Wavelet Transformation

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Transcript Wavelet Transformation

Wavelet Transform Modulus Maxima ridge and its application on Stratigraphic Profiling

STUDENT: R03521101 CHUN-HSIANG WANG LECTURER: JIAN-JIUN DING DATE: 2014/11/27 1

Outline

    Introduction Wavelet Transformation  Wavelet Zoom  Wavelet Transform Modulus Maxima Application-Stratigraphic profiling  SBT of CPT  Demonstration - Simulative case  Demonstration – Real case Conclusion 2

Outline

   

Introduction

Wavelet Transformation  Wavelet Zoom  Wavelet Transform Modulus Maxima Application-Stratigraphic profiling  SBT of CPT  Demonstration - Simulative case  Demonstration – Real case Conclusion 3

Introduction

 Why we need Wavelet Transform?

Time Frequency Analysis Wavelet

Short-time Fourier Transform Time & Frequency Wavelet Transform Transition & Scaling Characteristic of Frequency Distinguish local property 4

Outline

    Introduction

Wavelet Transformation

 Wavelet Zoom  Wavelet Transform Modulus Maxima Application-Stratigraphic profiling  SBT of CPT  Demonstration - Simulative case  Demonstration – Real case Conclusion 5

Continuous Wavelet Transform

6  Wavelet Transform = Dilation + Translation    CWT:    

D

  1

s

 * Translation  1

s

t

u s

Dilation

t

u s dt

|

u R s R

 → Convolution Form

CWT-cont.

 Basis characteristics  1.     (t)dt  0 2.      * (t)dt   3.  0    2 d    2  1 7

CWT-cont.

 Famous Wavelet Basis Type 0.25

0.2

Differential Gaussian function DerGaussian 0.5

0.4

0.15

0.3

0.1

0.05

0 -0.05

-0.1

-0.15

-0.2

-0.25

-10 -5 0 t (s) 5 10 0.2

0.1

0 -0.1

-0.2

-10 Mexican Hat Mexican hat -5 0 t (s) 5 10 8

CWT-cont.

 DerGaussain function as example 0.4

0.4

u=0, s=0.02

u=0.1, s=0.02

0.3

0.3

0.2

0.2

0.1

0.1

0 0 -0.1

-0.1

-0.2

-0.2

-0.3

-0.3

-0.4

-2 -1.5

-1 -0.5

0 0.5

1 1.5

2 u=0, s=0.02

u=0, s=0.04

-0.4

-2 -1.5

-1 -0.5

0 0.5

1 1.5

2 9

Wavelet Zoom

 Focus on localized signal structures with a zooming procedure that progressively reduces the scale parameter 10

Lipschitz Regularity

 such that,

p v

v m

 

t

, ( )

v

    

v

11

2 1.5

1 0.5

2.5

3

Lipschitz Regularity-Example

Lipschitz alpha =0 Lipschitz alpha =1 5 4.5

4 0 0 1 2 3 Jump 7 8 9 1.5

1 0.5

3.5

3 2.5

2 10 0 0 1 2 3 Cusp 7 8 4 5 t (s) 6 4 5 t (s) 6 9 10 12

Vanishing moment

 that,  

n

vanishing moments iff there exists with a fast decay such   

t k

  0,

k

  Ψ(t) with n vanishing moments can only “see” a change point with Lipschitz regularity α that is less than n .

13

Wavelet Transform Modulus Maxima(WTMM)

 WTMM = ridge     , 

As

  1 2  log  log

A

   1 2  ( The dip of equation of this ridge is 0.5 definitely.

jump

,  0) 14

WTMM-cont.

15

Outline

    Introduction

Wavelet Transformation

 Wavelet Zoom  Wavelet Transform Modulus Maxima

Application-Stratigraphic profiling

SBT of CPT

 Demonstration - Simulative case  Demonstration – Real case Conclusion 16

Geotechnical 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 ! 17

Cone Penetration Test

圓錐貫入試驗  In-situ Test  Main Measurement     Cone Resistance,q c Friction Sleeve,f s Pore Water Pressure,u 2 Target  Site investigation P.K. Robertson, 1990 18

Soil Behavior Type(SBT)

 P.K. Robertson,1998

F r

q v

f s

v

0

Q tn

 

q t P a

v

0 '     

P a v

0   1. Sensitive, fine grained 2. 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 sand 8. Very stiff sand to clayey sand 9. Very stiff, fine grained 19

I

c

imply SBT

 P.K. Robertson, 1998 20 𝐼 𝑐 = 3.47 − 𝑄 𝑡𝑛 2 + 1.22 + 𝐹 𝑟 2

SBT

Ic < 1.31

1.31< Ic < 2.05

2.05< Ic < 2.60

2.60< Ic < 2.95

2.95< Ic < 3.60

Ic > 3.60

Description

Gravelly sand to dense sand Sands: clean sand to silty sand Sand mixtures: silty sand to sandy silt Silt mixtures: clayey silt to silty clay Clays: silty clay to clay Organic soil

Insight 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! 6 8 0 2 4 10 12 14 16 18 20 -2 0 2 I c (z) 4 6 21

Transition 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 !

22

Outline

   Introduction

Wavelet Transformation

 Wavelet Zoom  Wavelet Transform Modulus Maxima

Application-Stratigraphic profiling

 SBT of CPT 

Demonstration - Simulative case

 

Demonstration – Real case

Conclusion 23

Simulated Case Demonstration

0 4 6 0 2 8 10 12 14 16 18 20 1 C A 2 I c (z) 3 B 4 -0.5

-1 -1.5

-2 -2.5

-3 -3.5

C A B -4 -1.5

-1 -0.5

log(s) 0 0.5

24

In-situ Case Demonstration-NGES

 Taxes A&M University 25 (National Geotechnical Experimentation Site,1993)

NGES-cont.

26

NGES-cont.

27

More difficult case

 Oslo Main airport station 28

Oslo in-situ case-cont.

29

Outline

    Introduction Wavelet Transformation  Wavelet Zoom  Wavelet Transform Modulus Maxima Application-Stratigraphic profiling  SBT of CPT  Demonstration - Simulative case  Demonstration – Real case Conclusion 30

Conclusion

 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. 31

Conclusion-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. 32

Reference

 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 33

Thanks for your listening!

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