### Wavelet Transformation

```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
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

4
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
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
Translation



 ( t ) dt  0 

1
tu
D   u , s ( t ) 
 
s  s




Dilation
CWT:
W f (u , s ) 



f (t )
tu
 
s
 s
1
*

 dt |u  R , s R 

→ Convolution Form
CWT-cont.

7
Basis characteristics
if  u , s ( t ) is a wavelet basis, then
1.
2.
3.








0
 (t)d t  0
 (t)  (t)dt   ( t )
*
 ( )

2
d  
2
1
CWT-cont.

8
Famous Wavelet Basis Type
0.25
0.5
Mexican hat
Differential Gaussian function
0.2
DerGaussian
0.4
Mexican Hat
0.15
0.3
0.1
0.05
0.2
0
0.1
-0.05
-0.1
0
-0.15
-0.1
-0.2
-0.25
-10
-5
0
t (s)
5
10
-0.2
-10
-5
0
t (s)
5
10
CWT-cont.

9
DerGaussain function as example
0.4
0.4
u=0, s=0.02
u=0.1, s=0.02
u=0, s=0.02
u=0, s=0.04
0.3
0.3
0.2
0.2
0.1
0
t (s)
t (s)
0.1
-0.1
-0.1
-0.2
-0.2
-0.3
-0.4
-2
0
-0.3
-1.5
-1
-0.5
0
0.5
1
1.5
2
Wavelet Zoom

Focus on localized signal structures with a
zooming procedure that progressively reduces
the scale parameter
10
Lipschitz Regularity

A function f is pointwise Lipschitz  at v , if there
exists K , and a polynomial p v of degree m  
such that,
 t   , f (t )  p v (t )  K t  v

11

The Lipschitz regularity of f at v or over [ a , b ] is the
least upper bound  of the such that f is
Lipschitz  .
Lipschitz Regularity-Example
Lipschitz alpha =0
12
Lipschitz alpha =1
5
3
4.5
2.5
4
3.5
3
Amptitude
Amptitude
2
1.5
Jump
1
2.5
Cusp
2
1.5
1
0.5
0.5
0
0
1
2
3
4
5
t (s)
6
7
8
9
10
0
0
1
2
3
4
5
t (s)
6
7
8
9
10
Vanishing moment

A wavelet  with a fast decay has n vanishing
moments iff there exists  with a fast decay such
that,




t  ( t ) dt  0, k  [0, n )
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
  ( u , s )  [a, b]   , W f ( u , s )  A s

1
2
1

 log W f ( u , s )  log A      log( s )
2


(
The dip of equation of this
ridge is 0.5 definitely.
jum p ,   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


Soil mechanics

Rock mechanics, Tunnel Engineering

Soil Dynamics, Geotechnical Earthquake
Engineering

Engineering Geology, Fault Detecting

Foundation Engineering, underground
Excavation

……

！是

17
Cone Penetration Test


In-situ Test

Main Measurement


Cone Resistance,qc

Friction Sleeve,fs

Pore Water Pressure,u2
18
Target

Site investigation
P.K. Robertson, 1990
Soil Behavior Type(SBT)

19
P.K. Robertson,1998
Fr 
fs
qv   v0
Q tn
 q t   v 0 '   Pa

 
 
Pa

 v 0




1.
2.
3.
4.
5.
6.
7.
8.
9.
Sensitive, fine grained
Organic soils (peats)
Clays (clay to silty clay)
Silt mixtures (clayey silt to silty clay)
Sand mixtures (silty sand to sandy silt)
Sands (clean sand to silty sand)
Gravelly sand to sand
Very stiff sand to clayey sand
Very stiff, fine grained
Ic imply SBT

20
P.K. Robertson, 1998
=
3.47 −
2
+ 1.22 +
2
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
Insight of Soil Layers
location at which the soil behavior type index
changes abruptly
0
2
4
SBT 6
SBT 3
6
But in reality……
There will be
some noise
definitely!
Depth z (m)

21
8
10
12
14
16
18
20
-2
0
2
Ic(z)
4
6
Transition Zone

22
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 !
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
0
2
B
C
-0.5
A
4
A
-1
8
logWI C(u,s)
Depth z (m)
6
B
10
12
-1.5
-2
-2.5
14
-3
16
C
-3.5
18
20
1
2
3
I c(z)
4
-4
-1.5
-1
-0.5
log(s)
0
0.5
24
In-situ Case Demonstration-NGES

Taxes A&M University
(National Geotechnical Experimentation Site,1993)
25
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
32
Reference
33

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
34