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