MODERN GLOBALSEISMOLOGYBODY WAVES AND RAY THEORY-2

3.1 The Eikonal Equation and Ray GeometryDirection cosine of associated with the ray is (dx1/ds,dx2/ds,dx3/ds) and must satisfy:

Connection between W and dx/dsUnit normal dx/ds

Compare (3.17) with eikonal equationDefine a-1=n=c0/c(x) (index of refraction) Normal equations

How does the normal to the wavefront change along the path of the ray ?

Ray-path equationRay-path equation describe how the normal to the wavefront change along the path of the ray it depends on the spatial change of velocity structure.

A simple case for ray-path equation: c=c(x3), i.e., the velocity only changes with depth.

Consider only x1x3 planeAt a given point, the direction cosine of the ray:

p ray parameter, or horizontal slowness. p varies from 0 to 1/c. Angle i angle of incidence the inclination of a ray measured from the vertical at any given depth. For a prescribed reference point and takeoff angle, a ray will have a constant ray parameter, p, for the entire path. This is also known as Snells law, which can be derived from Fermats principle.

Stretch your legs

Now, lets explore the story behind the second equation of (3.22)

The curvature of the ray is proportional to the velocity gradient (dc/dx3) The initial angle and the velocity structure determine the distance at which ray will emerge at the surface.

Travel time

Travel time equation travel time depends on X and Z and is separable ! vertical travel time and horizontal travel time p : horizontal slowness : vertical slowness

dT/dX = p (the change in travel time with horizontal distance is equal to ray parameter)

Amplitude of a seismic arrivalTotal energy on the hemispherical wavefront = KEnergy per unit area = K/2r2

dX cos i0 (suume source at surface)

dT/dX = p

Energy density is proportional to the change in ray parameter with distance.

Stretch your legs

3.2 Travel Times in a Layered Earth

Apparent velocity in the horizontal directionp: horizontal slowness

Critical reflection head wave

Travel time for direct/reflected/head waves in a flat layer

Head wave in layered structureTurning wave in a continuous structureTravel time equations

Determining the travel time curves

Xc : crossover distance

At the crossover distance, the travel times of the direct arrival and head wave are equal:Fig 3.11 Pn=head wave; Pg=direct arrival

From the seismic observations, draw the travel time curves From the slopes of direct arrival and headwave 1 and 2 The crossover distance thickness

In 1909, Andrija Mohorovicic analysed the records of an earthquake in Croatia.

Close to the epicenter he found one single arrival (Pg) the direct wave from the focus.

Beyond 200km there were two arrivals: a new arrival (Pn) overtaking Pg.

Mohorovicic identified that Pg and Pn travelled at a speed of ~5.4 and ~7.9 km/s respectively.

He also calculated that the jump in velocity in that part of the world occured at a depth of 54km.

Andrija Mohorovicic had just discovered the the existence of the mantle underneath the continental crust.

Moho discontinuity Andrija Mohorovicic

(1936) I. Lehmann (1888-1993) (1906-1914) B. Gutenberg (1889-1960) (1909) A. Mohorovicic (1857-1936)

Travel-time equation in 2-layered structure

Travel-time equation in 2-layered structure

Figure 3.13 Travel-time curve for a finely layered Earth. The first arrivel is comprised of short segments of the head-wave cruves for each layer, over the limited distance range between crossover points.

Pseudothickness = th1+th2(2/1)Low velocity layer

Blind zoneWhen T2

3.3 Travel-Time Curves in a Continuous Mediumn-layersContinuous medium

Ray pathsTravel timeP vs Xtriplication

3.4 Travel Times in a Spherical Earth

Anything new for the p in spherical Earth ?The ray parameter p is precisely the slope of the travel-time curve, as it was for the flat-Earth case except that distance is now measured in angular degrees.

A homogeneous sphereThe travel time curve is not a straight line even thought the velocity is constant

Travel-time equation in a Spherical EarthWith (3.64) and (3.65), we may remove ds

3.5 Wave Amplitude, Energy, and Geometric Spreading

For a homogeneous Earth, T=2r0sin(/2)/v0Energy decays as 1/R2 (Amplitude decays as 1/R)

At B and C, dp/dX ~ E ~ (Caustic) Ray theory breaks down here (WHY ?)

Stretch your legs

Total work done to a mass suspended in a springPotential energy in a small volume under stressSH plane wave propagating in the x1 direction, with all motion in x2 directionEnergy of seismic signalsAverage strain energy during a complete wavelength

The amplitude is modified during propagation by:Geometric spreading Reflection and refraction at a boundaryAttenuation

3.6 Partitioning of Seismic Energy at a Boundary Why we need SV ?P

Ray geometry is governed by Snells law(Mode conversion, Generalized Snells law)All of them have the same ray parameter, p

The ray geometry of the wave interaction is described by Snells law and ray theory

How about the amplitude partitioning ?

solidWelded interfaceContinuity across the interface Stress DisplacementBoundary conditions on a welded interfaceThe stresses and displacements must be transmitted across the interface.P waves alone can not satisfy B.C.solid

solidContinuity across the interface Normal traction Normal displacementfluidBoundary conditions on a solid-fluid interfaceOnly normal stresses and normal displacements can be transmitted across the interface.The tangential displacements are not continuous and the tangential tractions must vanish.

Boundary conditions on a free surfaceAll tractions must be zero, and no explicit restriction is placed on the displacements.

Potential of the incident plane P wave PPPSVSVx1x3

Similarly, for reflected/refracted wavesx1x3

A simple case: fluid-fluid

Displacement transmission and reflection coefficients for vertical incidence

Non-vertical incidence (2

Non-vertical incidence (2

Beyond the intramission angle, reflection coefficient is negative and decrease to a value of -1 at grazing incidence. (i=90)

If 2>1, head waves are produced at the critical angle. when i>ic Post-critical reflection

post-critical reflection

Track the behavior of a particular wavefront via constant phase argumentAn frequency-dependent apparent timeDispersion : wavespeed is frequency dependent (lower frequency signals arrive earlier)Wavefront is spread out

The change in the reflected pulse shape as a function of incidence angle given in degrees at the left of the pulse.

Fast slow mediaSlow fast media

P SV : mode conversionsSp and Ps are generated at a sediment-bedrock interface. Depth of interface and vp/vs ratio in the crust.

Post-critical reflection of a SH wave:Decays exponentially away from the interface in the media 2SH

Evanescent waves or Inhomogeneous waves surface waves

Attenuation The Earth is not perfectly elastic! Propagating waves attenuate with time due to various energy-loss mechanisms. Successive conversion of potential energy (particle position) to kinetic energy (particle velocity) as a wave propagates is not perfectly reversible, and other work is done: Internal friction, such as movements along mineral dislocations shear heating at grain boundaries Phenomenological description - seismic attenuation

Figure 3.31 Phenomenological model for seismic attenuation. Natural frequency

Introducing attenuation

Harmonic oscillation that decays exponentially with timeQuality factor Q

Q = the fractional loss of energy per cycle of oscillation.

Figure 3.32: Effects of attenuation on a seismic pulse. Geometrical spreading + attenuation amplitude decreases Attenuation pulse broadening. Why ?

Large Q weak attenuation (cold, high seismic velocity)Small Q strong attenuation. (warm, low seismic velocity) Q>Q Intrinsic attenuation occurs almost entirely in shear, associated with lateral movements of lattice effects and grain boundaries. (Q9/4 Q)Energy loss through non-elastic processes is usually measured by intrinsic attenuation and parameterized with Q.

Frequency dependency of Q.

Standard linear solid (Visco-elastic solid)

The response to harmonic waves depends on the product of the frequency and the relaxation time. For wave periods that are very short or very long compared to the relaxation time, there is little attenuation.

Absorption peak of a standard linear solid

Relaxation spectrum for a polycrystalline material showing attenuation peaks at different frequencies due to different microscopic mechanisms.

Schematic model to explain the observation that Q is roughly constant over a wide range of frequencies. The superposition of absorption peaks for different compositions at different temperatures and pressures yields a flat absorption band.

Dispersion in a standard linear solid

Dispersion due to attenuation

Scattering/coda wavesAPSE, the Apollo Passive Seismic Experiment. (1969-1972)Figure 3.B5.1 Three-comp seismograms recording the impact of and Apollo lander on the moon. Seismograms ring for more than 1h.

Advantage of using (p) ?((p) is a single value function)

Ray pathsTravel timep vs X vs ptriplication

Figure 3.26