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

1. Teoría de Rayos Básica por Carlos César Piedrahita

3. Ray method Wave propagation Ray Ansatz Ray path and Traveltimes Amplitudes

4. Ray method Wave propagation Ray Ansatz Ray path and Traveltimes Amplitudes

5. Ray method Wave propagation Ray Ansatz Ray path and Traveltimes Amplitudes

6. Ray method Wave propagation Ray Ansatz Ray path and Traveltimes Amplitudes

7. Ray method Wave propagation Ray Ansatz Ray path and Traveltimes Amplitudes

8. Wave propagation Wave propagation Ray Ansatz Ray path and Traveltimes Amplitudes Acoustic equation Elastodynamic equation Ray formulation

9. Wave propagation Wave propagation Ray Ansatz Ray path and Traveltimes Amplitudes Acoustic equation Elastodynamic equation Ray formulation

10. Wave propagation Wave propagation Ray Ansatz Ray path and Traveltimes Amplitudes Acoustic equation Elastodynamic equation Ray formulation

11. Wave propagation Wave propagation Ray Ansatz Ray path and Traveltimes Amplitudes Acoustic equation Elastodynamic equation Ray formulation

12. Wave propagation Acoustic equation Elastodynamic equation Ray formulation

13. Wave propagation Acoustic equation Elastodynamic equation Ray formulation

14. Wave propagation Acoustic equation Elastodynamic equation Ray formulation

15. Wave propagation Acoustic equation Elastodynamic equation Ray formulation

16. Wave propagation Observed wavefield: superposition of isolated events Acoustic equation Elastodynamic equation Ray formulation

17. Wave propagation Wave propagation Ray Ansatz Ray path and Traveltimes Amplitudes Acoustic equation Elastodynamic equation Ray formulation   

18. Ray method Wave propagation Ray Ansatz Ray path and Traveltimes Amplitudes 

19. Ray Ansatz  Wave propagation Ray Ansatz Ray path and Traveltimes Amplitudes Reflections Reflection event Ray formulation

20. Ray Ansatz Wave propagation Ray Ansatz Ray path and Traveltimes Amplitudes Reflections Reflection event Ray formulation 

21. Ray Ansatz Wave propagation Ray Ansatz Ray path and Traveltimes Amplitudes Reflections Reflection event Ray formulation 

22. Ray Ansatz Wave propagation Ray Ansatz Ray path and Traveltimes Amplitudes Reflections Reflection event Ray formulation 

23. Ray Ansatz Wave propagation Ray Ansatz Ray path and Traveltimes Amplitudes Reflections Reflection event Ray formulation 

24. Ray Ansatz Wave propagation Ray Ansatz Ray path and Traveltimes Amplitudes Reflections Reflection event Ray formulation 

25. Ray Ansatz Reflections Reflection event Ray formulation

26. Ray Ansatz  Reflections Reflection event Ray formulation t A

27. Ray Ansatz Reflections Reflection event Ray formulation t A 

28. Ray Ansatz Reflections Reflection event Ray formulation t A 

29. Ray Ansatz Reflections Reflection event Ray formulation t A 

30. Ray Ansatz  Reflections Reflection event Ray formulation t A

31. Ray Ansatz  Reflections Reflection event Ray formulation t A

32. Ray Ansatz  Reflections Reflection event Ray formulation t A

33. Ray Ansatz  Reflections Reflection event Ray formulation t A

34. Ray Ansatz  Reflections Reflection event Ray formulation t A

35. Ray Ansatz Reflections Reflection event Ray formulation Structure of reflected event A ( x ) = amplitude   ( x ) = traveltime Unknowns:  t A

36. Ray Ansatz Wave propagation Ray Ansatz Ray path and Traveltimes Amplitudes Reflections Reflection event Ray formulation    

37. Ray method Wave propagation Ray Ansatz Ray path and Traveltimes Amplitudes  

38. Ray path and traveltimes Wave propagation Ray Ansatz Ray path and Traveltimes Amplitudes Eikonal equation Ray equations Ray path   Traveltimes

39. Ray path and traveltimes Wave propagation Ray Ansatz Ray path and Traveltimes Amplitudes Eikonal equation Ray equations Ray path   Traveltimes

40. Ray path and traveltimes Wave propagation Ray Ansatz Ray path and Traveltimes Amplitudes Eikonal equation Ray equations Ray path   Traveltimes

41. Ray path and traveltimes Wave propagation Ray Ansatz Ray path and Traveltimes Amplitudes Eikonal equation Ray equations Ray path   Traveltimes

42. Ray path and traveltimes Eikonal equation Ray equations Ray path   Traveltimes

43. Ray path and traveltimes Eikonal equation Ray equations Ray path Traveltimes

45. Ray path and traveltimes Eikonal equation Ray equations Ray path Traveltimes Hamiltonian Method of characteristics Ray equations Ray jacobian

46. Ray path and traveltimes Eikonal equation Ray equations Ray path Traveltimes Hamiltonian Method of characteristics Ray equations Ray jacobian

47. Ray path and traveltimes Eikonal equation Ray equations Ray path Traveltimes Hamiltonian Method of characteristics Ray equations Ray jacobian

48. Ray path and traveltimes Eikonal equation Ray equations Ray path Traveltimes Hamiltonian Method of characteristics Ray equations Ray jacobian

49. Ray path and traveltimes Hamiltonian Method of characteristics Ray equations Ray jacobian

59. Ray path and traveltimes Eikonal equation Ray equations Ray path Traveltimes Without interfaces With interfaces

66. Ray path and traveltimes Wave propagation Ray Ansatz Ray path and Traveltimes Amplitudes Eikonal equation Ray equations Ray path   Traveltimes    

67. Ray method Wave propagation Ray Ansatz Ray path and Traveltimes Amplitudes   

68. Amplitudes Wave propagation Ray Ansatz Ray path and Traveltimes Amplitudes Transport equation Ray equations Geometric spreading   Amplitude 

69. Amplitudes Wave propagation Ray Ansatz Ray path and Traveltimes Amplitudes Transport equation Ray equations Geometric spreading   Amplitude 

70. Amplitudes Wave propagation Ray Ansatz Ray path and Traveltimes Amplitudes Transport equation Ray equations Geometric spreading   Amplitude 

71. Amplitudes Wave propagation Ray Ansatz Ray path and Traveltimes Amplitudes Transport equation Ray equations Geometric spreading   Amplitude 

72. Amplitudes Transport equation Ray equations Geometric spreading   Amplitude 

73. Amplitudes Transport equation Ray equations Geometric spreading Amplitude

81. Amplitudes Transport equation Ray equations Geometric spreading Amplitude Without interfaces With interfaces Basic problems General expression

82. Amplitudes Without interfaces With interfaces A ( s ) A ( s 0 ) Basic problems General expression

99. Amplitudes Without interfaces With interfaces Basic problems General expression

100. Amplitudes Wave propagation Ray Ansatz Ray path and Traveltimes Amplitudes Ray equations Transport equation Geometric spreading   Amplitude     

101. Ray method Wave propagation Ray Ansatz Ray path and Traveltimes Amplitudes     Detalles de la teoría de rayos:

102. T H E E N D

103. Briefing

104. Appendix

105. Method of Characteristics  

106. PDE of First Order

107. Elastic Medium

109. Acoustic Medium

115. Plane Wave

119. Hemlholtz Equation Fourier Transforming the Wave Equation we obtain Hemlholtz Equation

121. Acoustic Case

122. Elastodynamics : Eikonal Equation

123. Elastodynamics : Transport Equation

124. Elastodynamics : Transport Equation

125. General Ray Equations

130. PDE of First Order

131. Characteristic Equations Ordinary Differential Equation(ODE) System :

132. Ray Equations

133. Special Choices of  :

135. Tube of Rays

136. Ray Coordinates

137. Jacobian of the Transformation :

140. Amplitude due to a Point Source

141. Normalized Geometrical Spreading Factor

142. Line Caustic

143. Focus Caustic Point

147. Reflected Ray S G x z

148. Acoustic Reflection Coefficient

149. Transmitted Ray S x z

150. Acoustic Transmission Coefficient

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. Amplitude at the Geophone

153. Target Reflector Amplitude