wavelet transform modulus maxima ridge and its application on stratigraphic profiling student:...
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1
Wavelet Transform Modulus Maxima ridge and its application on Stratigraphic Profiling
STUDENT: R03521101 CHUN-HSIANG WANG
LECTURER: JIAN-JIUN DING
DATE: 2014/11/27
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2Outline
Introduction Wavelet Transformation
Wavelet Zoom Wavelet Transform Modulus Maxima
Application-Stratigraphic profiling SBT of CPT Demonstration - Simulative case Demonstration – Real case
Conclusion
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3Outline
Introduction Wavelet Transformation
Wavelet Zoom Wavelet Transform Modulus Maxima
Application-Stratigraphic profiling SBT of CPT Demonstration - Simulative case Demonstration – Real case
Conclusion
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4Introduction
Why we need Wavelet Transform?
Time Frequency Analysis Wavelet
Short-time Fourier Transform
Wavelet Transform
Time & Frequency Transition & Scaling
Characteristic of Frequency Distinguish local property
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5Outline
Introduction Wavelet Transformation
Wavelet Zoom Wavelet Transform Modulus Maxima
Application-Stratigraphic profiling SBT of CPT Demonstration - Simulative case Demonstration – Real case
Conclusion
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6Continuous Wavelet Transform
*
,
1( , ) ( ) |
u R s R
t uWf u s f t dt
ss
Wavelet Transform = Dilation + Translation
CWT:
→ Convolution Form
( ) 0t dt
,
1( )u s
t uD t
ss
Dilation
Translation
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7CWT-cont.
Basis characteristics
if is a wavelet basis, then
1.
2.
3.
(t)dt 0
2*(t) (t)dt ( ) 1t
2
0
( )d
, ( )u s t
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8CWT-cont. Famous Wavelet Basis Type
DerGaussian Mexican Hat
-10 -5 0 5 10-0.25
-0.2
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
0.2
0.25
t (s)
Differential Gaussian function
-10 -5 0 5 10-0.2
-0.1
0
0.1
0.2
0.3
0.4
0.5
t (s)
Mexican hat
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9CWT-cont. DerGaussain function as example
-2 -1.5 -1 -0.5 0 0.5 1 1.5 2-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
t (s
)
u=0, s=0.02u=0.1, s=0.02
-2 -1.5 -1 -0.5 0 0.5 1 1.5 2-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
t (s
)
u=0, s=0.02u=0, s=0.04
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10Wavelet Zoom Focus on localized signal structures with a
zooming procedure that progressively reduces the scale parameter
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11Lipschitz Regularity A function is pointwise Lipschitz at , if
there exists , and a polynomial of degree such that,
The Lipschitz regularity of at or over is the least upper bound of the such that is Lipschitz .
f vK m vp
, ( ) ( )vt f t p t K t v
f v f
[ , ]a b
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12Lipschitz Regularity-Example
0 1 2 3 4 5 6 7 8 9 100
0.5
1
1.5
2
2.5
3
t (s)
Am
ptitu
de
0 1 2 3 4 5 6 7 8 9 100
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
t (s)
Am
ptit
ud
e
Jump Cusp
Lipschitz alpha =0 Lipschitz alpha =1
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13Vanishing moment A wavelet with a fast decay has n vanishing
moments iff there exists with a fast decay such that,
Ψ(t) with n vanishing moments can only “see” a change point with Lipschitz regularity α that is less than n.
( ) 0, [0, )kt t dt k n
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14Wavelet Transform Modulus Maxima(WTMM) WTMM = ridge
The dip of equation of this ridge is 0.5 definitely.
1
2( , ) [a,b] , ( , )
1log ( , ) log log( )
2
u s Wf u s As
Wf u s A s
( , 0)jump
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15WTMM-cont.
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16Outline
Introduction Wavelet Transformation
Wavelet Zoom Wavelet Transform Modulus Maxima
Application-Stratigraphic profiling SBT of CPT Demonstration - Simulative case Demonstration – Real case
Conclusion
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17Geotechnical Engineering大地工程 Soil mechanics Rock mechanics, Tunnel Engineering Soil Dynamics, Geotechnical Earthquake
Engineering Engineering Geology, Fault Detecting Foundation Engineering, underground
Excavation ……
凡是地下的工程議題都是大地有關範
疇!
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18Cone Penetration Test圓錐貫入試驗 In-situ Test Main Measurement
Cone Resistance,qc
Friction Sleeve,fs
Pore Water Pressure,u2
Target Site investigation
P.K. Robertson, 1990
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19Soil Behavior Type(SBT) P.K. Robertson,1998
1. Sensitive, fine grained2. Organic soils (peats)3. Clays (clay to silty clay)4. Silt mixtures (clayey silt to silty clay)5. Sand mixtures (silty sand to sandy
silt)6. Sands (clean sand to silty sand)7. Gravelly sand to sand8. Very stiff sand to clayey sand9. Very stiff, fine grained
0
0
0
'
v
a
a
vttn
vv
sr
P
P
q
fF
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20Ic imply SBT P.K. Robertson, 1998
SBT Description
Ic < 1.31 Gravelly sand to dense sand
1.31< Ic < 2.05 Sands: clean sand to silty sand
2.05< Ic < 2.60 Sand mixtures: silty sand to sandy silt
2.60< Ic < 2.95 Silt mixtures: clayey silt to silty clay
2.95< Ic < 3.60 Clays: silty clay to clay
Ic > 3.60 Organic soil
𝐼 𝑐=√ (3.47−𝑄𝑡𝑛 )2+(1.22+𝐹𝑟 )
2
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21Insight of Soil Layers location at which the soil behavior type index
changes abruptly
SBT 6
SBT 3
But in reality……
There will be some noise definitely!
-2 0 2 4 6
0
2
4
6
8
10
12
14
16
18
20
Ic(z)
Dep
th z
(m
)
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22Transition Zone Cone can sense a layer boundary up to a distance
of 15 cone diameters ahead and behind.
That will make us more difficult to identify layers !
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23Outline
Introduction Wavelet Transformation
Wavelet Zoom Wavelet Transform Modulus Maxima
Application-Stratigraphic profiling SBT of CPT Demonstration - Simulative case Demonstration – Real case
Conclusion
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24Simulated Case Demonstration
1 2 3 4
0
2
4
6
8
10
12
14
16
18
20
Ic(z)
De
pth
z (
m)
-1.5 -1 -0.5 0 0.5-4
-3.5
-3
-2.5
-2
-1.5
-1
-0.5
0
log(s)lo
gW
I C(u
,s)
A
C
A
C
B
B
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25In-situ Case Demonstration-NGES Taxes A&M University
(National Geotechnical Experimentation Site,1993)
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26NGES-cont.
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27NGES-cont.
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28More difficult case Oslo Main airport station
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29Oslo in-situ case-cont.
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30Outline
Introduction Wavelet Transformation
Wavelet Zoom Wavelet Transform Modulus Maxima
Application-Stratigraphic profiling SBT of CPT Demonstration - Simulative case Demonstration – Real case
Conclusion
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31Conclusion WTMM is widely applied to detecting discontinuity,
like jump or cusp, in nowaday engineering. Using a series of scale, or narrowing windows, we
can grab the characteristic of a signal at some one local position.
It’s used to bore one or several holes at a construction site for investigation the stratigraphic property. If we enforce CPT and WTMM in field investigation, it will be more efficient and economical.
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32Conclusion-cont. In Taiwan we usually take USCS as main
principle of soil classification but not SBT of CPT. However, it must take lots of time and manpower if we still take USCS.
SBT of CPT has a clear and concise image of civil engineering application, because of the clear distinguishing principle of sand and clay. It will help us to realize a better design in engineering.
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33Reference P.K. Robertson, C.E. Wride, Evaluating cyclic liquefaction
potential using the cone penetration test, 1998 P.K. Robertson, Interpretation of cone penetration tests — a
unified approach, 2009 B. S. Chen, P.W. Mayne, Profiling the overconsolidation raito of
clays by Piezocone tests, 1994 Y. Wang, Probabilistic identification of underground soil
stratification using cone penetration tests, 2013 J. Benoît, A. J. Lutenegger, National Geotechnical
Experimentation Sites, 1993 Mallat, A Wavelet Tour of Signal Processing, 2008
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34
Thanks for your listening!