*** N.B. The material presented in these lectures is from the principal textbooks, other books on similar subject, theresearch and lectures of my colleagues from various universities around the world, my own research, and finally,numerous web sites. Some colleagues to whom I am grateful for the material I used are: B. Bolt, P. Wu, B. Kennett,E. Garnero, E. Calais and D. Dreger. I am thankful to many others who make their research and teachingmaterial available online; sometimes even a single figure or an idea about how to present a subject is a valuableresource. Please note that this PowerPoint presentation is not a complete lecture; it is most likely accompanied byan in-class presentation of main mathematical concepts (on transparencies or blackboard).***

LECTURE 4 - SeismologyHrvoje Tkalčić

Late Professor Bruce A. Bolt(1930-2005)with a model ofChang Heng’s seismoscope

Earthquakes as natural disasters: can we predict them?

• Victims in Banda Aceh, Indonesia, after theSumatra-Andaman earthquake and tsunami in 2004 Pakistan, 2005

San Francisco, 1906

Tokyo-Yokohama, 1923

A simulation of the San Simeonearthquake, CA, through a modelof 3D structure. This is achievedusing a numerical finitedifference method on a grid ofpoints.

The main wave front is visiblyrefracted or bent by contrasts inthe velocity across both theHayward and San Andreasfaults.

Concentrations of highamplitude standing wavespersist throughout the moviearound San Jose and in SanPablo Bay. These areas are low-velocity sedimentary basins andcause the amplitudes of groundmotion to be amplified as wellas extend the duration of themotions.

Both of these factors increasethe level of hazard to structures.

Strong motion simulation in SF Bay Area

Courtesy of Prof. Douglas Dreger, UC Berkeley and Dr. Shawn Larsen, LLNL

San

Francisco

Oakland

San Jose

San Andreas

Hayward

A simulation movie

Berkeley

Seismology as a tool for probing the internalstructure of the Earth

Tkalč ić , Romanowicz and Houy 2002

Li and Romanowicz 1996

Lithospheric structure under AustraliaGlobal shear velocity structure

C ompressional velocity

structure in the lowermost mantle

Some examples of seismic

tomography

van der Hilst, Kennett and Shibutani1998

The beginnings

An artist’s conception of the Chinese scholar Chang Hengcontemplating his seismoscope. Balls were held in the dragons’mouth by lever devices connected to an internal pendulum. Thedirection of the first main impulse of the ground shaking wasreputed to be detected by the particular ball that was released.

Early seismographs and advances in seismology

Emil Wiechert (1861-1928) The 1200 kg Wiechert seismograph for measuring horizontal displacements

• John Milne - constructed the first reliable seismograph in 1892

• F. Reid - elastic rebound model in 1906 after the Great San Francisco Earthquake and fire Earthquakes happen on preexisting faults

• A notion that the core is needed to explain seismic travel time proposed by R, Oldham in 1906

Probing the Earth with seismology: European discoverers of seismic discontinuities

Andrija Mohorovičić (1857-1936)

Crust-Mantle boundary 1910

Beno Gutenberg (1889-1960)

Mantle-Core boundary 1914

Inge Lehmann (1888-1993)

Inner Core 1936

Recipe for longevity: study the inner core!

The Earth’s Interior

INNER COREDiscovered by I. Lehmann

(1936)

CORE-MANTLE BOUNDARYDiscovered by B. Gutenberg

(1914)

CRUST-MANTLE BOUNDARYMohorovičić discontinuity (Moho)

(1910)

* For Comparison: Pluto discovered in 1931

Portion o seismograms recorded by the short-periodvertical-component seismograph at the Jamestownstation of the University of California Berkeley network. Thewave packet A is the core phase P4KP, and B isP7KP.These exotic seismic phases are multiple reflections fromthe lower side of the core mantle boundary.

The east-west component of ground motion at theBerkeley station recorded by the Bosch Omoriseismograph on March 10, 1922, from an earthquakesource near Parkfield, California.The recording is part of the basis of the "ParkfieldPrediction Experiment" (1988 ± 5 years). Reproduced on awine label printed for the Centennial Symposium, May28–30, 1987.

Berkeley Seismographic Station

•The first seismographs in the western hemisphereinstalled at the University of California Berkeleycampusin 1887 (largely due to the interest of astronomers).

•The occurrence of the San Francisco GreatEarthquake and Fire in 1906 began a new era inseismology.

APOLLO 11Astronaut Edwin E. Aldrin Jr., lunar module pilot, isphotographed during the Apollo 11 extravehicular activityon the Moon. He has just deployed the Early ApolloScientific Experiments Package (EASEP). In theforeground is the Passive Seismic Experiment Package(PSEP); beyond it is the Laser Ranging RetroReflector(LR-3); in the left background is the black and white lunarsurface television camera; in the far right background isthe Lunar Module. Astronaut Neil A. Armstrong,commander, took this photograph with a 70mm lunarsurface camera.

APOLLO 14Astronaut Alan B. Shepard Jr., foreground, Apollo 14commander, walks toward the Modularized EquipmentTransporter (MET), out of view at right, during the firstApollo 14 extravehicular activity (EVA-1). An EVA checklistis attached to Shepard's left wrist. Astronaut Edgar D.Mitchell, lunar module pilot, is in the background workingat a subpackage of the Apollo Lunar Surface ExperimentsPackage (ALSEP). The cylindrical keg-like object directlyunder Mitchell's extended left hand is the Passive SeismicExperiment (PSE).

Seismographs on the Moon

When a force is applied to a material, it deforms: stress inducesstrain– Stress = force per unit area– Strain = change in dimension

For some materials, displacement is reversible = elastic materials

– Experiments show that displacement is:• Proportional to the force and dimension of the solid• Inversely proportional to the cross-section

– One can write: ΔL ∝ FL/A– Or ΔL/L ∝ F/A– Strain is proportional to stress = Hooke’s law– Hooke’s law: good approximation for many Earth’s materials when ΔL is small

Hooke’s Law of elasticity

1660 Robert Hooke

Stress-strain relation:

Elastic domain• Stress-strain relation is linear• Hooke’s law applies

Beyond elastic domain• Initial shape not recovered when stress is removed• Plastic deformation• Eventually stress > strength of material => failure

Failure can occur within the elastic domain = brittle behavior

Strain as a function of time under stress• Elastic = no permanent strain• Plastic = permanent strain

What is the mathematical relation between stress and strain?

Stress and strain

Normal strain

The series expansion of u1:

x1

Stress and strain Shear strain

For small deformations:

x1

x2

and since u2(A)=0:

Similarly, for AD segment:

The series expansion of u2:

Shear tensor

Dilatation

For products of Δu, Δv, Δw ≈ 0

Stress and strainStress

σ ij =

σ ij

Normal to the surface upon which the stress acts

Direction of the stress component

Stress tensor:

Internal traction (stress):

The stress field is the distribution of internal "tractions"that balance a given set of external tractions and bodyforces.

σ xx = σ 11, σ xy= σ 12 etc. using the notation we used for strain

A cubic element in static equilibrium

For a medium to be in stableequilibrium, the moments must sumto zero. Moments are given by theproduct of a force times theperpendicular distance from the forceto a reference point. Let’s consider amoment around x3 axis first:

As Δx1, Δx2 -> 0, we have σ12= σ21Similarly, for the moments around x1 andx2 axes, σ13= σ31 and σ23= σ32.Thus, stress tensor is also symmetric,with 6 independent elements.

The most general form of Hooke’s law

The constants of proportionality, Cijkl are elastic moduli. We saw that the both strain and stresstensors are second-order tensors, which are symmetric and have 6 independent elements. Cijklis thus a third-order tensor and in its most general form consists of 81 elements. However,since the strain and stress tensors only have 6 independent elements, the number ofindependent elements in Cijkl can be reduced to 36.The first stress element is related to the strain elements by:!

" ij = Cijkl#kl

!

" ij = C1111#11

+ C1112#12

+ C1113#13

+ C1121#21

+ C1122#22

+ C1123#23

+ C1131#31

+ C1132#32

+ C1133#33

For an isotropic medium (material properties independent on direction or orientation ofsample), the number of elastic moduli can conveniently be reduced to only 2. Theseelastic moduli are called the Lamé constants λ and µ.

!

" ij = #$%ij + 2µ&ij

where δij is Krönecker delta function (δij=0 when i≠j and δij=1 wheni=j). This was formulated by Navier in 1821 and Cauchy in 1823.

Definitions of elastic moduli - from Lay and Wallace book