Dynamic Corner Frequency:
A new concept in stochastic finite fault
modeling
Dariush Motazedian
& Gail Atkinson
1
• Finite fault modeling is used to simulate ground motion of
large earthquakes.
• A fault is divided into a number of subfaults (N).
• Each subfault is considered as a point source.
• The source spectrum of each subfault is calculated based
on the Brune point source model:
• Aij (f)=CM0ij (2π f) 2/ [1+f/ f0ij) 2]
• f0ij=4.9e+6(∆σ/M0ij)1/3
2
• The corner frequency is the same for all subfaults and is
based on the size or seismic moment of each subfault.
• The acceleration time series of each subfault is calculated
based on the Stochastic point source modeling
• All of the obtained time series are summed at the
observation point with a proper time delay.
a(t)=
nl
nw
i =1
j 1
∑ ∑=
aij(t+ ∆ tij)
3
The heterogeneity of fault can be considered in the calculation
of the seismic moment of each subfault, based on the relative
amount of slip of each subfault.
nl
Moij = Mo Sij /
(
nw
∑ ∑
l =1
Skl ))
k =1
In this model the obtained ground motion, A(t), for a large fault
depends on the subfault size.
log l=-2+0.4M
M7.0 ==> l ≈ 6 km
18 km
4
48 km
A music analogy for finite Fault modeling
• String instruments, like a violin or a double-bass,
produce waves based on their length of the strings.
• A double-bass produces lower frequencies than a violin.
• When playing a double-bass we may place our finger at
one end of the string and gradually move our finger
towards the other end of the string.
5
A music analogy for finite Fault modeling
• At the beginning it produces high frequency music and
by varying the total free length of the string we vary the
frequency content.
• As we approach to the full length of the string we
produce the low frequency music of the double-bass.
6
A music analogy for finite Fault modeling
• The frequency content of Earthquake waves depends on
the size of the fault.
• Larger earthquakes produce richer lower frequency than
smaller earthquakes.
• The corner frequency is inversely proportional to the
ruptured area.
fo(t) ∝ 1/ S(t)
7
• If we consider a large earthquake as a double bass
and a subfault as a violin , it seems in finite fault
modeling we are asking an orchestra full of small
violins to produces double-bass music.
• An orchestra full of small violins produces high
frequency music but cannot produce the music of a
double-bass, no matter how we arrange the timing of
each player.
8
Dynamic corner frequency
• During an earthquake the rupture begins with high
frequencies and progresses to lower frequencies.
• We consider the corner frequency, which to an extend
represents the frequency content, as a function of time.
fo(t)
• The rupture begins with high corner frequency and
progresses to lower corner frequencies.
9
Dynamic corner frequency
• The dynamic corner frequency can be defined as a
function of the cumulative ruptured area.
• foij(t)= 4.9e+6(∆σ /( NR(t) Moij))1/3
• Aij (f)=C Moij NR 1/2 f2 /[1+(f/foij)2]
10
Variability of pulsing area
100 % Pulsing area
25% Pulsing area
11
Radiated Energy From a Fault
• FINSIM program (Beresnev and Atkinson, 1998)
• A vertical fault
• L = 40 km
• W=20km
• M7.0
• far field observation point
• Simulations for different subfault sizes1, 2, 5 and 10 km.
12
Far field received energy of a fault with different subfault
lengths, Using static corner frequency
log l=-2+0.4M
20
18
a(f)**2
16
14
1 km
12
2 km
10
5 km
10 km
8
6
4
2
0
0
5
10
f (Hz)
15
20
13
Far field received energy of a fault with different subfault
lengths, Using dynamic corner frequency
20
20
18
18
16
16
14
1 km
14
1 km
12
2 km
12
2 km
10
5 km
5 km
10
10 km
8
a(f)**2
a (f)**2
Far field received energy of a fault with different subfault
lengths, Using static corner frequency
6
6
4
4
2
2
0
0
0
5
10
f (Hz)
15
20
10 km
8
0
5
10
15
20
f (Hz)
14
Far field acceleration time series
Static Corner frequency
Dynamic Corner frequency
t (sec)
0
t (sec)
0
5
10
15
20
25
30
35
40
45
10
15
20
25
30
35
15
20
40
45
1km*1km
10
10
a(t) cm/s*s
15
a(t) cm/s*s
5
20
1km*1km
5
0
5
0
-5
-10
-5
-15
-10
-20
-15
-20
0
5
10
15
20
25
30
35
40
45
20
0
5
10
15
20
25
30
35
40
45
15
20
a(t) cm/s*s
10
a(t) cm/s*s
2km*2km
10
15
2km*2km
5
0
5
0
-5
-10
-5
-15
-10
-20
-15
-20
0
0
5
10
15
20
25
30
35
40
15
10
a(t) cm/s*s
15
a(t) cm/s*s
10
5km*5km
5
0
-5
5
10
15
20
25
30
35
40
45
20
45
20
5km*5km
5
0
-5
-10
-10
-15
-15
-20
-20
0
0
5
10
15
20
25
30
35
40
20
0
-5
-10
-15
-20
10
10km*10km
a(t) cm/s*s
a(t) cm/s*s
5
10
15
20
25
30
35
40
45
15
15
10
5
20
45
5
10km*10km
0
-5
-10
-15
-20
15
Far field PSA with different subfault lengths, Using
Static Corner frequency
Far field PSA with different subfault lengths, Using
Dynamic Corner frequency
100
1 km
10
2 km
5 km
10 km
1
0.1
A c c e l e r a ti o n R e s p o n s e S p e c tr a
A c c e l e ra ti o n R e s p o n s e S p e c tra
100
1 km
10
2 km
5 km
10 km
1
0.1
0.1
1
10
f (Hz)
100
0.1
1
10
100
f (Hz)
16
Near source acceleration time series
Static Corner frequency
Dynamic Corner frequency
t (sec)
0
5
10
15
20
t (sec)
25
30
35
40
0
1000
800
800
600
1km*1km
400
200
0
-200
-400
25
30
35
40
45
1km*1km
0
-400
-600
-800
-1000
0
5
.
.
.
.
.
.
0
.
.
5
10
15
20
25
30
35
40
45
1000
800
800
2km*2km
400
200
0
-200
-400
600
a(t) cm/s*s
600
a(t) cm/s*s
20
-200
-800
-600
2km*2km
400
200
0
-200
-400
-600
-800
-800
-1000
-1000
0
5
10
15
20
25
30
35
40
45
0
1000
1000
800
800
5km*5km
400
200
0
-200
-400
5
10
15
20
25
30
35
-600
40
45
5km*5km
600
a(t) cm/s*s
600
a(t) cm/s*s
15
200
-600
400
200
0
-200
-400
-600
-800
-800
-1000
-1000
0
5
10
15
20
25
30
35
40
0
45
1000
1000
800
800
400
200
0
-200
-400
-600
10km*10km
600
a(t) cm/s*s
600
a(t) cm/s*s
10
400
-1000
1000
5
600
a(t) cm/s*s
a(t) cm/s*s
45
1000
400
5
10
15
20
25
30
35
40
45
10km*10km
200
0
-200
-400
-600
-800
-800
-1000
-1000
17
Near source PSA with different subfault lengths,
Using Static Corner frequency
Near source PSA with different subfault lengths,
Using Danamic Corner frequency
10000
1 km
1000
2 km
5 km
10 km
100
10
0.1
1
10
f (Hz)
100
A c c e le ra tio n R e s p o n s e S p e c tra
A c c e le ra tio n R e s p o n s e S p e c tra
10000
1 km
1000
2 km
5 km
10 km
100
10
0.1
1
10
100
f (Hz)
18