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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+
8/18/2019 1 1 Campillo
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5(67 5(67"0)+ =&7>1**6+ B6CD+
8/18/2019 1 1 Campillo
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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
8/18/2019 1 1 Campillo
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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\
8/18/2019 1 1 Campillo
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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
8/18/2019 1 1 Campillo
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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$$%$0 0:,== 2"%';7 C%$
#%:%;$,67. YvMo\ -'# #%% =,$;2 C%$ :%"8#%$8"; Y;%,= e MNLc\
8/18/2019 1 1 Campillo
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a2#2()"; , 0:,== (7,";2 %C 0280:8( 0622&@ (%&, 3,520
?%:6,$8"; , #$,(2 38#7 , $2C2$2"(2 '"&2$ #72 ,00':6)%" %C ," 7%:%;2"2%'0 (7,";2
J72 ‘&%'-=2#’ :2#7%&@ :%58"; 38"&%3 ($%00 062(#$,= ,",=.080 Y67,02 :2,0'$2:2"#0\
Alternative technique: stretching
8/18/2019 1 1 Campillo
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Precision of the measure of delay/velocity variations in the coda
= 7 7 C = & Y= = \
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92,0'$8"; 0=8;7# (7,";20 %C 0280:8( 52=%(8#. '08"; (%&, 3,520 Y=%"; #$,52= ):2\
V':2$8(,= 08:'=,)%"0 8" , 0(,A2$8"; :2&8':
=6*67M1+ =L&>C)+ 81**"(9 ") &*#+ ,-:. 10 >("99
Qa 062(#$,= 2=2:2"#0P ,"80%#$%68( 8"#2"08#. %C 0%'$(20
?%:6,$80%" %C
(%$$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
8/18/2019 1 1 Campillo
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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!)
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