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1
Basic Theoryfor CCA Method
May 11, 2012
IISEE, BRI, Japan
By T.Yokoi
2
Microtremor (Ambient Noise)
Freq.
Amplitude
1 Hz 10 Hz0.1 Hz
(10.sec) (1sec) (0.1sec) (Period)
Urban Area
Rural Area
IslandContinent
microseisms
Caused by oceanic waves Caused by human activities
Surface sources
3
Preparation: Phase Velocity & Group Velocity
f(x+Dx,t)
f(x,t)
Propagation of Energy: Group Velocity v.
Propagation of Information: Phase Velocity c.
Dependency of Phase Velocity on the Frequency f (or Wave Number k): Dispersion.
c is different from v in Dispersive Media.
dk
dvcv +=
ck
=
4
Preparation: Phase Velocity Determination
Definition of Cross-correlation ( ) ( ) ( ) dtyxtCxy +
( )y
( )x
( )+ty
t
Product & Integration Cross-correlation
5
Preparation: Phase Velocity Determination by Cross-Correlation
Cosine Function with the angular frequency (one cycle only)
( ) ( ) cos,0 0ff =
( ) ( )krfrf −= cos, 0
t
( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( )krtf
krtdkrtdf
dkrtftrCc
−
−−−=
−+=
cos
sinsincoscoscos
coscos,,0
2
0
22
0
2
0
Time lag t gives coincidence. If distance is r, the phase velocity c is given by r/t.
The maximum value of Cc corresponds to this time lag.
6
In the frequency domain:
( ) ( ) ( ) ( ) ( ) ( )( ) D== irFFrFFrCc exp,,0,,0,,0
+=
+
c
rti
c
rti
expexp
c
r =D
( ) ( ) ( )
=
c
rirFFrCc
exp,,0,,0
( )( )
( ) ( )
=
=
=
c
r
c
ri
rFF
rCcrCoh
cosexpRe
,,0
,,0Re,,0
Phase lag due to wave propagation is
because
Therefore,
Coherence
Here, c is the phase velocity measured along the measurement line.
( ) ( ) ( ) ( ) ( ) ( ) 22,,,,,,0,0,0,0 rFrrCcrAcFCcAc ====
Auto-Correlation
7
Zero and 1st order Fourier transforms of the wave field along
the circle over the azimuth.
Zero Order 1st Order
( )( )
( ) ( )
( )( )crJ
crJ
dirtZPSD
drtZPSD
rsCCA
2
1
2
0
exp,,
,,
, =
−
=
−
−
CCA (Cho et al., 2006, Tada et al., 2007)
PSD<> denotes the power spectral density
What’s CCA ?
8Schematic graphs for comparison of SPAC with CCA.
J0(kr): SPAC coefficient
0.1
1
0.01 0.1 1 10kr
Observed SPAC coefficient
Estimated kr
[J0(kr)/J1(kr)]^2: CCA coefficient
0.1
1
10
100
1000
10000
100000
0.01 0.1 1 10kr
Observed CCA coefficient
Estimated kr
( )( )
( ) ( )
( )( )crJ
crJ
dirtZPSD
drtZPSD
rsCCA
2
1
2
0
exp,,
,,
, =
−
=
−
−
CCA (Cho et al., 2006, Tada et al., 2007)
SPAC (Aki, 1957;1965; Okada et al. 1987; Okada, 2003)
( )( )
( ) ( ) ( )krJd
CECE
rCEr
mmcc
mc
0
2
0 ,,
, ,Re
2
1, ==
9
Analysis in the frequency domain
PSD: Power Spectral Density, C mm’(r,) : Cross correlation
( ) ( )
( ) ( ) ( )
= ===
−
=
=
M
m
M
m
mm
M
m
m
M
m
m
ZZ
rCEM
rZM
rZM
PSD
drtZPSDrrG
1 1'
',2
2
1'
*
'
1
00
,4
,,2
,,2
,,,,
( ) ( ) ( )
( ) ( ) ( ) ( )
( ) ( )
= =
==
−
−−=
−
−
−=
M
m
M
m
mmmm
M
m
mm
M
m
mm
ZZ
irCEM
irZM
irZM
PSD
dirtZPSDrrG
1 1'
'',2
2
*
1'
''
1
11
exp,4
exp,,2
exp,,2
exp,,,,
10
Analysis in the frequency domain
CCA coefficient
( )( )
( ) ( )
( )( )crJ
crJ
irCE
rCE
rsM
m
M
m
mmmm
M
m
M
m
mm
CCA
2
1
2
0
1 1'
'',
1 1'
',
'
exp,
,
,
−−
= =
= =
11
Analysis in the frequency domain
CSD: Density of Cross Spectra
( ) 2
,,10
−= rrGArg ZZR
Incoming azimuth
( ) ( ) ( ) ( )
( ) ( ) ( )
( ) ( )
= =
==
−−
−=
−
−=
M
m
M
m
mmm
M
m
mm
M
m
m
ZZ
irCEM
irZM
rZM
E
dirtZdrtZCSDrrG
1 1'
',2
2
*
11'
'
10
exp,4
exp,,2
,,2
exp,,,,,,,
12
SPAC coefficient
( ) ( ) ( )
( ) ( ) ( ) ( )
( ) ( )
=−
−−
−−
=
=
=
=
M
m
m
ZZ
rCEM
drCE
drZZEdrZZE
drtZdtZCSDrG
1
,0
2
,0
**
00
,4
,2
,,0,0,2,,0,0,2
,,,0,0,,,0
( ) ( ) ( )
( ) ( ) ( )
,40,0,0,0,4
0,0,,0,0,,0,0
0,0
2*2
00
rCEZZE
dtZdtZCSDG ZZ
==
= −−
( )( ) ( )
( )( )
( )
( ) ( )crJ
CE
rCE
M
G
rG
rCE
drCEr
M
m
m
ZZ
ZZ
/,0
,1
,0,0
,,0
,
,
2
1,
0
0,0
1
,0
00
00
0,0
,0
=
==
=
−
13
Correction for incoherent noise (Cho et al. 2006)
( )( )
( ) ( )
( ) ( )
( )( )( )
( )
( ) ( )( )
( ) ( ) ( ) ( ) ( ) ( ).24
,11
,1
2
,
,,0,0,,
,,0
2
2
2
2
2
2
22
0000
2
002
ACABB
cohC
NcohB
A
GrrG
rGcoh
ZZZZ
ZZ
−−−=
−=
−−=
−=
=
( )( ) ( )( ) ( ) NcrJ
NcrJrsCCA
+
+=
2
1
2
0,
14
System Correction for CCA in the frequency domain.
Yokoi (2012)
Record of i-th channnel
( ) ( ) ( ) ( )fHfGfdfd iii
obs
i =
System Characteristics Very Local Effect
( ) ( ) iii jfGfG −= exp ( ) ( ) iii jfHfH −= exp
Ground motion of propagating waves
15
( ) ( ) ( )
( ) ( ) ( ) ( ) ( )**
*
fHfHfGfGfC
fdfdfC
kikiik
obs
k
obs
i
obs
ik
=
=
Cross spectra between i-th and k-th channnels’ records
Cross spectra between i-th and
k-th channnels’ ground motion
CCA coefficient is defined using cross spectra of ground motion
( )( )
( ) ( )
= =
= =
−−
=M
i
M
k
kiik
M
i
M
k
ik
CCA
jCE
CE
rs
1 1
1 1
exp
,
16
Phase difference among the system characteristics
( ) ( ) ( )( ) ( )
( ) ( ) ( )
( ) ( ) ( )
( ) ( ) ( )
( ) ( ) ( ) ( )
( )
( ) ( )
huddle
kk
huddle
ii
huddle
ik
k
huddle
i
huddle
ki
huddle
ki
huddle
ki
huddle
ki
kiki
CC
C
GdGd
GGd
GGd
GGd
GG
GGj
=
=
=
=−−
2222
*2
2
*2*
exp
Correction factor for phase difference among the system characteristics
( ) ( ) ( ) ( )
( )
huddle
ik
huddle
kk
huddle
ii
ki
huddle
ikC
CCjCor
=−= exp
17
Corrected cross spectra
( )( ) ( )
( )
=
fC
fCfCArgjEfCor
huddle
ik
huddle
kk
huddle
iihuddle
ik exp
For a stable processing, average over time blocks is applied
( ) ( ) ( )
( ) ( ) ( ) ( ) ( )*
kikiik
huddle
ik
obs
ik
cor
ik
HHGGC
CorCC
=
=
18
Complex coherence between i-th and k-th channels’ corrected records
An interim quantity Rik()
( )( )
( ) ( ) ( )
( ) ( ) ( ) ki
kkii
ik
cor
kk
cor
ii
cor
ikcor
ik jCECE
CE
CECE
CECoh
−−== exp
( ) ( ) ( ) ( )( )
( ) ( )
( )( ) ( )
( ) ( ) ( ) ( )
( ) ( ) ( )
( ) ( )
obs
kk
obs
ii
huddle
ik
obs
ik
obs
huddle
kk
obs
kk
huddle
ii
obs
ii
huddle
ik
obs
ikobs
cor
kk
cor
ii
cor
ikcorcor
ik
cor
ik
CECE
CorCEC
CorCECorCE
CorCEC
CECE
CECCohCR
00
00
0000
=
=
==
On the other hand,
( ) ( )( )
( ) ( ) ( ) ki
kkii
ikobs
ik jCECE
CECR
−−= exp00
:Description using observed records
19
Assumptions:
1) Phase difference due to very local effect is negligible
2) Power spectra of ground motion is same for all channels
0− ki
Contents of ( ) is common for all channel pairs, then
( ) ( ) ( ) kkii CECEP ==
( )( )
( )( )
ik
obs
ik CEP
CR
00
Complete definition of CCA coefficient is obtained by using Rik() in place
of Cikobs()
( )( )
( ) ( )
( )
( ) ( )
= =
= =
= =
= =
−−
−−
=M
i
M
k
kiik
M
i
M
k
ik
M
i
M
k
kiik
M
i
M
k
ik
CCA
jR
R
jCE
CE
s
1 1
1 1
1 1
1 1
expexp
20
SPAC coefficient (Aki 1957, Okada 2003)
( )( ) ( ) ( )
( ) ( )
obs
kk
obs
ii
huddle
ik
obs
ik
obs
ik
CECE
CorCECR 00=
Neglect phase difference of system characteristic
Okada et al. (1987)’s formula is derived by two assumptions in the
previous slide
( ) 1exp −= ki
huddle
ik jECor
( )( )
( ) ( )crJ
CE
rCE
Mr
M
m
m
/,0
,1
, 0
0,0
1
,0
=
=
( )( )
( )
( )
( )
( ) ( ) ( )
=
== =M
mobs
mm
obs
obs
m
M
m
m
M
m
m
CECE
CE
MR
R
MCE
rCE
Mr
1,0,0
,0
0,0
1
,0
0,0
1
,011
,0
,1
,
Use in place of( )mR ,0 ( ) mCE ,0
Or perform system correction
( ) ( ) ( ) obs
ik
huddle
ik
obs
ik CECorCE
21
CCA Analysis
1.E-05
1.E-04
1.E-03
1.E-02
1.E-01
1.E+00
1.E+01
1.E+02
1.E+03
1.E+04
0.1 1 10 100
Frequency (Hz)
Pow
er
Spe
ctr
a
GZ0Z0
GZ1Z1
GZ0Z0/GZ1Z1
Total Power
A real example of CCA Analysis
(BRI, Tsukuba, Japan)
( )( )( )
( )( )crJ
crJ
rrG
rrGrs
ZZ
ZZCCA
2
1
2
0
11
00
,,
,,, ==
22 22
CCAExample using 2Hz
seismometers
0.001
0.010
0.100
1.000
0.1 1 10 100
Frequency (Hz)
Po
wer
of
G(Z
0)
or
G(Z
1)/
All
0.1
1.0
10.0
100.0
1000.0
0.1 1 10 100
Frequency (Hz)
CC
A c
oef
fici
ent
1.E-11
1.E-10
1.E-09
1.E-08
1.E-07
1.E-06
1.E-05
0.1 1 10 100
Frequency (Hz)
Po
wer
Sp
ectr
a
CCA CoefficientPower Spectra
Power Partition Ratio
23
System Correction
Effect of System Correction
0
50
100
150
200
250
300
350
400
0 2 4 6 8 10
Frequency (Hz)
Ph
ase
Vel
oci
ty (
m/s
)
Full CorrectionPartly CorrectedNo Correction
Phase velocity
24
( )( )( )crJ
crJrs
2
1
2
0, =
Long wavelength apploximation
(Small value of kr) (Cho et al,2006)
( ) ( ) 2,2
+=
rsr
c
Improved approximation:
Yokoi(2012)
c1Comparison of Long Wavelength Approx.GS-1, Gain x16, Huddle Test Cor., Coh.
0
100
200
300
400
0.0 5.0 10.0 15.0 20.0
Frequency (Hz)
Phas
e V
elo
city
(m/s)
Improved Approx.
Conventional Approx.
c2
s(r,omg)
1.00E-01
1.00E+00
1.00E+01
1.00E+02
1.00E+03
1.00E+04
0 5 10 15 20
Frequency (Hz)
CC
A C
oef
fici
ent
GZ0Z0/GZ1Z1
s(c1(f),f)
s(c2(f),f)
0.0138-0.0015245)2.0003
0.97221)s(r,1.0003()/rc( +
+=
25
Dispersion Curve
Heuristic Search: VFSA-DHSM
Vs Structure