Embed Size (px)
Citation preview
A Wavelet Analysis of Ground Motion
Characteristics
R. Z. Sarica
M. S. Rahman
Outline
Introduction
Wavelet transform (WT) vs. Fourier transform (FT)
Wavelet analysis of ground motion characteristics (Chi-Chi earthquake motions)
Example analysis
Effect of distance
Effect of ‘softening’
Closure
Introduction
Ground Motion
Time History
Fourier Transform, Response Spectrum
Wavelet Transform
Applications to Geotechnical Earthquake Engineering
Study of Ground Motions
Simulation of Ground Motion
Wavelet domain analysis of site response (Including liquefaction)
Fourier Transform
a1, w1
a3, w2
a3, w3
deFtfdtetfF titi )()(;)(2
1)(
Fourier Transform
0 0.2 0.4 0.6 0.8 1-5
0
5S
tati
on
ary
0 0.2 0.4 0.6 0.8 1-5
0
5
No
n-s
tati
on
ary
0 50 100 150 2000
0.5
FT
Frequency (Hz)
Wavelet Transform
dt
s
kttfsW
TransformWaveletIntegral
ksf )()(2/1
),(
2
2/1
),( )(1
)(
Re
s
dkds
s
ktsW
Ctf
TransformWaveletfromonconstructi
ksf
dC 2ˆ
2
)()(2/1
),( s
ktst
functionsWaveletBasis
ks
10 15 20 25 30-0.5
0
0.5
1db10 Wavelet
level 7level 5
Wavelet Analysis: Evaluation of Coefficients
0 20 40 60 80 100-2
-1
0
1
2
level 5
0 20 40 60 80 100-2
-1
0
1
2
level 7
Discrete Wavelet Representation
a function aj(t) is resolved to several level(j) functions
a0(t) can be expressed in the following series of detail functions dj(t)
Thus,
kjkjj Ctd ,,)(
1
0 )()(j
j tdta
ao
a1 d1
a2 d2
a3 d3
ao = a1 + d1
a1 = a2 + d2
a2 = a3 + d3
ao = a3 + d3 + d2 + d1
where,
with , )2(2 /, ktjkjkj
kj
n
j kkjCta ,
1,0 )(
Energy Calculation
0 1
2
0
20 )()(
t
n
jj
t
tdtaE
2,, kjkj ddE
k
kjkj dEE1
,,
Total energy :
Energy density :
Cumulative energy :
Total energy in terms of level cumulative energies :
Frequency range : with
n
jkjEE
1,
][ 2,1 jjj
)2/(1;)2/(1 21
1 tt jj
jj
Frequency and period ranges
Level Period range (seconds) Frequency range (Hz)
1 0.01 - 0.020 100 - 50.0
2 0.02 - 0.040 50.0 - 25.0
3 0.04 - 0.080 25.0 - 12.5
4 0.08 - 0.160 12.5 -6.3
5 0.16 - 0.320 6.3 - 3.1
6 0.32 - 0.640 3.1 - 1.6
7 0.64 - 1.280 1.6 - 0.8
8 1.28 - 2.560 0.8 - 0.4
9 2.56 - 5.120 0.4 - 0.2
Wavelet Analysis of Chi-Chi motion TCU129-N
0 10 20 30 40 50 60 70 80
-600
-400
-200
0
200
400
600
Ground Motion,TCU129-N
Time (sec)
Acce
lera
tio
n (
cm
/se
c2)
0 5 10 15 20 25 30 35 400
100
200
300
400
500
600
Period (sec)
Fo
uri
er
Am
plitu
de
(cm
/se
c)
-1000
0
1000Approx. coeffs. btw level 1 and 9 from up to bottom
-10
0
10Det. coeffs. btw level 1 and 9 from up to bottom
-2000
0
2000
-50
0
50
-2000
0
2000
-500
0
500
-2000
0
2000
-1000
0
1000
-2000
0
2000
-2000
0
2000
-2000
0
2000
-1000
0
1000
-1000
0
1000
-2000
0
2000
-500
0
500
-1000
0
1000
0 20 40 60 80 100-500
0
500
Time (sec)0 20 40 60 80 100
-500
0
500
Time (sec)
TCU129-N
Wavelet Coefficients
0
500
Approximations for level 1 to 9 from up to bottom
0
500
Details for level 1 to 9 from up to bottom
0
500
0
500
0
500
0
500
0
500
0
500
0
500
Accele
ration
(cm
/sec2 )
0
500
Accele
ration
(cm
/sec2 )
0
500
0
500
0
500
0
500
0
500
0
500
0 20 40 60 80 1000
500
Time (sec)0 20 40 60 80 100
0
500
Time (sec)
TCU129-N
Approximations (aj) and Detatils (dj)
0
200
400level5original
0
200
400level6original
0
200
400
Fo
uri
er
am
plit
ud
e (
cm/s
ec)
level7original
0
200
400level8original
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50
200
400
Period (sec)
level9original
Fourier amplitude spectra of TCU129-N and dj (t)
Fourier Spectra of a0 and dj
Energy density dEj,k
Change this figure
Wavelet Analysis : Energy contents
0 10 20 30 40 50 60 70 80 900
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Time (sec)
No
rma
lize
d c
um
ula
tive
en
erg
y
h) TCU129-N
level 5 (0.16-0.32 sec)
level 4 (0.08-0.16 sec)
level 6 (0.32-0.64 sec)
level 7 (0.64-1.28 sec)
level 3level 8
Cumulative energy profile normalized wrt the total energy
0 10 20 30 40 50 60 70 80 900
1
2
3
4
5
6
7x 10
7 e) TCU129-N
Time (sec)
To
tal e
ne
rg
y (
cm
2 /se
c4 )
Total energy profile
Effect of Site Distance
Motion Location Site Classification Site Condition
Closest distance to
fault rupture (km)
Distance to epicentre
(km)
TCU078 NantouGeomatrix or CWB (1)
USGS (C) 7.5 7.1
TCU031 NantouGeomatrix or CWB (1)
USGS (C) 26.78 78.3
Alluvial deposit composed of cemented or poorly cemented clay, silty sand and gravel distributed on valleys and alluvial plateau
Motion Location Site Classification Site Condition
Closest distance to
fault rupture (km)
Distance to epicentre
(km)
TCU078 NantouGeomatrix or CWB (1)
USGS (C) 7.5 7.1
TCU031 NantouGeomatrix or CWB (1)
USGS (C) 26.78 78.3
Alluvial deposits composed of
cemented or poorly cemented clay, silty
sand, sand and gravel distributed on valleys and alluvial plateau.
Selected Motions
0 10 20 30 40 50 60 70 80 90-300
-200
-100
0
100
200
300
Time (sec)
Accele
ration (
cm
/sec
2 )
a) TCU078-N
0 10 20 30 40 50 60 70 80 90-300
-200
-100
0
100
200
300
Time (sec)
Accele
ration (
cm
/sec
2 )
b) TCU031-N
Effect of Site Distance
0 10 20 30 40 50 60 70 80 900
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5x 10
7 e) TCU078-N
Time (sec)
To
tal e
ne
rgy (
cm
2 /se
c4 )
0 10 20 30 40 50 60 70 80 900
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5x 10
7 f) TCU031-N
Time (sec)
To
tal e
ne
rgy (
cm
2 /se
c4 )
0 10 20 30 40 50 60 70 80 900
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
Time (sec)
No
rma
lize
d c
um
ula
tive
en
erg
y
h) TCU078-N
level 5 (0.16-0.32 sec)
level 6 (0.32-0.64 sec)
level 4 (0.08-0.16 sec)
level 7 (0.64-1.28 sec)
level 8 (1.28-2.56 sec)
0 10 20 30 40 50 60 70 80 900
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
Time (sec)
No
rma
lize
d c
um
ula
tive
en
erg
y
h) TCU031-N
level 9 (2.56-5.12 sec)
level 7
level 8
level 6level 5
level 4
Effect of Site Distance
Selected Motions
Wavelet Analysis : Effect of softening
Motion LocationSite
Classification Site Condition
Closest distance to
fault rupture (km)
Distance to epicentre
(km)
TCU129 Nantou
Geomatrix or CWB (1)
USGS (C) 1.18 11.90
TCU076 Nantou
Geomatrix or CWB (1)
USGS (C) 1.95 13.70
TCU120 Yuanlin
Geomatrix or CWB (1)
USGS (B) 8.10 23.20
TCU110 Yuanlin
Geomatrix or CWB (3)
USGS (D) 12.56 26.00
Alluvial deposit composed of
cemented or poorly cemented clay, silty
sand, sand and gravel distributed on valleys and alluvial plateau.
Situated on on a thick alluvial deposit. The
depth of bedrock increases from east to west, reaching a depth
of more than 200m. GWT is quite shallow, 0.5-4.0 m below GS.
0 10 20 30 40 50 60 70 80 90-800
-600
-400
-200
0
200
400
600
800
Time (sec)
Accele
ration (
cm
/sec
2 )
a) TCU129-N
0 10 20 30 40 50 60 70 80 90-800
-600
-400
-200
0
200
400
600
800
Time (sec)
Accele
ration (
cm
/sec
2 )
b) TCU076-N
Wavelet Analysis : Effect of softening
0 10 20 30 40 50 60 70 80 900
1
2
3
4
5
6
7x 10
7 e) TCU129-N
Time (sec)
Tota
l energ
y (
cm
2 /sec4 )
0 10 20 30 40 50 60 70 80 900
1
2
3
4
5
6
7x 10
7 f) TCU076-N
Time (sec)
Tota
l energ
y (
cm
2 /sec4 )
0 10 20 30 40 50 60 70 80 900
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Time (sec)
Nor
mal
ized
cum
ulat
ive
ener
gy
h) TCU129-N
level 5 (0.16-0.32 sec)
level 4 (0.08-0.16 sec)
level 6 (0.32-0.64 sec)
level 7 (0.64-1.28 sec)
level 3level 8
0 10 20 30 40 50 60 70 80 900
0.1
0.2
0.3
0.4
0.5
0.6
Time (sec)
Nor
mal
ized
cum
ulat
ive
ener
gy
h) TCU076-N
level 4
level 6
level 7
level 8level 4level 3level 9
Wavelet Analysis : Effect of softening
0 10 20 30 40 50 60 70 80 90-200
-150
-100
-50
0
50
100
150
200
Time (sec)
Acce
lera
tio
n (
cm
/se
c2 )
a) TCU120-N
0 10 20 30 40 50 60 70 80 90-200
-150
-100
-50
0
50
100
150
200
Time (sec)
Acce
lera
tio
n (
cm
/se
c2 )
b) TCU110-N
Wavelet Analysis : Effect of softening
0 10 20 30 40 50 60 70 80 900
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
Time (sec)
Nor
mal
ized
cum
ulat
ive
ener
gy
h) TCU120-N
level 6
level 5
level 7
level 4
level 8
level1level2level3level4level5level6level7level8level9
0 10 20 30 40 50 60 70 80 900
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
Time (sec)
Nor
mal
ized
cum
ulat
ive
ener
gy
h) TCU110-N
level 8
level 7
level 6
level 9
level1level2level3level4level5level6level7level8level9
0 10 20 30 40 50 60 70 80 900
0.5
1
1.5
2
2.5
3x 10
7 f) TCU110-N
Time (sec)
To
tal e
ne
rgy (
cm
2 /se
c4 )
0 10 20 30 40 50 60 70 80 900
0.5
1
1.5
2
2.5
3x 10
7 f) TCU120-N
Time (sec)
To
tal e
ne
rgy (
cm
2 /se
c4 )
Wavelet Analysis : Effect of softening
Liquefiable layer
Base motion (TCU120)
Recorded surface motion (TCU110)
0 10 20 30 40 50 60 70 80 90-400
-300
-200
-100
0
100
200
300
400
Time (sec)
Accele
ration (
cm
/sec
2 )
a) TCU120-N
0 10 20 30 40 50 60 70 80 90-400
-300
-200
-100
0
100
200
300
400
Time (sec)
Accele
ration (
cm
/sec
2 )
b) TCU120-Go
0 10 20 30 40 50 60 70 80 90-400
-300
-200
-100
0
100
200
300
400
Time (sec)
Accele
ration (
cm
/sec
2 )
b) TCU120-Gf
Computed surface motion with reduced G,(TCU120-Gf)
Computed surface motion (TCU120-Go)
0 10 20 30 40 50 60 70 80 90-400
-300
-200
-100
0
100
200
300
400
Time (sec)
Acce
lera
tio
n (
cm
/se
c2)
TCU110-N ground motion
Wavelet Analysis : Effect of softening
Wavelet Analysis : Effect of softening
Recorded Computed
0 10 20 30 40 50 60 70 80 900
0.5
1
1.5
2
2.5
3x 10
7 e) TCU110-N
Time (sec)
Tota
l ene
rgy (
cm2 /s
ec4 )
0 10 20 30 40 50 60 70 80 900
0.5
1
1.5
2
2.5
3x 10
7 f) TCU120-Gf
Time (sec)
Tota
l ene
rgy (
cm2 /s
ec4 )
0 10 20 30 40 50 60 70 80 900
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
Time (sec)
Norm
aliz
ed c
um
ula
tive e
nerg
y
h) TCU120-Gf
level 6
level 7
level 8
level 9level 5
level 4
0 10 20 30 40 50 60 70 80 900
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
Time (sec)
Norm
aliz
ed c
um
ula
tive e
nerg
y
h) TCU110-N
level 8
level 7
level 6
level 9
level 5
Wavelet Analysis : Effect of softening
Recorded Computed
Closure
Wavelet analysis has been used to study the characteristics of several ground motions recorded during Chi-chi earthquake.
Effect of site distance and site softening has been investigated. Reduction in energy and redistribution of energy contents observed.
Wavelet analysis has potential of providing more insight into ground motion characteristics including the energy content in both frequency and time domain simultaneously.
