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.

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.

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

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

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

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.

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.

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.

The slowness vector Properties Components in 2D Ray equation

Depth dependent velocity model A B Depth dependent velocity model ⌫ ★ Cartesian Coordinates

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

Heterogeneous Earth Models

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

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.

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

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

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.

Two homogenous medium in contact Snell Law v1 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 =

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.

Two homogenous medium in contact P-SV waves

Two homogenous medium in contact SH waves x2 x1 x3 sin(j1) b1 sin(j2) b2 =

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

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

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

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

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

Amplitudes SH wave

Multiple layers

Realistic complexity

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

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