# AW avelet Analysis of Ground Motion Characteristics

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```					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

                                      
1
F ( ) 
2   

f (t )e i t dt   ;   f (t )    

F ( )ei t d
Fourier Transform

5
Stationary
0

-5
0   0.2        0.4        0.6     0.8    1
5
Non-stationary

0

-5
0   0.2        0.4        0.6     0.8    1

0.5
FT

0
0          50        100         150     200
Frequency (Hz)
Wavelet Transform
IntegralWaveletTransform                                       Re construction fromWaveletTransform
1 / 2                    t k                        1                     1 / 2 t  k ds dk
W f ( s ,k )  s              
f (t ) (
s
)dt         f (t ) 
C   W f ( s ,k ) s  (
s
) 2
s


2
BasisWavelet functions                                                      ˆ
C  2     d
t k                                   
1 / 2
 ( s ,k ) (t )  s        (            )
s
db10 Wavelet
1
level 7
level 5

0.5

0

-0.5
10                  15            20                25                30
Wavelet Analysis: Evaluation of Coefficients

2
level 5
1

0

-1

-2
0   20            40     60   80   100

2
level 7
1

0

-1

-2
0   20            40     60   80   100
Discrete Wavelet Representation
a function aj(t) is resolved to                              ao                ao = a1 + d1
several level(j) functions
a1              d1            a1 = a2 + d2

a0(t) can be expressed in the
a2            d2                         a2 = a3 + d3
following series of detail
functions dj(t)                          a3          d3                        ao = a3 + d3 + d2 + d1


a 0 (t )    d
j 1
j   (t )

where,                    d j (t )   C j ,k j ,k


with ,               j , k  2 j / k  (2 j t  k )


 Thus,                                         n
a 0 (t )         C        j ,k j ,k
j 1 k  
Energy Calculation

                    n
Total energy :            E   a (t )   d 2 (t )
2
0     j
t 0            t  0 j 1

Energy density :     dE j ,k  d 2,k
j

k
Cumulative energy : E j ,k   dE j ,k
1

n
Total energy in terms of level cumulative energies :                     E   E j ,k
j 1

Frequency range :         j  [1 j ,2 j ]       with

1 j  1/(2 j 1 t )          ;     2 j  1/(2 j t )
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

Ground Motion,TCU129-N
Acceleration (cm/sec2)                 600

400

200

0

-200

-400

-600
0   10   20        30         40        50        60        70    80
Time (sec)

600
Fourier Amplitude (cm/sec)

500

400

300

200

100

0
0    5        10         15        20        25        30        35    40
Period (sec)
Wavelet Coefficients
TCU129-N
Approx. coeffs. btw level 1 and 9 from up to bottom      Det. coeffs. btw level 1 and 9 from up to bottom
1000                                                     10
0                                                      0
-1000                                                    -10
2000                                                     50
0                                                      0
-2000                                                   -50
2000                                                   500
0                                                      0
-2000                                                  -500
2000                                                  1000
0                                                      0
-2000                                                  -1000
2000                                                   2000
0                                                      0
-2000                                                  -2000
2000                                                   1000
0                                                      0
-2000                                                  -1000
1000                                                   2000
0                                                      0
-1000                                                  -2000
500                                                   1000
0                                                      0
-500                                                  -1000
500                                                    500
0                                                      0
-500                                                   -500
0   20       40      60       80      100              0   20       40      60        80      100
Time (sec)                                             Time (sec)
Approximations (aj) and Detatils (dj)
TCU129-N
Approximations for level 1 to 9 from up to bottom                                       Details for level 1 to 9 from up to bottom
500                                                                            500

0                                                                              0
500                                                                            500

0                                                                              0
500                                                                            500

0                                                                              0
500                                                                            500
Acceleration (cm/sec 2)

Acceleration (cm/sec 2)
0                                                                              0
500                                                                            500

0                                                                              0
500                                                                            500

0                                                                              0
500                                                                            500

0                                                                              0
500                                                                            500

0                                                                              0
500                                                                            500

0                                                                              0
0    20      40      60        80      100                                     0         20       40      60        80          100
Time (sec)                                                                           Time (sec)
Fourier Spectra of a0 and dj

Fourier amplitude spectra of TCU129-N and d j (t)
400
level5
200                                                                           original

0
400
level6
200                                                                           original
Fourier amplitude (cm/sec)

0
400
level7
200                                                                           original

0
400
level8
200                                                                           original

0
400
level9
200                                                                           original

0
0    0.5   1        1.5       2       2.5        3     3.5         4   4.5         5
Period (sec)
Wavelet Analysis : Energy contents
h) TCU129-N
0.7

0.6

Normalized cumulative energy
0.5
level 5 (0.16-0.32 sec)

0.4

0.3

0.2                                              level 4 (0.08-0.16 sec)
level 6 (0.32-0.64 sec)
0.1                                                      level 7 (0.64-1.28 sec)

Change this figure                                     level 3
level 8
0
0       10   20    30    40      50   60      70      80          90
Time (sec)

Cumulative energy profile normalized wrt the total energy
x 10
7                   e) TCU129-N
7

6

5

Total energy (cm2/sec4)
4

3

2

1

0
0       10   20    30    40      50   60      70      80          90
Time (sec)

Total energy profile
Energy density dEj,k
Effect of Site Distance

Selected Motions

Closest
distance to    Distance to
fault rupture    epicentre
Motion   Location    Site Classification     Site Condition            (km)           (km)

Alluvial deposit
composed of
Alluvial deposits
cemented or
composed of
poorly cemented
Geomatrix or CWB (1)
cemented or poorly
clay, silty sand
TCU078    Nantou        USGS (C)                                        7.5            7.1
cemented clay, silty
and gravel
sand, sand and gravel
distributed on
distributed on valleys
valleys and
and alluvial plateau.
alluvial plateau
Geomatrix or CWB (1)
TCU031    Nantou        USGS (C)                                      26.78           78.3
Effect of Site Distance
a) TCU078-N                                                                          b) TCU031-N
300                                                                                  300

200                                                                                  200
Acceleration (cm/sec2)

Acceleration (cm/sec2)
100                                                                                  100

0                                                                                    0

-100                                                                                 -100

-200                                                                                 -200

-300                                                                                 -300
0   10   20   30    40      50   60   70   80   90                                   0   10   20   30    40      50   60   70   80   90
Time (sec)                                                                           Time (sec)
Effect of Site Distance
h) TCU078-N                                                                                                h) TCU031-N
0.45                                                                                                       0.45
level 5 (0.16-0.32 sec)
level 9 (2.56-5.12 sec)
0.4                                                                                                        0.4

0.35                                                                                                       0.35
Normalized cumulative energy

Normalized cumulative energy
0.3                                                                                                        0.3

level 6 (0.32-0.64 sec)
0.25                                                                                                       0.25

0.2                                    level 4 (0.08-0.16 sec)                                             0.2                                              level 7
level 8
0.15                                                                                                       0.15
level 6
level 7 (0.64-1.28 sec)                                                                                              level 5
0.1                                                                                                        0.1

0.05                                                                                                       0.05
level 8 (1.28-2.56 sec)                                                                                              level 4

0                                                                                                          0
0       10   20   30    40      50      60      70         80   90                                         0       10   20   30    40      50   60   70         80    90
Time (sec)                                                                                                 Time (sec)

x 10
7                  e) TCU078-N                                                                         x 10
7                  f) TCU031-N
5                                                                                                          5

4.5                                                                                                        4.5

4                                                                                                          4

3.5                                                                                                        3.5
Total energy (cm2/sec4)

Total energy (cm2/sec4)
3                                                                                                          3

2.5                                                                                                        2.5

2                                                                                                          2

1.5                                                                                                        1.5

1                                                                                                          1

0.5                                                                                                        0.5

0                                                                                                          0
0       10   20   30    40      50      60      70         80   90                                         0       10   20   30    40      50   60   70         80    90
Time (sec)                                                                                                 Time (sec)
Wavelet Analysis : Effect of softening
Selected Motions

Closest
distance to    Distance to
Site                                   fault rupture    epicentre
Motion   Location   Classification      Site Condition             (km)           (km)

Alluvial deposit
Geomatrix or
composed of
CWB (1)
cemented or poorly
TCU129    Nantou     USGS (C)                                      1.18           11.90
cemented clay, silty
sand, sand and gravel
distributed on valleys
Geomatrix or      and alluvial plateau.
CWB (1)
TCU076    Nantou     USGS (C)                                      1.95           13.70

Situated on on a thick
Geomatrix or       alluvial deposit. The
CWB (1)             depth of bedrock
TCU120   Yuanlin     USGS (B)        increases from east to        8.10           23.20
west, reaching a depth
of more than 200m.
GWT is quite shallow,
Geomatrix or      0.5-4.0 m below GS.
CWB (3)
TCU110   Yuanlin     USGS (D)                                     12.56           26.00
Wavelet Analysis : Effect of softening
a) TCU129-N                                                                          b) TCU076-N
800                                                                                   800

600                                                                                   600

400                                                                                   400
Acceleration (cm/sec2)

Acceleration (cm/sec2)
200                                                                                   200

0                                                                                     0

-200                                                                                  -200

-400                                                                                  -400

-600                                                                                  -600

-800                                                                                  -800
0   10    20   30    40      50   60   70   80   90                                   0   10   20   30    40      50   60   70   80   90
Time (sec)                                                                           Time (sec)
Wavelet Analysis : Effect of softening
h) TCU129-N                                                                                                      h) TCU076-N
0.7

0.6                                                                                                              0.6
Normalized cumulative energy

Normalized cumulative energy
0.5
level 5 (0.16-0.32 sec)                                  0.5

0.4                                                                                                              0.4
level 6

0.3                                                                                                              0.3

level 4
0.2                                             level 4 (0.08-0.16 sec)                                          0.2
level 7
level 6 (0.32-0.64 sec)
level 8
0.1                                                     level 7 (0.64-1.28 sec)                                  0.1                                                  level 4
level 3
level 3
level 9
level 8
0                                                                                                                0
0       10   20   30    40      50   60      70      80          90                                              0       10   20   30    40      50   60   70         80   90
Time (sec)                                                                                                       Time (sec)

x 10
7                  e) TCU129-N                                                                               x 10
7                  f) TCU076-N
7                                                                                                                7

6                                                                                                                6

5                                                                                                                5
Total energy (cm2/sec4)

4
Total energy (cm2/sec4)   4

3                                                                                                                3

2                                                                                                                2

1                                                                                                                1

0                                                                                                                0
0       10   20   30    40      50   60      70      80          90                                              0       10   20   30    40      50   60   70         80   90
Time (sec)                                                                                                       Time (sec)
Wavelet Analysis : Effect of softening
a) TCU120-N                                                                          b) TCU110-N
200                                                                                  200

150                                                                                  150

100                                                                                  100
Acceleration (cm/sec2)

Acceleration (cm/sec2)
50                                                                                   50

0                                                                                    0

-50                                                                                  -50

-100                                                                                 -100

-150                                                                                 -150

-200                                                                                 -200
0   10   20   30    40      50   60   70   80   90                                   0   10   20   30    40      50   60   70   80   90
Time (sec)                                                                           Time (sec)
Wavelet Analysis : Effect of softening
h) TCU120-N                                                                                                   h) TCU110-N
0.45                                                                                                          0.45
level1                                                                                     level 8                   level1
level2                                                                                                               level2
0.4                                                             level3                                        0.4                                                                    level3
level4                                                                                                               level4
level5                                                                                                               level5
level6                                                                                                               level6
0.35                                                                                                          0.35
level7                                                                                                               level7
Normalized cumulative energy

Normalized cumulative energy
level8                                                                                                               level8
level9                                                                                     level 7                   level9
0.3                                             level 6                                                       0.3

0.25                                             level 5                                                      0.25

0.2                                                                                                           0.2
level 7
0.15                                                                                                          0.15
level 4                                                                                                               level 6
0.1                                         level 8                                                           0.1
level 9

0.05                                                                                                          0.05

0                                                                                                             0
0       10   20   30    40      50   60         70   80            90                                         0       10   20   30    40       50       60     70         80            90
Time (sec)                                                                                                    Time (sec)

x 10
7                  f) TCU120-N                                                                            x 10
7                  f) TCU110-N
3                                                                                                             3

2.5                                                                                                           2.5
Total energy (cm2/sec4)

Total energy (cm2/sec4)
2                                                                                                             2

1.5                                                                                                           1.5

1                                                                                                             1

0.5                                                                                                           0.5

0                                                                                                             0
0       10   20   30    40      50   60         70   80            90                                         0       10   20   30    40       50       60     70         80            90
Time (sec)                                                                                                    Time (sec)
Wavelet Analysis : Effect of softening
b) TCU120-Go

Computed surface motion
400

300

400
TCU110-N ground motion
200                                                          (TCU120-Go)

Acceleration (cm/sec2)
300
100

200
0                                                                                                                b) TCU120-Gf
400
Acceleration (cm/sec2)

100                                                                                   -100
300

0                                                                                   -200
200

Acceleration (cm/sec2)
-100                                                                                  -300
100

-200                                                                                  -400
0   10   20   30     40     50    60   70   80   90                                 0
Time (sec)
-300
-100

-400
0   10   20   30    40      50    60   70   80   90                                                                                                                     -200
Time (sec)

-300

Recorded surface                                                                                                                                                             -400
0   10   20   30    40      50
Time (sec)
60   70   80   90

motion (TCU110)
Computed surface
motion with reduced
Liquefiable layer
G,(TCU120-Gf)

a) TCU120-N
400

300

200
Acceleration (cm/sec2)

100
Base motion
0

-100
(TCU120)
-200

-300

-400
0   10   20   30     40     50    60   70   80   90
Time (sec)
Wavelet Analysis : Effect of softening

Recorded                  Computed
Wavelet Analysis : Effect of softening
Recorded                                                                                             Computed
h) TCU110-N                                                                                          h) TCU120-Gf
0.5                                                                                                  0.5

level 8                                                                                                    level 6
0.45                                                                                                 0.45

0.4                                                                                                  0.4
Normalized cumulative energy

Normalized cumulative energy
0.35                                                                                                 0.35
level 7
level 7
0.3                                                                                                  0.3

0.25                                                                                                 0.25

0.2                                                                                                  0.2

0.15                                                                                                 0.15
level 6
0.1                                                                                                  0.1                                               level 8
level 9
level 9
0.05                                                                                                 0.05                                                              level 5
level 5
level 4
0                                                                                                    0
0       10   20   30    40      50   60   70         80   90                                         0       10   20   30    40      50    60   70         80         90
Time (sec)                                                                                           Time (sec)

x 10
7                  e) TCU110-N                                                                   x 10
7                  f) TCU120-Gf
3                                                                                                    3

2.5                                                                                                  2.5
Total energy (cm2/sec4)

Total energy (cm2/sec4)
2                                                                                                    2

1.5                                                                                                  1.5

1                                                                                                    1

0.5                                                                                                  0.5

0                                                                                                    0
0       10   20   30    40      50   60   70         80   90                                         0       10   20   30    40      50    60   70         80         90
Time (sec)                                                                                           Time (sec)
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.

```
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