The seismogramThe seismogram U = Source * Propagation * Site U = Source * Propagation * Site
POINT SOURCE APPROXIMATIONPOINT SOURCE APPROXIMATION
Distance rWavelengthFault dimensionL
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un(r x ,t) = Mpq ∗Gnp, q
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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
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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
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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
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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