plane SH incident at the base of a dipping layer and per pendicular to strike has been developed and numerical examples presented. For waves propagating in the up~dip direction with angle of incidence in the range of that for teleseismic S waves (45°<^ oi <C 75°), it is found that the reflected wave solution closely approximates the com plete solution for small dip angles as the boundary con ditions are approximately satisfied. However, for waves propagating in the down-dip direction, the displacement discontinuity along the edge of the final wave which does not collide with the interfaces becomes large. In this case the wave has reverberated only a very few times within the wedge and hence is still of significant amplitude. The size of this discontinuity lias been determined and hence serves as a guide as to whether the ray solution 106 is applicable. For a transient input to the wedge, the reflected waves will arrive earlier than the diffracted waves and hence even for large discontinuities, this type of solution should apply to the composition of the initial section of a seismogram. The diffracted wave, must provide continuity in displacement and stress along the edge of the final wave as well as those imposed at. the surface and the boundary between the media. In Chapter 3, the behaviour of P and SV waves in cident at the base of a dipping layer and perpendicular to strike has been investigated by means of a reflected wave solution developed using a cylindrical coordinate system. Due to the complexity of this problem, a series solution is not presented as was done for the SH problem; however, a computational scheme is given by which the amplitudes and propagation directions of all the contri buting waves are determined. In this way the displacement at any point in the wedge due to reflected waves may be found. Numerical examples of displacements and displace ment ratios at the surface are presented for incident waves propagating in both up-dip (oC, ^ = 60°) and down-dip (oL, ^ = 120°) directions. It is found that the displacement ratios versus frequency curves for constant depth to interface become flat for incident P and SV waves propagating in the down-dip 107 direction for dip angles greater than 15°. This is very different from the case of up-dip direction. For the P wave propagating in the up-dip direction ( = 60°), the peaks are large for large dip angles and for dip angles greater than 10° the peaks shift to lower frequency and become narrower with decreasing dip. A feature of parti cular note is that the H/V displacement ratio curves for incident SV are much more sensitive to small changes of dip at small dip angles than are the V/H displacement ratio curves for incident P waves. It appears therefore that a study of SV waves would be more likely to yield informa tion concerning dipping interfaces than would P waves. For waves- propagating in the down-dip direction, it is found that the displacement discontinuity may be large even for small dip angles indicating that the diffracted wave is of significant amplitude. However, since the reflected waves will arrive earlier than the diffracted waves for a transient input to the wedge, the reflected wave solution should again apply to the composition of the initial sec tion of the seismogram. The complex propagation direction used in this chapter has been interpreted in Appendix II using the example of a free Rayleigh wave to show that the real part of the angle indicates the propagation direction and the imaginary part gives the decrease of amplitude. In Chapter 4, the propagation of, SH waves from a 108 line source in a clipping layer overlying an elastic medium has been investigated using multiple reflection formulation. A formal solution which does not include diffracted waves has been obtained. The first two series terms of the in tegral have been evaluated using the method of steepest descent to obtain displacements for both a harmonic and an aperiodic time variation and contributions have been inter preted using ray paths in terms of head and reflected waves. If in the integral the branch points are smaller than the saddle points, head waves do not appear. Hence the range of existence of the various types of head waves may be determined. Using the same technique, the solution in the case of a horizontal layer has also been found and compari son made with the dipping layer through numerical examples. The wave forms of the arrivals do not. differ greatly; however, the character of the synthetic seismogram markedly changes due to changes in arrival times. Discontinuities in displacement which are associated with the diffracted wave have been studied. For special cases it is found that the reflected wave solution is the complete solution. In the other cases, this solution can be applied to the initial section of the seismogram. 109 5,2 Suggestions for Further Studies As a result of this study, the following lines of investigation are suggested.: (1) The calculation of a synthetic seismogram at a station in a wedge with an elastic base for an incident plane wave pulse. (2) The calculation of the amplitude characteris tics of a multiple reflection in the case of both dipping and horizontal layers by a combination of the technique developed in this thesis and Haskell's method (Haskell, 1953). (3) The problem of P and SV line sources in a wedge overlying an elastic medium in terms of head and reflected waves neglecting the diffracted waves. (4) The exact solution in terms of multiply re flected waves and multi-reflected head waves in the case of a line source in a dipping layer with an elastic base for transient time variations using the method of Cagniard (1962). (5) An investigation of surface wave propagation in the presence of a dipping layer overlying an elastic medium. In the case of a horizontal layer, surface waves appear from a contribution of poles. When the dip angle approaches zero, the solution found in Chapter 4 reduces 110 to the case of a horizontal layer. Hence surface waves in the presence of a dipping layer could be investigated if the finite series solution can be written in a compact form which, corresponds to a normal mode expression. (6) An.attack on the problem of diffracted waves using the multiple reflection wave solution and the dis continuities found in this solution which are related to the diffracted, waves. Ill BIBLIOGRAPHY Cagniard, L., 1962. Reflection and refraction of progres  sive seismic waves, McGraw-Hill, New York. Clowes, R. M., Kanasewich, E. R., and Cumming, G. L,, 1968. Deep crustal seismic reflections at near-vertical incidence, Geophysics, 3_3, 441-451 . Ellis, R. M. and Basham, P. W. , 1968. Crustal characteris tics from short-period P waves, Bull. Seism. Soc. Amer.. 3 58_, 1681-1700. Emura, K., 1960. Propagation of the disturbances in the medium consisting of semi-infinite liquid and solid, Sci. Rep. Tohoku Univ., Ser. 5, Geophysics, 12, 63-100. Ewing, W. M., W. S. Jardetzky, and F. Press, 1957. Elastic  waves in layered media, McGraw-Hill, New York. Fernandez, L. M. and Careaga, J., 1968. The thickness of the crust in central United States and La Paz, Bolivia, from the spectrum of longitudinal seismic waves, Bull. Seism. Soc. Amer., 58_, 711-741. Fuchs, K., 1966. Synthetic seismograms of P waves propaga ting in solid wedges with free boundaries, Geophysics, 3_1_, 524-535 . Haskell, N. A., 1953. The dispersion of surface waves in multilayered media, Bull. Seism. Soc. Amer., 43, 17-34. Haskell, N. A., 1960. Crustal reflection of plane SH waves, J. Geophys. Res., 65_, 4147-4150 . Haskell, N. A. 1962. Crustal reflection of plane P and SV waves, J. Geophys. Res., 6J7 , 4751-4767 . Honda, H. and Nakamura, K., 1954. On the reflection and refraction of the explosive sounds at the ocean bottom II, Sci. Rep. Tohoku Univ., Ser. 5, Geophysics, 6, 70-84. Hudson, J. A., 196 3. SH waves in a wedge - shaped medium, Geophys. J. R.A.S., 7, 517-546. 112 Hudson, "J. A. and Knopoff, L. , 1964. Transmission and reflection of surface waves at a corner 2, Rayleigh waves, J. Geophys. Res., 69, 281-289. Ibrahim, A. B. , 1969. Determination of crustal thickness from spectral behavior of SH waves, Bull. Seism. Soo. Amer., S9_, 1247-1258. Jeffreys, H. and Jeffreys, B. S., 1956. Methods of mathe matical physics, Cambridge Uriiv. Press, Cambridge, England. Kane, J. and Spence, J., 1963. Rayleigh waves transmission on elastic wedges, Geophysics, 28_, 715-723 . Kane, J., 1966. Teleseismic response of a uniform dipping crust (Part I of a series on crustal equalization of seismic arrays), Bull. Seism. Soo. Amer., 56, 841-859. Keller, J. B., 1962. Geometrical theory of diffraction, J. Aooust. Soo. Am., 5_2, 116-130. Lapwood, E. R., 1961. The transmission of a Rayleigh pulse round acorner, Geophys. J. R. Astr. Soo., 4_, 174-196. McGarr, A. and Alsop, L. E., 1967. Transmission and reflec tion of Rayleigh waves at vertical, boundaries, J. Geophys. Res., 72_, 2169-2180. Nagumo, S. , 1961. Elastic wave propagation in a liquid layer overlying a sloping rigid bottom, J. Seism. Soo. Japan, 14_, 189-197. Nakamura, K., 1960. Normal mode waves in an elastic plate (1), Sci. Rep. Tohoku Univ., Ser. 5, Geophysics, 12, 44-62. Phinney, R. A., 1964. Structure of the earth's crust from spectral behavior of long period body waves, J. Geophys. Res., 69, 2997-3017. Sato, R., 1963. Diffraction of SH waves at an obtuse-angled corner, J. Phys. Earth, 11_, 1-17. , 113 APPENDIX I ENERGY RELATIONS As a check on the amplitude relations derived in the text, the method used by Ewing et al (1957) has been used to derive expressions for energy partition between the incident, reflected and refracted waves. To calculate kinetic energies, we note that the velocities 1X are related to the displacements IA. by IL— LCOU. and hence may be obtained directly from equations (3.17) and (3.18). The energy flux for the waves can then be obtained by multiplying the kinetic energy per unit volume, ~~^f(^r~^~ l^e') ^y the velocity of propagation and the area of wavefront involved. For a P wave incident on the surface, its energy flux per unit area must be equal to the sum of the energies in the reflected and refracted waves. We have X c*2 ctn \s\n(oLi~dd)\ ~ ~Z?2. Caz Crjt ^2. (SI n (oirsi- 0<0| + 2 ?z cti Drx cbx I s f n QO| (A-1.1) A following computationally more useful form is obtained 114 using (3.28) 1 ( Cm/ HvJlCiJ |Sin(6k-o6)| •+ >Y 11 A/Wf Yi-Ci/v^) ^cos^Ce^-oO The corresponding equation for S waves is •'" W^WVPW |sin(6k-p)| (A-1.2) + 4§\M I Din/ I Sln(6d-P)| For P and SV waves incident on the boundary from medium (1), the relations are respectively 115 sin(e^-OL)| , c ^/Cr*A v71- ^£COS%d(k-d5 (A-1.4) B + Si Vbi/ 1 Bin, (A-1.5) 116 APPENDIX II EXPRESSION OF A FREE RAYLEIGH WAVE USING COMPLEX ANGLES In the calculation of the displacements, complex angles have been used in order that the cases of total reflection and incident angles greater than the critical are involved in the results. Although Rayleigh waves are not produced in this problem, the expression of Rayleigh waves in terms of complex angles is of interest. Consider an elastic half-space with free surface 0~O (Figure A-l) The solution in the medium can be written as (A-2.1) 0>K- iV.c^TCOS(°-^ The boundary conditions at S — 0 are ee = o T0=O Substituting (A-2.1) into the boundary conditions using (3.15) and (3.16), we have (l-^VbtcosVOA^+ BJ?SI n^cos^= 0 (A-2.3) (A-2.2) 118 and Vb[COSd^ COSpg. (A-2.4) From (A-2.3), we have ~ ^v^s\noi?<cosd,cis\n^cos$^o (A'2-5) Substituting (A-2.4), and writing % — COSbL^and V — t^i gives |G(i-v)?c3+(i6-^)^+-^^-i73=0 (A.2.6) Assuming Poisson's relation, Z>~J^- , yields The real root of this equation is %=3. which cor responds to COS olfi= ± 1,88^ and using (A-2.4), COS ± 1.0 88 • Recall ing the relations a.rccos(r B)=JC-arccosn arc cos p = C&rccoshp (p=rea\ >i) we obtain oCp.= 1.-2^-7 L or jo-\.2.<t7 I (A-2.8) Pa=o.^o68L or jo™o.4-oG8o 119 If in equations (A-2.1), we use cos(p± 1%) = cospcosh%:f" ls\n?s\nh% Sin(p±L^)= SinpcoshST icosps\r\]r\% we have OA ^±l^rCi.S8^cose±i i.F^7sma) - o +Ltblr(|.o88CoS0±Lo.^73sfn6) (A-2.9) 60 _ 0.^2.78., %.aA (A-2.10) We see that the dilatation and rotation propagate with the velocity 0.cJ{c|'j-Cfc>| which coincides with the velocity of the free Rayleigh wave. As a result we see that for a Rayleigh wave written in terms of complex angles, the real part of the angle indicates the propagation direction and the imaginary part gives the decrease of amplitude with the two solutions of (A-2.8) representing waves propa gating in opposite directions ( 0 O^Hoi JC ). 120 APPENDIX III EVALUATION OF THE SECOND SERIES TERMS OF THE INTEGRAL As a guide to computation of higher order terms in the series, a summary of the evaluation procedures and results for the contributions by waves twice reflected from the boundary between the elastic media are evaluated here. The second terms of the series have the form -LOO where ^_ ASin(^+o6L)-§yi-A"cos^^o6L)  1 A s i n( 0r+ou)+S/1 -tfcosXtf*-* ou) z A s i n (0^ oa)+S /1 -tfcosXA+oil) and 1= and 771 = + , — From equations (4.17) and (4.18) (A-3.1) (A-3.2) (A-3.3) and 0^ have been given by equations (4.29) to (4.32)*. 121 As the integrands of (A-3.1) contain the expres sions and (A- 3.4) which are both two-valued, a four-sheeted Riemann surface is required for their representation. The branch cuts, along which the four sheets coalesce are defined by R.s(A's0=O and (P^sz)= 0 • F°r evaluation purposes the medium is assumed to be very slightly absorptive as before. The sheets I, II, III and IV are defined corresponding to the combinations „ „, (A-3.5, (Re(Asi)<o,Re(AS2)>o) , (ReM>o, Re(AS2.)>o); (Re(ASi)>o,Re(Asa)<o) , (Re(Asl)<o,Re(A«a)<o). respectively. The original path of integration can be shifted on any sheet of the Riemann surface for the factor 0 i&b\fi-2Si.COS(oLi, ^^vanishing along the path at a large distance from the origin. The original path [_J is taken on sheet II where the relations Im(S\)T\oLl)<iO , Im(Asi)*C0 and hold along L . As an example, when QQ ^> Qc, 122 where (A- 3.6) Q0= ATCCOSO/A) the original path can be shifted to Lis , ( L3 } Ly- ) and (bi , Ltjj, ) as shown in Figure A-2. [jS passes through the saddle point £> , and the contours ( L»3 ; LyO and ( L1; ) go around the branch points £ and 3 respectively, each one of them being drawn along the path of steepest descent given by cos(x-d?i)cosh*&= I (A-3.7) cosoc-e^coshty = cos(eo- eS) . (A-3.8) and COS(X-0j£)COSh ^= COS(0B- 0^) (A-3.9) where 123 n m Re Xs, < o Re XS2 >o Re XSI >o ReXS2 >o Re XS) >o Re XS2 <o Re Xs, <o Re XS2<o Eig. A-2. The oci-plane showing branch cuts and integral paths for evaluation of the second series term of the integrals. Notation: B,C - branch points; S - saddle point; L - original path of integration; Ls - path of steepest descent through saddle point; and L{, ("i= 1,.2 ••••) - paths of branch line integral* 124 Integral Around C The contour integrals along ( L3,L»f) can be evaluated by the same procedure as before. By noting the relations CA-3.10) for the path Vu\. near C on sheet I, for a harmonic time variation the contribution to the displacement is found to be where Integral Around B The contour integrals along ( \jK } L^,) can also be evaluated by the same procedure as before. By noting the 125 relations He(XsO>0 , Im(AsO>0, (A-3.13) for the path L.z near B lying on sheet II, for a harmonic time variation the contribution to the displacement is found to be :-::^\':f . - = AJ/IJGS | i • (A-3.14) Integral through S The contour integral along L5 can also be eva luated by the same procedure as before. For a harmonic time variation the contribution to the displacement is found to be us-KJ^& AT(C) A:(C) otfeM!Ci ^l (A-3.15) 126 where for V* ^B V' ^c Al(C> A Sin (0r+ O - h J\-tfcos^+ e%) A S in ($zT+ 65) + iJ\-£?cos*(0r+eZ) A s i n (ej).+ g/i-^Cos^^+ eS.) (A-3.16) if eB>e^>0c : At@£) = same as (A-3.16) (A-3.17) tanf,= Asinc^+e^) (A-3.18) if 6& > 6c > dza tan^ Asin(^a^) (A-3.19) (A-3.20) 127 Aperiodic Solution When the motions are aperiodic and vary as </>(t)=- t4c* A>0 , OO the operation 0 -<?o applied to (A-3.11), (A-3.14) and (A-3.15) yields the fol lowing solutions: Head waves, jE{\-\/£?r* (R-)3A |Sjn(ec_• * (A-3.21) X A Sin (f^+ ds)-Sj 1-^005"^+ 6s) Jl+pz___J"jl/f-A _sinC0r+aB)+§/i-A^ost$r-feB) (A-3.23) 128 where u T,„n = — —— & Gb| Reflected waves, ;7"' f': fj . A A;(O-AM) V JT: ,.. for eB>027>0c il = -A- 30 A 1 Y COS +^ + 3ft 129 for eB 70c>e_jL . m nw- l ............ where RL^= /Cbl (A-3. If the branch points are smaller than the saddle points head waves do not appear. The ray paths for arrivals which travel along part of the path as head waves are shown in Figure A-3. Hence, except for diffracted waves, we can formally obtain a complete synthetic seismogram in the case of a dip ping layer by applying this procedure to the third and higher order series terms of the formal integral solution. 130 From branch point B From branch point C m = + in (A-3.-1). PUBLICATIONS Nakamura, K. and Ishii, H., 1965. Refraction of explosive sound waves from a line source in air into water, Sci. Rep. Tohoku Univ., Ser. 5, Geophysics, 16, 90-107. Tohoku Univ. Aftershocks Observation Group, 1966. Observa tion of aftershocks of an earthquake happened off Oga-Peninsula on 7th, May, 1964, Tohoku Disaster Prevention Research Group Report, 85-101. Ishii, H. and Takagi, A., 1967. Theoretical study on the crustal movements, Part I. The influence of surface topography (Two-dimensional SH torque source), Sci. Rep. Tohoku Univ., Ser. 5, Geophysics, 19, 77-94. Ishii, H. and Takagi, A., 1967. Theoretical study on the crustal movements, Part II. The influence of horizontal discontinuity, Sci. Rep. Tohoku Univ., Ser. 5, Geophysics, 19, 95-106. Ishii, H. and Ellis, R. M., Multiple reflection of plane SH waves by a dipping layer, Bull. Seism. Soc. Amer. (accepted for publication). 

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