Basic Ray Theory

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Brief heuristic, description of the application of ray theory to seismology and seismic exploration

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Basic Ray Theory

  1. 1. Teoría de Rayos Básica por Carlos César Piedrahita
  2. 2. Overview <ul><li>Introduction </li></ul><ul><li>Wave propagation </li></ul><ul><li>Ray method </li></ul><ul><ul><li>Ray path </li></ul></ul><ul><ul><li>Traveltimes </li></ul></ul><ul><ul><li>Amplitudes </li></ul></ul>
  3. 3. Ray method Wave propagation Ray Ansatz Ray path and Traveltimes Amplitudes
  4. 4. Ray method Wave propagation Ray Ansatz Ray path and Traveltimes Amplitudes
  5. 5. Ray method Wave propagation Ray Ansatz Ray path and Traveltimes Amplitudes
  6. 6. Ray method Wave propagation Ray Ansatz Ray path and Traveltimes Amplitudes
  7. 7. Ray method Wave propagation Ray Ansatz Ray path and Traveltimes Amplitudes
  8. 8. Wave propagation Wave propagation Ray Ansatz Ray path and Traveltimes Amplitudes Acoustic equation Elastodynamic equation Ray formulation
  9. 9. Wave propagation Wave propagation Ray Ansatz Ray path and Traveltimes Amplitudes Acoustic equation Elastodynamic equation Ray formulation
  10. 10. Wave propagation Wave propagation Ray Ansatz Ray path and Traveltimes Amplitudes Acoustic equation Elastodynamic equation Ray formulation
  11. 11. Wave propagation Wave propagation Ray Ansatz Ray path and Traveltimes Amplitudes Acoustic equation Elastodynamic equation Ray formulation
  12. 12. Wave propagation Acoustic equation Elastodynamic equation Ray formulation
  13. 13. Wave propagation Acoustic equation Elastodynamic equation Ray formulation
  14. 14. Wave propagation Acoustic equation Elastodynamic equation Ray formulation
  15. 15. Wave propagation Acoustic equation Elastodynamic equation Ray formulation
  16. 16. Wave propagation Observed wavefield: superposition of isolated events Acoustic equation Elastodynamic equation Ray formulation
  17. 17. Wave propagation Wave propagation Ray Ansatz Ray path and Traveltimes Amplitudes Acoustic equation Elastodynamic equation Ray formulation   
  18. 18. Ray method Wave propagation Ray Ansatz Ray path and Traveltimes Amplitudes 
  19. 19. Ray Ansatz  Wave propagation Ray Ansatz Ray path and Traveltimes Amplitudes Reflections Reflection event Ray formulation
  20. 20. Ray Ansatz Wave propagation Ray Ansatz Ray path and Traveltimes Amplitudes Reflections Reflection event Ray formulation 
  21. 21. Ray Ansatz Wave propagation Ray Ansatz Ray path and Traveltimes Amplitudes Reflections Reflection event Ray formulation 
  22. 22. Ray Ansatz Wave propagation Ray Ansatz Ray path and Traveltimes Amplitudes Reflections Reflection event Ray formulation 
  23. 23. Ray Ansatz Wave propagation Ray Ansatz Ray path and Traveltimes Amplitudes Reflections Reflection event Ray formulation 
  24. 24. Ray Ansatz Wave propagation Ray Ansatz Ray path and Traveltimes Amplitudes Reflections Reflection event Ray formulation 
  25. 25. Ray Ansatz Reflections Reflection event Ray formulation
  26. 26. Ray Ansatz  Reflections Reflection event Ray formulation t A
  27. 27. Ray Ansatz Reflections Reflection event Ray formulation t A 
  28. 28. Ray Ansatz Reflections Reflection event Ray formulation t A 
  29. 29. Ray Ansatz Reflections Reflection event Ray formulation t A 
  30. 30. Ray Ansatz  Reflections Reflection event Ray formulation t A
  31. 31. Ray Ansatz  Reflections Reflection event Ray formulation t A
  32. 32. Ray Ansatz  Reflections Reflection event Ray formulation t A
  33. 33. Ray Ansatz  Reflections Reflection event Ray formulation t A
  34. 34. Ray Ansatz  Reflections Reflection event Ray formulation t A
  35. 35. Ray Ansatz Reflections Reflection event Ray formulation Structure of reflected event A ( x ) = amplitude   ( x ) = traveltime Unknowns:  t A
  36. 36. Ray Ansatz Wave propagation Ray Ansatz Ray path and Traveltimes Amplitudes Reflections Reflection event Ray formulation    
  37. 37. Ray method Wave propagation Ray Ansatz Ray path and Traveltimes Amplitudes  
  38. 38. Ray path and traveltimes Wave propagation Ray Ansatz Ray path and Traveltimes Amplitudes Eikonal equation Ray equations Ray path   Traveltimes
  39. 39. Ray path and traveltimes Wave propagation Ray Ansatz Ray path and Traveltimes Amplitudes Eikonal equation Ray equations Ray path   Traveltimes
  40. 40. Ray path and traveltimes Wave propagation Ray Ansatz Ray path and Traveltimes Amplitudes Eikonal equation Ray equations Ray path   Traveltimes
  41. 41. Ray path and traveltimes Wave propagation Ray Ansatz Ray path and Traveltimes Amplitudes Eikonal equation Ray equations Ray path   Traveltimes
  42. 42. Ray path and traveltimes Eikonal equation Ray equations Ray path   Traveltimes
  43. 43. Ray path and traveltimes Eikonal equation Ray equations Ray path Traveltimes
  44. 44. Ray path and traveltimes <ul><li>Traveltimes (Ray Tracing) </li></ul><ul><li>Partial differential equation(PDE) of first order </li></ul><ul><li>Nonlinear PDE </li></ul><ul><li>Hamilton-Jacobi type </li></ul>Eikonal equation Ray equations Ray path Traveltimes
  45. 45. Ray path and traveltimes Eikonal equation Ray equations Ray path Traveltimes Hamiltonian Method of characteristics Ray equations Ray jacobian
  46. 46. Ray path and traveltimes Eikonal equation Ray equations Ray path Traveltimes Hamiltonian Method of characteristics Ray equations Ray jacobian
  47. 47. Ray path and traveltimes Eikonal equation Ray equations Ray path Traveltimes Hamiltonian Method of characteristics Ray equations Ray jacobian
  48. 48. Ray path and traveltimes Eikonal equation Ray equations Ray path Traveltimes Hamiltonian Method of characteristics Ray equations Ray jacobian
  49. 49. Ray path and traveltimes Hamiltonian Method of characteristics Ray equations Ray jacobian
  50. 50. Ray path and traveltimes Hamiltonian Method of characteristics Ray equations <ul><li>Eikonal as a Hamiltonian </li></ul>Ray jacobian
  51. 51. Ray path and traveltimes Hamiltonian Method of characteristics Ray equations <ul><li>A special change of variables </li></ul>where, along curves  , <ul><li>  is called characteristic </li></ul>Ray jacobian
  52. 52. Ray path and traveltimes Hamiltonian Method of characteristics Ray equations <ul><li>Along characteristics curves  </li></ul>  Ray jacobian
  53. 53. Ray path and traveltimes Hamiltonian Method of characteristics Ray equations Ray jacobian <ul><li>System of 7 ordinary differential equations (ODE) </li></ul>
  54. 54. Ray path and traveltimes Hamiltonian Method of characteristics Ray equations Ray jacobian <ul><li>Point source initial condition </li></ul>
  55. 55. Ray path and traveltimes Hamiltonian Method of characteristics Ray equations <ul><li>Ray coordinates </li></ul> 2  1 p 0 (  1 ,  2 , s ) Ray jacobian
  56. 56. Ray path and traveltimes Hamiltonian Method of characteristics Ray equations Ray jacobian <ul><li>Ray jacobian </li></ul>
  57. 57. Ray path and traveltimes Hamiltonian Method of characteristics Ray equations Ray jacobian <ul><li>Example of ray jacobian: Point source and constant velocity </li></ul>
  58. 58. Ray path and traveltimes <ul><li>System of 7 ordinary differential equations (ODE) </li></ul>Eikonal equation Ray equations Ray path Traveltimes
  59. 59. Ray path and traveltimes Eikonal equation Ray equations Ray path Traveltimes Without interfaces With interfaces
  60. 60. Ray path and traveltimes Without interfaces With interfaces <ul><li>Requires a smooth velocity model and one initial condition </li></ul><ul><li>4 th order Runge-Kutta can be used </li></ul>x S x p 0
  61. 61. Ray path and traveltimes Without interfaces With interfaces <ul><li>Requires new initial conditions for each interface </li></ul><ul><li>Snell’s law must be used </li></ul>First initial condition Second initial condition Third initial condition Incident ray reflected ray transmitted ray
  62. 62. Ray path and traveltimes Without interfaces With interfaces <ul><li>Snell’s law </li></ul> j  j+1 c j c j+!
  63. 63. Ray path and traveltimes Without interfaces With interfaces <ul><li>Example: Six initial conditions </li></ul>
  64. 64. Ray path and traveltimes Without interfaces With interfaces <ul><li>Example: Eight initial conditions (multiple) </li></ul>
  65. 65. Ray path and traveltimes Eikonal equation Ray equations Ray path Traveltimes <ul><li>Traveltime is computed along the ray  </li></ul>
  66. 66. Ray path and traveltimes Wave propagation Ray Ansatz Ray path and Traveltimes Amplitudes Eikonal equation Ray equations Ray path   Traveltimes    
  67. 67. Ray method Wave propagation Ray Ansatz Ray path and Traveltimes Amplitudes   
  68. 68. Amplitudes Wave propagation Ray Ansatz Ray path and Traveltimes Amplitudes Transport equation Ray equations Geometric spreading   Amplitude 
  69. 69. Amplitudes Wave propagation Ray Ansatz Ray path and Traveltimes Amplitudes Transport equation Ray equations Geometric spreading   Amplitude 
  70. 70. Amplitudes Wave propagation Ray Ansatz Ray path and Traveltimes Amplitudes Transport equation Ray equations Geometric spreading   Amplitude 
  71. 71. Amplitudes Wave propagation Ray Ansatz Ray path and Traveltimes Amplitudes Transport equation Ray equations Geometric spreading   Amplitude 
  72. 72. Amplitudes Transport equation Ray equations Geometric spreading   Amplitude 
  73. 73. Amplitudes Transport equation Ray equations Geometric spreading Amplitude
  74. 74. Amplitudes Transport equation Ray equations Geometric spreading Amplitude <ul><li>Uses the ray solution </li></ul><ul><li>Partial Differential Equation (PDE) of First Order </li></ul>
  75. 75. Amplitudes Transport equation Ray equations Geometric spreading Amplitude <ul><li>Alternative expression: </li></ul><ul><li>Applying Gauss’ theorem: </li></ul>
  76. 76. Amplitudes Transport equation Ray equations Geometric spreading Amplitude dS 0 dS 1 <ul><li>Ray tube </li></ul>D
  77. 77. Amplitudes Transport equation Ray equations Geometric spreading Amplitude <ul><li>Ray tube </li></ul>
  78. 78. Amplitudes Transport equation Ray equations Geometric spreading Amplitude <ul><li>Ray tube </li></ul>
  79. 79. Amplitudes Transport equation Ray equations Geometric spreading Amplitude <ul><li>Ray tube </li></ul>
  80. 80. Amplitudes Transport equation Ray equations Geometric spreading Amplitude <ul><li>Ray tube </li></ul><ul><li>Geometrical spreading </li></ul>
  81. 81. Amplitudes Transport equation Ray equations Geometric spreading Amplitude Without interfaces With interfaces Basic problems General expression
  82. 82. Amplitudes Without interfaces With interfaces A ( s ) A ( s 0 ) Basic problems General expression
  83. 83. Amplitudes Without interfaces With interfaces <ul><li>Point source initial condition: </li></ul>Basic problems General expression
  84. 84. Amplitudes Without interfaces With interfaces <ul><li>Requires new initial conditions for each interface </li></ul><ul><li>Snell’s law must be used </li></ul><ul><li>Reflection coefficient must be used (boundary conditions) </li></ul>Basic problems General expression
  85. 85. Amplitudes Without interfaces With interfaces <ul><li>Acoustic reflection coefficient </li></ul><ul><li>Acoustic transmission coefficient </li></ul>Basic problems General expression
  86. 86. Amplitudes Without interfaces With interfaces S M G Basic problems General expression <ul><li>Reflected ray </li></ul><ul><li>Known </li></ul>
  87. 87. Amplitudes Without interfaces With interfaces <ul><li>Reflected ray </li></ul>S M G Basic problems General expression
  88. 88. Amplitudes Without interfaces With interfaces Basic problems General expression S M G <ul><li>Reflected ray </li></ul>
  89. 89. Amplitudes Without interfaces With interfaces Basic problems General expression S M G <ul><li>Reflected ray </li></ul>
  90. 90. Amplitudes Without interfaces With interfaces Basic problems General expression S M G <ul><li>Reflected ray </li></ul><ul><li>Boundary condition </li></ul>
  91. 91. Amplitudes Without interfaces With interfaces Basic problems General expression S M G <ul><li>Reflected ray </li></ul>
  92. 92. Amplitudes Without interfaces With interfaces <ul><li>Transmitted ray </li></ul>Basic problems General expression S M G <ul><li>Known </li></ul>
  93. 93. Amplitudes Without interfaces With interfaces <ul><li>Transmitted ray </li></ul>Basic problems General expression S M G
  94. 94. Amplitudes Without interfaces With interfaces <ul><li>Transmitted ray </li></ul>Basic problems General expression S M G
  95. 95. Amplitudes Without interfaces With interfaces <ul><li>Transmitted ray </li></ul>Basic problems General expression S M G
  96. 96. Amplitudes Without interfaces With interfaces <ul><li>Transmitted ray </li></ul>Basic problems General expression S M G <ul><li>Boundary condition </li></ul>
  97. 97. Amplitudes Without interfaces With interfaces <ul><li>Transmitted ray </li></ul>Basic problems General expression S M G
  98. 98. Amplitudes Without interfaces With interfaces Basic problems General expression 1 2 3 S G 5 4 <ul><li>To obtain a general amplitude expression, the two basic problems can be combined </li></ul>
  99. 99. Amplitudes Without interfaces With interfaces Basic problems General expression
  100. 100. Amplitudes Wave propagation Ray Ansatz Ray path and Traveltimes Amplitudes Ray equations Transport equation Geometric spreading   Amplitude     
  101. 101. Ray method Wave propagation Ray Ansatz Ray path and Traveltimes Amplitudes     Detalles de la teoría de rayos:
  102. 102. T H E E N D
  103. 103. Briefing
  104. 104. Appendix
  105. 105. Method of Characteristics  
  106. 106. PDE of First Order
  107. 107. Elastic Medium
  108. 108. Elastodynamic Wave Equation <ul><li>Vector variables : </li></ul><ul><li>Wave Equation in Isotropic Media : </li></ul>
  109. 109. Acoustic Medium
  110. 110. Fourier Analysis <ul><li>Before stating Zero-Order Ray hypothesis lets briefly review some </li></ul><ul><li>Fourier concepts </li></ul>
  111. 111. Fourier Transform Pair <ul><li>Time to Frequency Fourier Transform : </li></ul><ul><li>Frequency to Time Transform : </li></ul>
  112. 112. Analytic Signal <ul><li>Real and Causal signal: </li></ul><ul><li>Fourier representation: </li></ul>
  113. 113. Analytic Signal <ul><li>Extend the definition to the Complex domain and define an Analytic function: </li></ul><ul><li>Hilbert Transform Pairs: </li></ul>
  114. 114. Analytic Signals <ul><li>Hilbert Transform pairs and Signals : </li></ul>
  115. 115. Plane Wave
  116. 116. Plane Wave <ul><li>Source Signal: </li></ul><ul><li>Fourier Transform : </li></ul>
  117. 117. Zero-Order Ray Theory(ZORT) <ul><li>Elastodynamic case: </li></ul><ul><li>Acoustic case: </li></ul>
  118. 118. Transient Analytic ZORT <ul><li>Elastodynamic case: </li></ul><ul><li>Acoustic case: </li></ul>
  119. 119. Hemlholtz Equation Fourier Transforming the Wave Equation we obtain Hemlholtz Equation
  120. 120. Asymptotic Aproximation <ul><li>High frequency assumption: </li></ul><ul><li>Asymptotic Series, ZORT is the first term of the series: </li></ul>
  121. 121. Acoustic Case
  122. 122. Elastodynamics : Eikonal Equation
  123. 123. Elastodynamics : Transport Equation
  124. 124. Elastodynamics : Transport Equation
  125. 125. General Ray Equations
  126. 126. Eikonal Equation <ul><li>Slowness Vector: </li></ul>
  127. 127. Eikonal Solution (Travel Times) <ul><li>Characteristic Curve : </li></ul><ul><li>Eiconal equation in Hamiltonian form : </li></ul>
  128. 128. <ul><li>Traveltimes( Ray Tracing). </li></ul><ul><li>Partial Differential Equation(PDE) of First Order. </li></ul><ul><li>Nonlinear PDE. </li></ul><ul><li>Hamilton-Jacobi type. </li></ul>Eikonal Equation
  129. 129. Method of Characteristics <ul><li>Solution Surface </li></ul> Initial Curve  , Characteristic Curves
  130. 130. PDE of First Order
  131. 131. Characteristic Equations Ordinary Differential Equation(ODE) System :
  132. 132. Ray Equations
  133. 133. Special Choices of  :
  134. 134. First Transport Equation <ul><li>Integrating along a tube of rays : </li></ul>
  135. 135. Tube of Rays
  136. 136. Ray Coordinates
  137. 137. Jacobian of the Transformation :
  138. 138. Amplitude Formula <ul><li>Integrating along the Ray Tube: </li></ul>
  139. 139. Initial Conditions <ul><li>Point Source : </li></ul>
  140. 140. Amplitude due to a Point Source
  141. 141. Normalized Geometrical Spreading Factor
  142. 142. Line Caustic
  143. 143. Focus Caustic Point
  144. 144. Caustic Points <ul><li>Line Caustic Points: </li></ul><ul><li>Focus Caustic Point: </li></ul>
  145. 145. Maslov Theory <ul><li>Explain Ray theory in a Caustic Neighborhoods </li></ul>
  146. 146. R/T Coefficients(Snell´s Law) <ul><li>Reflection </li></ul><ul><li>Condition : </li></ul><ul><li>Transmission </li></ul><ul><li>Condition : </li></ul>
  147. 147. Reflected Ray S G x z
  148. 148. Acoustic Reflection Coefficient
  149. 149. Transmitted Ray S x z
  150. 150. Acoustic Transmission Coefficient
  151. 151. Elastic Layered Medium Interface 1 Interface i Interface j Interface N-1 S = O 0 O 1  i +  i - O i G = O N  j +  j - O j
  152. 152. Amplitude at the Geophone
  153. 153. Target Reflector Amplitude
  154. 154. Differential Scalar Operators <ul><li>Gradient of u: </li></ul><ul><li>Laplacian of u : </li></ul>
  155. 155. Polynomial Equation in  <ul><li>Equating coefficients of  : </li></ul>

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