Tracking the origin
of the seismic noise
Tracking the origin
of the seismic noise
Tracking the origin of the seismic noise
Apparent origin of the noise
Average sea wave height
winter
summer
Importance of scattering:
a simple laboratory experiment with a few sources
C(A,X)
Correlations in water
G(A,X)
travel time
A
Xs
sources
receivers
Correlations in presence of scattering
H(2)n(kr)=Jn(kr) − iYn(kr)
THE 2D SCALAR CASE
1
G22 (x, y, t ) 
2

G
22
(x, y,  ) exp( it )d 

2 t 2  r 2 /  2
x3
Illumination by plane waves
n
H (t  r /  )
1
x
θ
y
x1
ψ
Azimuthal average over ψ leads to
1
v(y,  )v (x,  )  F ( )
2
*
2
2
 exp( ikr cos[   ])d  F ( ) J 0 (kr)
2
0
v(y,  )v * (x,  )  ESH J 0 (kr)  4ESH ImG22 (x, y,  )
SPAC method  k  C
A isotropic distribution of plane waves in an homogeneous body : the local approach
THE 2D SCALAR CASE
SH waves in a homogeneous elastic medium
Propagation takes place in the x1-x3 plane. Therefore, the antiplane (out-of-plane) displacement v(x,t) fulfils the wave equation
 2v  2v
1  2v


x12 x32  2 t 2
where β = shear wave velocity and t = time.
A typical harmonic, homogeneous plane wave can be written as
v(x,  , t )  F ( , ) exp(  i

x n ) exp(i  t )
 j j
where, F(ω, ψ) = complex waveform, ω=circular frequency, xT = (x1, x3) = Cartesian coordinates
Keiiti Aki
x1 = r cosθ = , x3 = r sinθ = , with r, θ = polar coordinates
nj = direction cosines (n1= cos ψ , n3 = sin ψ )
Consider the auto-correlation of the motion, evaluated at positions x and y, respectively. For simplicity assume y at the origin:
x3
n j x j  rn j j  r cos[   ]
n
v(y,  )v* (x,  )  F ( , ) F * ( , ) exp( ikr cos[   ])
ψ
y
x
θ
x1
Cross-correlations of seismic noise: ANMO - CCM
(from Shapiro and Campillo, GRL, 2004)
30 days of vertical motion
Dispersion analysis
global model by
Ritzwoller et al. 2002
PHL - MLAC 290 km
correlations computed over four
different three-week periods
bandpassed
15 - 30 s
bandpassed
5 - 10 s
repetitive measurements provide
uncertainty estimations
Causality
1
( 2)   r 

G
H 0 
4i
  
r/
H  t  r 

1

G
2 t 2  r 2
2

r/
r/
t
G/2
(Re)
(Im, Re)
(Im)
r/
t