1. The Asymptotic Ray Theory
Green functions for the asymptotic high frequency approximation
In order to compute the seismic waves radiated by an earthquake, it is necessary to appropriately compute the Green Functions to model the wave propagation in a given crustal medium.
The accuracy for computing Green Functions depends on the detailed knowledge of the Earth crust. It is evident that computing high frequency
(f > 1 Hz) Green Functions requires the knowledge of the complex 3-D structure of the crust.
It is assumed that the high frequency component of seismic waves propagates along particular trajectories called rays; therefore, it relies on the substitution of the wave front with its normal, which is the seismic ray.

2. The Eikonal equation EQUATION OF MOTION TENTATIVE SOLUTION SOLUTION
APPROXIMATION
We have derived the first important equation in the framework of ray theory: the travel time of a seismic wave follows the Fermat’s principle
In optics, Fermat's principle or the principle of least time is the idea that the path taken between two points by a ray of light is the path that can be traversed in the least time.

3. The ray equation wave front travel time ray parametric equation
The second important equation is the equation of the seismic ray; that is the equation that allows the identification of the trajectory followed by a seismic wave to propagate from x to x.
travel time
ray parametric equation
tentative solution

4. The ray equation
This equation is relatively simple to solve, but it has only a kinematic meaning. In other words, by solving equation given a wave velocity profile it is possible to find the path followed by the seismic ray.
is parallel to
c is constant
constant along the ray
p is the ray parameter

5. Plane waves in 3D homogenous medium
v = w / k = l / T
Plane waves in 3D homogenous medium

6. The ray coordinate system
In order to study the ray-tracing problem, it is more convenient to avoid the Cartesian coordinate system and to use a more appropriate local reference framework.

7. The ray coordinate system
In order to study the ray-tracing problem, it is more convenient to avoid the Cartesian coordinate system and to use a more appropriate local reference framework.

8. The ray coordinate system
In order to study the ray-tracing problem, it is more convenient to avoid the Cartesian coordinate system and to use a more appropriate local reference framework.

9. The slowness vector
Properties
Components in 2D
Ray equation

10. Depth dependent velocity modelA B
Depth dependent velocity model ⌫ ★
Cartesian Coordinates

11. Seismic Waves in a spherical Earth
Ray parameter in a spherical coordinate system i
v(r)
r being the distance from the centre of symmetry

12. Heterogeneous Earth Models

13. THE GEOMETRICAL SPREADING
in a homogenous medium (that is, constant body wave velocity) the rays are lines and the amplitudes scale with a geometrical factor being proportional to
the radiation pattern of P-waves
radiated in different directions
time function related to the body force
at the source giving a displacement pulse
in the longitudinal direction
Cartesian
3 coordinate system
the general form of the geometrical
spreading.
local mobile
for a homogeneous case
Local spherical

14. THE GEOMETRICAL SPREADING
Geometrical spreading of four rays at two different values of travel time (to, t)
THE GEOMETRICAL SPREADING
ds(to) and ds(t) are the two
elementary surfaces describing
the section of the ray tube on the
wave front at different times.
The geometrical spreading factor in inhomogeneous media describes the focusing and defocusing of seismic rays. In other words, the geometrical spreading can be seen as the density of arriving rays; high amplitudes are expected where rays are concentrated and low amplitudes where rays are sparse.
The focusing or defocusing of the rays can be estimated
by measuring the areal section on the wave front at
different times defined by four rays limiting an elementary
ray tube.
Each elementary area at a given time is proportional to
the solid angle defining the ray tube at the source , but
the size of the elementary area varies along the ray tube.

15. displacement field radiated by a double couple
homogeneous
The analytical expression of the geometrical spreading factor depends on the general properties of the orthogonal curvilinear coordinates adopted to define the local framework system
spherically symmetric

16. Seismic wave Energy The strain energy density
This relation comes out from considering that the mechanical work ( W) is a function of strain components and is equal to
The kinetic energy
P wave
S wave
For a steady state plane wave incident
on a boundary between two homogeneous half spaces the energy flux leaving the boundary must equal that in the incident wave.
That is there is no trapped energy at the interface

17. Reflection and transmission coefficients for seismic waves
The internal structure of solid Earth is characterized by the distribution of physical properties that affect seismic wave propagation and, therefore, can be studied by analyzing seismic waves.
For seismological purposes, this is done by assigning the distribution of elastic properties and density or equivalently of seismic wave velocity and density.

18. Two homogenous medium in contact Snell Lawv1 v2
i1 = i3
i3 = reflection angle
i1 = incidence angle
REFLECTION
i2 = refraction angle
REFRACTION
Ray parameter
Amplitudes are distributed between different waves (reflection and refraction coefficients)
sin(i1) v1
sin(i2) v2
= p =

19. THE SNELL LAW
Suppose that a plane P-wave is travelling with horizontal slowness in a direction forming and angle with the normal to the interface.
A P-wave incident from medium 1 generates reflected and transmitted P-waves. In addition, part of this P-wave is converted into a reflected SV-wave and a transmitted SV wave.

20. Two homogenous medium in contact P-SV waves

21. Two homogenous medium in contact SH wavesx2 x1 x3
sin(j1) b1
sin(j2) b2 =

22. Two homogenous medium in contact SV waves
Incident SV wave
Reflected & refracted SV i2
sen(i2)= (b2/b1) sen(i1)
a2, b2 SV
a1, b1 SV i1
a2<a1
b2<b1 SV

23. Two homogenous medium in contact SV waves
Reflected & Refracted P waves
sen(i2)= (a2/b1) sen(i1)
a2, b2 i2 P P
a1, b1 i3 i1
a2<a1
b2<b1
sen(i3)= (a1/b1) sen(i1) SV

24. Critical Incidence If the second medium has a higher velocity,
Because in any medium P-waves travel faster than S-waves ( ), Snell’s law requires that Moreover, the angle of incidence for refracted P-wave is related to that of the incident P-wave by
S-wave reflected
If the second medium has a higher velocity,
the transmitted P-wave is farther from the vertical
(i.e., more horizontal) than the incident wave.
As the incidence angle increases, the transmitted waves approach the direction of the horizontal interface. The incidence angle reaches a value

25. Critical Incidence incident P-waves on a faster medium
incident SH wave only generates reflected
and transmitted SH waves

26. The ray parameter again!
It is useful to remind that the ray parameter is
horizontal wavenumber
apparent velocity
For a P-wave the slowness vector is given by

27. Amplitudes
SH wave

28. Multiple layers

29. Realistic complexity

30. Seismic Wave propagation.
As seismic waves travel through Earth, they interact with the internal structure of the planet and:
Refract – bend / change direction
Reflect – bounce off of a boundary (echo)
Disperse – spread out in time (seismogram gets longer)
Attenuate – decay of wave amplitude
Diffract – non-geometric “leaking” of wave energy
Scatter – multiple bouncing around

31. Waves Amplitude Attenuation
Wave amplitudes decrease during propagation
Causes:
geometrical spreading (elastic)
scattering (elastic)
Impedance contrast (elastic)
attenuation (anelastic)