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!"#$%&'()%"

+,-%$,#%$. /(%'0)(0

1"&2$3,#2$ /(%'0)(0

4,52 67.08(09,#72:,)(0

!:,;8";

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/$$,.0F (%")"'%'0 $2(%$&8";0

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   G  r  o  u  n   d  v  e   l  o  c

   i   t  y

time

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J72 %$8;8" %C #72 "%802 8" #72 62$8%& -,"& KLMN0

+,"&O0 2# ,=IP QNMNR/B!/S+T

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0

-2

2

    1    0  -

    4

   v    i   t   e   s   s   e

-0,0

-0,5

00300205100105 250

0,5

1,0

temps (s)

onde P

onde S

CODANOISE

NOISE CODA

00300205100105 250temps (s)

   v    i   t   e

   s   s   e    x

/ #.68(,= $2(%$&0 %C , =%(,= 2,$#7W',X2 YNIMLMNZ[\

3/"7307 )3!2 %8"!!-.-5 )"9-%: !"$307 "59"0!"7- 4;

%8"!!-.-5 )"9-%:< !"#$% '( -* )3##3"/% "05 /* ;30$

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Propagation regimes and energy description

single diffraction (short times)

diffusion (long lapse times)

radiative transfer Equation

temps

Time

   E   N   E   R   G

   Y

   D   E   N   S   I   T   Y

time/mft

!"#$% '( .* )-"9-.1 =* 5- .4%0( 1 2* %"!4 "05 -* #".4%-

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Searching for a marker of the regime of scattering...

Equipartion principle for a completely randomized (diffuse) wave-field: in average, all the

modes of propagation are excited to equal energy.

Implication for elastic waves (Weaver, 1982, Ryzhik et al., 1996): P to S energy ratio stabilizes

at a value independant of the details of scattering!

RTE Monte CarloDiffusion equation

 Numerical simulationObservations

T0]T6 %$ T7]T5 (," -2 6$2&8(#2&

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low frequency

limit: half space

high frequency

limit: half space

Theory for S-P energy ratio

J72%$. ,"& &,#, C%$ 52$)(,= #% 7%$8[%"#,= 2"2$;. $,)%

This leads to an inversion method to extract the layering from asingle measurement (Margerin et al., 2009; Sanchez-Sesma et

al., 2011;!

.).

!"#$ '( ;* %"082->?%-%/"

&,#,

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!"#"$%"&()*"

" #

Source in A ⇒ the signal recorded in B characterizes the

propagation between A and B.

Green function between A and B: G AB

!"#"$%"&

" #

!"#"$%"&

G AB can be reconstructed by the correlation of noise or« diffuse » (equipartitioned) fields recorded at A and B (C 

AB)

 A way to provide new data with control on source location and origin time

How, when and why?

+%"; $,";2 (%$$2=,)%"0

!"#$% '( =* 7".03-.1 /* 5- 244,1 9* !%"3

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B26$202"#,)%" #72%$2: C%$ (%$$2=,)%"@ 6,00852 8:,;8";

/$-8#$,$. :2&8':@ ," 8"#2;$,= $26$202"#,)%" 3$8A2" 8" #72 C$2W'2"(. &%:,8"

R%=':2 #2$:

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Surface term: 

and we obtain a widely used integral relation: 

!a2$%&2 2# ,=IP QNNb@ /",=%;. 38#7 J8:2 $252$0,= :8$$%$0

!4,62",,$ QNNc

d%$ 0'$C,(2 3,520@ &80#,"# 0%'$(20 %C "%802 ,# #72 0'$C,(2 %C #72 0672$2 YeQa 6$%-=2:\

e

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(a)

(c)

(b)

(d)

Stationary phase and end fire lobes: actual data

From Gouédard et al., 2008

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Contributions to direct wavesin the GF

Contributions to scattered wavesIn the GF

End fire lobes!source noise kernels

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 An issue for surface wave tomography:

In practice, the noise sources are not evenly distributed and the field is

not made fully isotropic by scattering.

The absence of isotropy of the intensity of the field incident on the

receivers results in a bias on the measurements of direct path travel

times.

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/"80%#$%68( 8"#2"08#. %C #72 "%802@ #72 2>,:6=2 %C #72

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 B(! ) =1+  B2 cos(2! )

" t  =1

2t # 02

 B(0)

d 2 B(! )

d ! 

2

! =0

Bias in the travel time

Increasing anisotropy of the source intensity B

?%$$2=,)%" %C &8$2(# 3,520@

/[8:'#7,=

&80#$8-')%" %C

0%'$(2 8"#2"08#.

J$,52= ):2

2$$%$ 3$# #72

%-02$52& i$22"C'"()%"

5,=8& 38#7 # Y#$,52= ):2\ j J Y62$8%&\

5(67 %"&'"(+ 5(67"0)+ =&7>1**6 ?,--@A &02 5(67"0)+ =&7>1**6+ B6CD+

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5(67 5(67"0)+ =&7>1**6+ B6CD+

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5(67 5(67"0)+ =&7>1**6+ B6CD+

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5(67 G1$60"+ H"0IG160+ =&7>1**6 &02 B6CD+ ,-:.

ba 072,$ 52=%(8#.

La,:,;2& C,'=# [%"2

Ld=%32$L=8X2 6,A2$"0

La8h'02 0280:8(8#. ,00%(8,#2& 38#7 =%3L52=%(8#. Y&,:,;2&\ ,$2, -2#322" 3401 2* ("41 "05 %* 8"!2-#30-

4!2-. ",,#38"!340% 4; 7; .-840%!.+8!340< !"#$ '( 7* '-.4>"

8/18/2019 1 1 Campillo

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?%"08&2$ "%3 #72 6$%-=2: %C -%&. 3,520 ,# #72 ;=%-,= 0(,=2 38#7 "%8020%'$(20 ,# #72 0'$C,(2@

/ 6$%-=2: %C , &8h2$2"# ",#'$2P ,=#7%';7 8"&22& #72 '"252" &80#$8-')%"

0'$C,(2 "%802 0%'$(20 80 0)== #72$2I

J780 $26$202"#,)%" 80 "%# C%$:,==. 5,=8& %" #72 C$22 0'$C,(2@ #72 8"#2;$,= 5,"80720I

id $2(%"0#$'()%" 3%'=& $2W'8$2 , :%$2 (%:6=2> 6$%(2&'$2 YB'8;$%X 2# ,=IP QNNl\

Z2$2 ,=0%P #72 (%$$2=,)%" %C :'=)6=. 0(,A2$2& 3,520 07%'=& =2,& #% #72 i$22" C'"()%"I

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9.5

10

10.5

11

11.5

12

θ (°)

   t  m  e  m

  n   )

58 60 62 64 66 68−12

−11.5

−11

−10.5

−10

−9.5

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E%=8 2# ,=I

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5(67 J6*1+ KL67&9+ =&7>1**6 &02 J"2"(9"0 ,-:.

?%$2 67,020 E(E ,"& E&E

aHH@ L &8h2$2"# 7.6%#72020 C%$ #72

",#'$2 %C #72 =,.2$

L E&E &8^('=# #% %-02$52

L =,(X %C 2,$#7W',X2 &,#,

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Time to P [s]

   S   l  o  w  n  e  s  s   t  o   P

  w  a  v  e   [  s   /   d  e  g   ]

−10 0 10 20 30 40 50−3

−2.5

−2

−1.5

−1

−0.5

0

0.5

A)

P

PdP   PcP

Time to P wave [s]

   S   l  o  w  n

  e  s  s   t  o   P

  w  a  v  e   [  s   /   d  e  g   ]

 

−10 0 10 20 30 40 50−3

−2.5

−2

−1.5

−1

−0.5

0

0.5

PdP

P

PcP

B)

5(67 J6*1+ KL67&9+ =&7>1**6 &02 J"2"(9"0 ,-:.

/&5,"#,;2 %C "%802 50 2,$#7W',X2 $2(%$&0@

L0'$C,(2 #% 0'$C,(2

L8:6'=0852 3,52=2#L&%'-=2 -2,: C%$:8";

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GLOBAL TELESEISMIC CORRELATIONS (periods 25-100s vertical components)

Boué, Poli et al., GJI 2013

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V':2$%'0 67,020 (," -2 8&2")q2&

R2$)(,==. 8"(8&2"# <

3,520 %" #72 52$)(,=

(%:6%"2"#DD

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+%"; 62$8%&0 YQKLMNN0\

E$%(2008";@ 026,$,)"; Tn ,"& (%&, C$%: ,:-82"# "%802

+%3 &,8=. (%72$2"(2 Z8;7 &,8=. (%72$2"(2

YTn0\

/r!

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d$%: S%'t 2# ,=IP QNMc

+%"; 62$8%&0 YQKLMNN0\

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9%"8#%$8"; #2:6%$,= (7,";20 8" #72 0%=8& T,$#738#7 0280:8( 52=%(8)20

T$'6)%" ,# E8#%" &2 =, d%'$",802

J7$'0# %C #72 42"(7'," TnZ.&$,'=8( =%,&0

i2%#72$:,= F C$,(X8";II

J8&20

!"#$% '( -* ;4.7+-%1 $* %"'."1 ,* 7+-7+-01 ;* '.-07+3-.

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V%802 -,02& 0280:8( 52=%(8#. #2:6%$,= (7,";20

S2(,'02 0280:8( "%802 80 (%")"'%'0 8" ):2P 8# 80 6%008-=2 #% $2(%"0#$'(# CJSJ@NAU VKCDO@I

GJKGHKL GMOCLJG ,"& 62$C%$: LMANAOMOG HMAKDMCKAU MP GJKGHKL VJIMLKNJGI

uR

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?%$$2=,)%" C'"()%"0 ,0 ,66$%>8:,#2 i$22" C'"()%"0

a8$2(# 3,520 ,$2 02"08)52 #% "%802 0%'$(2 &80#$8-')%" Y2$$%$0 0:,== 2"%';7 C%$

#%:%;$,67. YvMo\ -'# #%% =,$;2 C%$ :%"8#%$8"; Y;%,= e MNLc\

8/18/2019 1 1 Campillo

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a2#2()"; , 0:,== (7,";2 %C 0280:8( 0622&@ (%&, 3,520

?%:6,$8"; , #$,(2 38#7 , $2C2$2"(2 '"&2$ #72 ,00':6)%" %C ," 7%:%;2"2%'0 (7,";2

J72 ‘&%'-=2#’ :2#7%&@ :%58"; 38"&%3 ($%00 062(#$,= ,",=.080 Y67,02 :2,0'$2:2"#0\

Alternative technique: stretching

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Precision of the measure of delay/velocity variations in the coda

= 7 7 C = & Y= = \

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92,0'$8"; 0=8;7# (7,";20 %C 0280:8( 52=%(8#. '08"; (%&, 3,520 Y=%"; #$,52= ):2\

V':2$8(,= 08:'=,)%"0 8" , 0(,A2$8"; :2&8':

=6*67M1+ =L&>C)+ 81**"(9 ") &*#+ ,-:. 10 >("99

Qa 062(#$,= 2=2:2"#0P ,"80%#$%68( 8"#2"08#. %C 0%'$(20

?%:6,$80%" %C

(%$$2=,)%"0 38#7i$22" C'"()%"

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Th2(# %C 0(,A2$8"; Y08";=2 0%'$(2\

=6*67M1+ =L&>C)+ 81**"(9 ") &*#+ ,-:. 10 >("99

S2,: C%$:8";

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92,0'$2 %C #72 -8,0 8"&'(2& -. , 0#$%"; ,"80%#$%6.

%C #72 3,52 q2=&

Y&2=,. 38#7 $2062(# #% #72 i$22" C'"()%"\

=6*67M1+ =L&>C)+ 81**"(9 ") &*#+ ,-:.

S='2@ &2=,.

B2&@ $2=,)52 &2=,.

d='(#',)%"0 %C ]# %C #72 %$&2$ %C MNLc 

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B26$202"#,)%" %C (%&, 3,520 ,0 #72 0': %C (%"#$8-')%"0 %C "':2$%'0 6,#70

d%$ , 08";=2 6,#7@

We have to compute the contributions of paths with first scatterers at all distances l  f  and

all azimuths! 

 

l  f  

t  f =l  f /V

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We have to consider that the distribution of distance between scattering events is exponential:

where l  is the mean free path< l f    > = l   t  f    = l f   / V 

ratio of max average "t  over

"t  of the average l  f   (=l the mean free path) 

valid for l  f >  # 

We remove the singularity for l  f   to 0 in a conservative way ( "t  ! t ) and define:

Finally we consider all directions:

We make use of

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/66=8(,)%"0

V':2$8(,= 08:'=,)%"0

! 

g,6," Y0,:2 (7,";2 %C ,"80%#$%6.\

! 

B = ) = 8# 7 Y 8 o\ & 8 #7 - & N M N w Z

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B2=,)52 52=%(8#. (7,";2 Y 8" o\ :2,0'$2& 8" #72 -,"& NIMLNIw Z[

?,=2"&,$ ):2 :2,0'$2& 8" &,.0 38#7 $2062(# #% 9,$(7 MM Y9w J%7%X' Tn\

5(67 H("0$C1"(+ =&7>1**6+ K&N"2&+ O6N1+ /L&>1(6+ H(1&02+ P76)6 

&02 Q1R&N"+ /S1"0S" ,-:.

+05-.%!"05307 !2- 82"07-%< !"#$% '( ,* =420%40 "05 "* %82+'0-#

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?%'=& 32 07%3 #7,# #72 =,#2 6,$# %C #72 (%$$2=,)%" C'"()%" (%"#,8"0

#72 0(,A2$2& 3,520D

L2"2$;. &2(,. Y2I;I

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S R

Coherent backscattering/weak localization

and the multiple scattering regime

Energy density is represented by the sum of contributions of scattering paths

RTE for , DE for ,! but the actual signal results from

deterministic waves

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.

S R

If this path exists..

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.

S R

If this path exists, the reciprocal path exists too.

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.

S R

Phase difference: location of the scatterers! 

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.

S, R

Phase difference: location of the scatterers! 

Except if R and S are at the same place

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.

S, R

Coherent summation!Spot of intensity enhancement at the source: factor 2

Consequence of first principles, namely reciprocity.

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Consider now correlations as virtual Green functions (Julien Chaput etal. 2015 –see Poster).

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Chaput et al., JGR 2015

Erebus volcano: icequakes

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44 ‘large’ events All 3318 events

Coda Correlations

kk (%$$2=,)%"0@ $2(86%$(8#. 7%=&0

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Energy vs distance at a given time

Weak localization can be observed in correlations!(!mean free path!)