the seismogram u = source * propagation * site

The seismogramThe seismogram U = Source * Propagation * Site U = Source * Propagation * Site

POINT SOURCE APPROXIMATIONPOINT SOURCE APPROXIMATION

Distance rWavelengthFault dimensionL

€

un(r x ,t) = Mpq ∗Gnp, q

€

r >> λ

λ >> L

€

r >> L

Far field terms dominates because r is relatively largeFar field terms dominates because r is relatively large

NUCLEATION POINT POSITION

depth

surface

fault

EXTENDED SOURCEFAULT PARAMETERS

dip

N

Strike

wid

th W

length L

Hanging wallfoot wall

Fault azimuth

Fault dip

EXTENDED SOURCEFAULT PARAMETERS

surface

EXTENDED SOURCE PARAMETERIZATION

An extended source is represented by the distribution of point sources at the each grid point

surface

fault

Rupture velocity (vr)

EXTENDED SOURCEFAULT PARAMETERS: Rupture Velocity

surface

fault

d rakey

),( tyD

t

yDmax

rv

d

EXTENDED SOURCEFAULT PARAMETERS: Slip

€

tr =ξ

vr

€

ξvr

+ Tr

barriersbarriers

asperitiesasperities

COMPLEX SOURCE PHENOMENAAsperities and barriers

Depth

Into the

earth

Surface of the earth

Distance along the fault plane 100 km

KINEMATICS EXTENDED SOURCESlip on an earthquake fault

KINEMATICS EXTENDED SOURCESlip on an earthquake fault: second 2.0

KINEMATICS EXTENDED SOURCESlip on an earthquake fault: second 4.0

KINEMATICS EXTENDED SOURCESlip on an earthquake fault: second 6.0

KINEMATICS EXTENDED SOURCESlip on an earthquake fault: second 8.0

KINEMATICS EXTENDED SOURCESlip on an earthquake fault: second 10.0

KINEMATICS EXTENDED SOURCESlip on an earthquake fault: second 12.0

KINEMATICS EXTENDED SOURCESlip on an earthquake fault: second 14.0

KINEMATICS EXTENDED SOURCESlip on an earthquake fault: second 16.0

KINEMATICS EXTENDED SOURCESlip on an earthquake fault: second 18.0

KINEMATICS EXTENDED SOURCESlip on an earthquake fault: second 20.0

KINEMATICS EXTENDED SOURCESlip on an earthquake fault: second 22.0

KINEMATICS EXTENDED SOURCESlip on an earthquake fault: second 24.0

Rupture on a Fault

Total slip during the 1992 Landers earthquake

KINEMATICS EXTENDED SOURCEFinal dislocation on the fault

• Rupture velocity is few km/s. By default, seismologist uses 3 km/s

• The maximum duration d of the rupture is :

• The slip amplitude on the fault scales with the length.

• Slip velocity is around 1 m/s

€

T =L

vr

EXTENDED SOURCEFAULT PARAMETERS: Slip Velocity

surface

fault

Rupture velocity (vr)

L

)(tD

t

Tr = rise time

maxD

D(t).

t

maxD

tr

CAVEAT: Using Appropriate Source Time Functions

SOURCE TIME FUNCTIONS:

The slip velocity history on each point on the fault is determined by the shape of the a priori assumed source time function.

Examples of single-window STF’s:

Examples of multi-window STF’s:time

Kinematic relations:

N.B. This parameterization allow us to constrain the time of positive slip acceleration, i.e. time of Vpeak

Finite duration

Fast initial acceleration

Asymmetric shape

Large peak value

Focal Mechanism

Focal Sphere around the source

A. Kelly, USGS

azimuth

S. Stein and M. Wysession

Displacement Field from a double coupleDisplacement Field from a double couple x1

x2

x3

x1

x2

x3

x2

x1

NODAL PLANE AND POLARITIESNODAL PLANE AND POLARITIES

+ -

- +

x3

x1

x2

dilatationcompression

x3

x2

x1

The focal mechanism

• Polarities of first arrivals

+

-

-

+

FOCAL MECHANISM

DISPLACEMENT DISLOCATION

+ -

-+

Dilatationcompression

Focal Mechanism & Radiation pattern

b) Polarities of first P wave arrival• Stereographic projection

Focal Mechanims & Radiation pattern

Calculation1) From polarities of first arrivals P-

waves

2) From waveform modeling through moment tensor

Radiation pattern

Radiation pattern

Far Field

Onde P

Onde S

Radiation pattern

Far Field

Nodal Planes

S

P

directive

antidirective

Non directive

COMPLEX SOURCE PHENOMENA

Directivity

Hirasawa (1965)

COMPLEX SOURCE PHENOMENA

Directivity effect on radiation

Fraunhofer ApproximationFraunhofer Approximation

€

r =r x −

r ξ = ro 1+

ξ 2

ro2

−2

r ξ ⋅ ˆ γ ( )

ro

= ro −r ξ ⋅ ˆ γ ( ) +

1

2

ξ 2

ro

−

r ξ ⋅ ˆ γ ( )

2

2ro

€

r ≈ ro −r ξ ⋅ ˆ γ ( )

The error in this approximation is

€

∂r =1

2

1

ro

ξ 2

−r ξ ⋅ ˆ γ ( )

2 ⎡ ⎣ ⎢

⎤ ⎦ ⎥<<

λ

4

€

L2 <<1

2λro