1. 4. Reflection/transmisson coefficients Introduction R/T coefficient – reflectivity/transmissivity Zoeppritz type equations Critical angles/complex R/T Weak-contrast approximation Parametrization Second order approximation Reflection from single layer Reflection from free surface

2. Introduction Reflection/transmission coefficient describes the effect of interface between two half spaces: Solid, Liquid, Vacuum There are 5 different cases to be considered: - Solid-solid - Solid-liquid - Solid-vacuum - Liquid-liquid - Liquid-vacuum

3. Introduction There are two types of boundary conditions at the interface: -Strains -Stress components They have be continuous (boundary conditions) or discontinuous (slip effect)

4. Introduction There are two types reflection/transmission problems: -Amplitudes -Energy (energy flux), symmetries+geometrical spreading Reflection/transmission coefficients are frequency independent for pre-critical elastic reflections from flat smooth interface

5. R/T coefficient – reflectivity/transmissivity Figure 4.1. The R/T coefficients

6. Zoeppritz type equations assumptions Plane wave Isotropic elastic medium Plane interface

7. Zoeppritz type equations (4.1) (4.2)(4.3)

8. Zoeppritz type equations Figure 4.2. The real part of Zoeppritz reflection coefficients Critical angle Pre-critical reflection Post-critical reflection

9. Zoeppritz type equations Figure 4.3. The imaginary part of Zoeppritz reflection coefficients Post-critical reflection

10. Complex reflection coefficient  is the phase shift at interface (4.4)

11. Zoeppritz type equations Figure 4.4. The real part of Zoeppritz reflection coefficients (index interchange)

12. Zoeppritz type equations Figure 4.5. The real part of Zoeppritz reflection coefficients (no post-critical)

13. Zoeppritz type equations Figure 4.6. The real part of Zoeppritz reflection coefficients (weak contrast)

14. Energy flux (4.5)

15. Weak-contrast approximation Zoeppritz equation – 6 medium parameters. Weak-contrast approximation – 3 medium parameters. All angles are real (pre-critical): (4.6) Contrast in medium parameter m (4.7) What is now happened with index interchange?

16. Weak-contrast approximation Figure 4.7. Weak contrast model

17. Weak-contrast approximation Figure 4.7. Strong contrast model

18. Parametrization 1.Velocities 2.Impedances 3.Stiffness coefficients (elastic moduli) (4.9) (4.8)

19. Parametrization Figure 4.8. Parametrization effect for the strong contrast model

20. Parametrization (4.10) Exact expression for normal incidence reflection coefficient

21. Second order approximation Stovas&Ursin, 2002 (4.12) (4.11)

22. Second order approximation Figure 4.9. Interpretation of second order R/T

23. Second order approximation Figure 4.10. Second order R/T for weak-contrast model

24. Second order approximation Figure 4.11. Second order R/T for strong-contrast model

25. Reflection from single layer Figure 4.12. Sketch of the models (Helle, Stovas & Carcione, 1999)

26. Reflection from single layer Figure 4.13. Reflection for model A (frequency 40 Hz)

27. Reflection from single layer (4.15) (4.14) (4.13) Figure 4.13. Frequency dependent normal incidence reflection coefficient

28. Reflection from free surface water air (4.16) Source Receiver