Earthquake Engineering GE / CEE - 479/679 Topic 13. Wave Propagation 2

March 4, 2008 1 John Anderson: GE/CEE 479/679: Lecture 13

Earthquake EngineeringGE / CEE - 479/679

Topic 13. Wave Propagation 2

John G. Anderson

Professor of Geophysics


Combining in F=ma

• In this equation, Xi is a body force acting on the point, if any.

•

( ) iii

i Xuxt

uρμ

θμλρ +∇+

∂

∂+=

∂

∂ 22

2

23

2

22

2

21

22

x

u

x

u

x

uu

∂∂

+∂∂

+∂∂

=∇


The Free Surface

• S-waves can have two polarizations:– SH - wave motion is parallel to

the surface. Causes only horizontal shaking.

– SV - wave motion is oriented to cause vertical motion on the surface.

• Amplitudes are approximately doubled

SH

SV

Motion in and out of the plane of this figure - hard to draw.

Motion perpendicular to the direction of propagation causes vertical motion of the free surface.


Two Media in Contact

• This way of drawing is consistent with horizontal layers in the Earth.

• Lower velocities near the surface imply wave propagation direction is bent towards the vertical as the waves near the surface.

111 ,, βμρ

222 ,, βμρ

i1

i2

2

2

1

1 sinsin

ββii

=


Two Media in Contact• For an incoming SV

wave, the situation gets even more complex.

• In this case, both P- and SV-waves are transmitted and reflected from the boundary.

• The P- and SV-waves are coupled by the deformation of the boundary.

1111 ,,, αβμρ

1222 ,,, αβμρ

i1

i2 i2

Incoming SVReflected SV

Transmitted SV

Transmitted P

Reflected Pj2

j1

Generalized Snell’s Law

2

2

1

1

2

2

1

1 sinsinsinsin

ααββjjii

===


Realistic Earth Model

• Eventually, as the velocity increases with depth, rays are bent back towards the surface.

• Waves cannot penetrate into layers where β is too large.

111 ,, βμρ

222 ,, βμρ

i1

i2

2

2

1

1 sinsin

ββii

=

βi

psin

=

p is the “ray parameter. It is constant along the ray

β increases


Body Waves: Discussion

• The travel time curves of body waves can be inverted to find the velocity structure of the path.


Seismic Refraction

• Because velocity increases with depth, rays are bent back towards the surface.

• Apparent velocity at the array of sensors is the same as the velocity of the refracted ray along the top of the refracting layer.

• Records from a profile of sensors radial from an explosion can thus be inverted to find velocity with depth.

111 ,, βμρ

222 ,, βμρ

i1

i2

2

2

1

1 sinsin

ββii

=

βi

psin

=

p is constant along the ray

β increases

* ΔΔΔΔΔΔΔΔΔΔΔΔ

Refracted wave


Realistic Earth Model

• Due to Snell’s law, energy gets trapped near the surface.

• This trapped energy organizes into surface waves.

111 ,, βμρ

222 ,, βμρ

i1

i2

2

2

1

1 sinsin

ββii

=

β increases


Four types of seismic wavesBody WavesP Waves Compressional, Primary

S Waves Shear, Secondary

Surface WavesLove Waves

Rayleigh Waves


Surface Waves

• Love waves: trapped SH energy.

• Rayleigh waves: combination of trapped P- and SV- energy.


Surface Waves

• For surface waves, geometrical spreading is changed.– For body waves, spreading is ~1/r.– For surface waves, spreading is ~1/r0.5.


Surface Waves: Discussion

• Body waves are not dispersed.• Surface waves are dispersed, meaning that

different frequencies travel at different speeds.• Typically, low frequencies travel faster. These

have a longer wavelength, and penetrate deeper into the Earth, where velocities are faster.

• Typically, Love waves travel faster than Rayleigh waves.


Surface Waves

• Surface wave dispersion curves can be inverted to find the velocity structure of the path crossed by the surface waves.


Surface Waves: Discussion• Particle motion in S-waves is normal to the direction of

propagation.

• This is also true of Love waves.

• However, Love waves would show changes in phase along the direction of propagation that would not appear in vertically propagating S waves.


Surface Waves: Discussion

• Motion of Rayleigh waves is “retrograde elliptical”.


Site Response

• What is site response

• What causes it

• What are it’s characteristics.


Classic example of site effect : Mexico City

• Mexico City, Mexico


Figure 2


Physics of Site Response

• Layer over half space

• Multiple layers over half space

• Basins

• Topography


Multiple flat layers


Basins: major phenomena

• Amplification

• Energy trapped

• Conversion to surface waves at basin edge

• Longer duration


• Basin edge

• Kobe, Japan earthquake disaster.


Liu and Heaton, ~1980

Study of strong motion from the San Fernando earthquake.

Published in Bull. Seism. Soc. Am.

Demonstration of a basin effect.


Site Characterization• Goal: characterize the average effect of geology on strong motion,

and use this to improve predictions.

• The shallow geology is an almost miniscule part of the total path from the earthquake to the station.

• However, it has a strong effect on the ground motions, because it is the closest to the station.

• Geophysical measurements, using wave propagation techniques, are used to measure near-surface site characteristics.

• Also need to know basin geometry, depth.


Geotechnical Site Classification

• Many schemes to classify the site.

• Encroaching into the territory that Prof. Siddharthan will discuss later.

• But it’s good to introduce the subject from the viewpoint of the seismologist.


Seed and Idriss (1982)

• 1. Rock sites• 2. Stiff soil sites (< 60 m deep)• 3. Deep cohesionless soil sites (> 75 m deep)• 4. Sites underlain by soft to medium stiff clays

Problem with this approach:Does not recognize that the spectral shape also depends on the earthquake magnitude.


Geotechnical Classification Schemes

• Geology– Material on a geological map

• For example, for California one simple approach is the “QTM” approach, using the age of the material.

• Q = Quaternary• T = Tertiary• M = Mesozoic

– Whether the location is “erosion-dominated” or “sedimentation-dominated” (rock, soil)


NEHRP Category

Description Shear velocity

(m/s)

A Hard rock >1500

B Firm to hard rock 760-1500

C Dense soil, soft rock 360-760

D Stiff soil 180-360

E Soft soil <180

F Special studies soils

NEHRP ClassificationShear velocity of near-surface materials


Empirical site response and comparison with measured site conditions at ANSS sites in the

Reno area

Pancha, Anderson, Biasi, Anooshepor, Louie


Results from Pancha’s ReMi studies in the Central Truckee Meadows


Figure 1a


Figure 3


Figure 1b


Figure 5


Alternate Figure 9


Figure 3


Figure 2


Figure 6


OLD Figure 7

Earthquake Engineering GE / CEE - 479/679 Topic 13. Wave Propagation 2

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