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µ PR+Q(-)$!)+#$!+!%( %!+'$!* +*!= % )'+# +'&%-! 6 PR+Q6 % '*!%)! ( % ++!% +'&%-! #!$+'%!% ;'+'%'%$-+'%!%!<:%( )%+'&%-!L PQ*-)$" P2Q L = µ 6 6 '#!$+7-( 7(!:µ N :!*-&'!=1π2N >J R+6 )&)%#$$"#*$+((#1%$,>,*.9:(&'+$9#%=.$ 9%(!(-)$$!!)+#$!+!%( %!+'$!9')!= %)'+#-!)$!& PKR+Q: 9' (*%&!$!+ 7!+!%(!*)'$&!*$!*:)'$')-$*-#'$'(!%*-(!) $'%(! )%( )% % PR+ Q:!!+ 7!+!%(!)'$&!*-!*:)'$')-$*-#'$'(!%*-(! ) $'%(!(-#')!+!%; (!#'$*'%< % PR+ Q6

30 ? "#$%&%'#(#)"*$+(*,#)%,)"*$,,-"(.-')%(*(% <<>< 5' *!99!( %)'+#-!)$!:!+ 7!+!%(!*)'$&!*$!*!*)'$')-$*-#'$ ')%( )7--!)$!σ PR+Q*!%'(+" PQ % = σ& 6 ')%( )7-#!?$!)%*(-$-!)++! %!&$'%(! $)+#!,!" PIQ Z = 26 σ = σy+ σx6!9$+'*+!$!9!!9'!!*)'$&!*$!*%!$-#%(!%#'*%-)!**'$!+!% %*'%'%-+!% ' )'+# -!)$!6 % #% (! 7! #-%+-%&!: ' #'$! +'&%'$!(!')%( )7-#! $!%($!)+#!(!'7*)*-(!*9 (!*;('%*!)'*(! ')%( )7--!)$.!<: (!*)*%*!%$!#'$)!*('%*'+'$)!6'%*)! )'*:!*) $'%*( *',)'$&!*$!*%!* 7!%# *%*'%'%-+!%!*7'$'%*( )'+#-!)$!6 <<>> 5' *!99!( %)'+#-!)$!:!+ 7!+!%(!)'$&!*-!*!*)'$')-$*-#'$ '#!$+7-(-!)$! ε P/R+Q: (-)$!# 7$(!#'$*'%( +! 6! 7!)! $%( )%-!)$! PR+ Q*-)$" P1Q = ε& 6 '%*!7(!:+'&$-'*!%)!(!)'$&!*-!*:!(-#')!+!%-!)$!%!*#'* % 6%(-9%'$*'#!$+7-(-!)$!( 7(!:ε N:!*-&'!=M:M12N >2 /R+6 ' (! )%*!$7'% (!* )'$&!* #!$+! (' $ = ' $!'%!,#$+'%!* ) $'%*(!(-#')!+!%%!%9%)%( )'+#-!)$!" & PHQ % = ε 6 %*'$(!')%( )7--!)$!: ε #!?$!)%*(-$-!)++! %!&$'%(! $ )+#!,!" PLQ ε = εy εx6!9$+'*+!$!9!!9'!!*)'$&!*-!*;)++!#'$!,!+#!!*+-)!*= ++!%(#'$!%%%!!*!!*+-)!*(!' <:%!$-#%(!%#'*%-)!**'$!+!% %*'%'%-+!%!% '($' $! (! #'*! ' )'+# -!)$!6 % #% (! 7! #-%+-%&!: %#! )+#$!%($! '#'$! +'&%'$!(! '#!$+7-)++! )'$')-$*'%!* #!$!* -%!$&-!*!%&!%($-!* #'$!* +-)'%*+!* (! #'$*'% ;)++!#'$!,!+#! ')%7!$*%#'$!!(! -%!$&!-!)$+'&%-!!%-%!$&!

31 :(&#%$5$,,-"(.'+$9#%)(-"'#=.$,$"+$.-')+)-=.$ +-)'%!# *!$+!$*(!'$'%(!*+-)!*(!' <6 <<>?/- $$ '*++!(!'(!%*-(!) $'%(!)%( )%!(!'(!%*-(!) $'%(! (-#')!+!%(-9%'(!%*-(!) $'%'!% PR+ Q" PJQ % = % + % 6 &!* %)'+#'$+%!: # *&-%-$'!+!%:*'$!'%PJQ!*!,#$+-! ('%*!(+'%!(!/ $!$:!*!,#$!**%** 7'%!*#! 7!%?$!-'!*" & PMQ % = σ& + ε : P3Q % = {( σ Y+ ωεx) + ( σx+ ωεy) }& : σx σy & P2NQ % = εy + εx+ : ω ω Z P*Q!*!!+#*:!ω P$'(R*Q!*'# *'%; 9$-!%)!'%& '$!<6!! $!#$-*!%'%!* %-$!**'%!)'$!!(-+%$!! '#'$! +'&%'$!(! ' #!$+7-A!!$U!( %!)%( )7-:!!'#'$!+'&%'$!(!')%( )7- A!!$U!( %!#!$+7-6!*-&'!+!%=%!$!('%*!*!,#$!**%*P3Q!P2NQ: %%! 9'!,#)!+!%# * '(*%)%!%$!) $'%*(!)%( )%!) $'%*(! (-#')!+!%6 ##'$'@'$*'%%(!#!$+7-!(!)%( )7-!99!)7!*; ε P/R+Q! σ PR+Q<" P22Q = σ & = ( σ Y+ σ X)& : % & & % = : P2Q ε = ( ε Y ε X) P2IQ '7!) P21Q P2HQ σ = ωε: σ Y= σy+ ωεx:σ X= σx+ ωεy: σx σy εy = εy+ :εx = εx+ 6 ω ω!*9$+ '%*P22Q!P2Q*%%(?$!'$9)!!*6%!99!:!,#-$+!%'!+!%: %%!*' '%9!$!(!*#'$'+$!*!99!)9*#'$!'*(!'+!* $!(!')+#*'%!!% #'*!!% '($' $! (! #'*! ( ) $'% ':! )!):!*! *!%!* #-%+%!*#.*!*+*!%A! ;(-#')!+!%*(!)'$&!*$!* -!*<6%(' $!*

32 "#$%&%'#(#)"*$+(*,#)%,)"*$,,-"(.-')%(*(%!$+!*:!*+#**!(!(99-$!%)!$!*)%$ %*(!σy!εx( %!#'$:!(!εy! σx(' $!#'$6 '$!'%!,*'%!%$! %!& #!?$!(-)$!(!+'%$!&-+-$!('%*! #'%(!/$!*%!('%*!!!7!)! $ & (-9%',!(!*$-!*# $*;9& $!262<6! 7!)! $ % 9$+!'%&! δ P$'(Q'7!)',!(!*+'&%'$!*# $*6!'%&!!*'##!- '%&!(!#!$!" P2LQ '% ( ) εx σy δ = = ε Y σ X!* %#'$'+$! *!$' $* '$&!+!% *-# $)'$')-$*!$!*#-%+%!* ('-% '%('%*'*!(!)!+-+$!6 imaginaires e ' j e " E réels /& $!262" -9%%(!'%&!(!#!$!δ ('%*!#'%(!/$!*%!6 =.(#)",*$(@$+ 9% (! )+#$!%($! -7 % (%! %(! -!)$+'&%-! ('%* % +! #.*!:!*-'%*#$#*-!*)>(!** *%!* 99*!%#'*69' %'++!%)'$')-$*!$!*!%* %**'%!*#'$'+$!*(-)$7'%!*#-%+%!*+'&%-!*;6 :L : µ :!)6<! )!,(-)$7'%!*#-%+%!*-!)$!*;& : :ε :σ :!)6<6!*-'%*(!',F! (-)$7!%)!*!%*" L P2JQ & = : P2MQ = ρ : P23Q 6 = % + :

33 :(&#%$5$,,-"(.'+$9#%)(-"'#=.$,$"+$.-')+)-=.$ PNQ L = N: Z ρ!*'(!%*-(!)'$&!*$!*#'$ %-(!7 +!PR+ I Q6 2=.(#)"*$*<<.,)"&%)&(-(#)"!*- '%*(! ',F!:)+#--!*#'$!* $!'%* P2Q: PQ! P1Q:#!$+!!% (-'$!*- '%*(!(99 *%>#$#'&'%" & & P2Q & = σµ + εµ : L L PQ L = σµ + εµ 6 '%*'*!(!)!!,#*-:% *%!(-7!##!$%*!!*$!'%*$!'7!*', )'+#*& # *!)!*%!**!%!!+!%(!*#-%+%!*-!)$!* *%+!* $-* '7!)!*(*#*9*$'('$6!# *:!)'+#+'&%-!*!(-( ( )'+#-!)$!#'$!'*(!*- '%*(!',F!6,#$+-!('%*!(+'%!(!/ $!$:!,#$!**%P2Q#! *!$--)$$!" PIQ & = γ &: P1Q γ = ωµ ( σ + ωε ) Zγ!*!9')! $(!+GP$'(R+ Q6 %)'$')-$*! %#-%+%!('%* %+'-$'!= %!9$-!%)!!*! ωε << σ:!**!%!(! ' $'%*+**%(! -%!$&!*#$!*!% % +(!(!(99 *%: #'$) $'%(!)%( )%6'%*- '%P2Q:!!$+! σµ & (+%!6 %7!$*!:*%)'$')-$*! %#-%+%!('%* %+'-$'!= %!9$-!%)!!*! ωε >> σ:!**!%!(!'$'%*+**%(!-%!$&!*#$!*!% %+(!(! #$#'&'%:#'$) $'%(!(-#')!+!%6'%*- '%P2Q:!!$+! εµ & (+%!6 % *'%!9$+'*+!(!*#'$'+$!*!99!)9*!'(-9%%(!'%&!(!#!$! P2LQ:'##'$'@!'#$#'&'%!*(+%'%!*:= %!9$-!%)!(%%-!: '% δ << 2:! =%7!$*!:'(99 *%!*(+%'%!* '% δ >> 26 '%*!*+!,&-&!*!',9$-!%)!* *-!*#'$!*&-$'('$*:' #$#'&'% (+%! &-%-$'!+!% * $ ' (99 *%6 * $!7!%($%*!% (-' ' #'$'&$'#!26* $!*$($!*(!&$'%(! $(!*#'$'+$!*-!)$+'&%-!*(!*+!, &-&!*',9$-!%)!*$'('$6!#!%('%:'*!( (-7!##!+!%*!$'($!*!(-A= $!%-!7!$* %9$+'*+!)'$')-$*'%(!*#-%+%!*Z'#$#'&'%!*(+%'%!: *'%*# $' '%%-&&!$!*#-%+%!*(!(99 *%6

34 2 "#$%&%'#(#)"*$+(*,#)%,)"*$,,-"(.-')%(*(% '* %(! - '% PIQ*-)$:('%*!(+'%!(!/ $!$!# $ %! %(! #'%!#'$*-!('%*'($!)%NP+Q*!#$#'&!'%('%*'($!)%7 P+Q" γ PHQ &( ω 7 ) = & ( ω: 7 = N) 7 = & ( ω: 7 N) O7 : N N = Z O P$'(R+Q!*! %+$! (%(!6!! &$'%(! $!* # * * 7!% *-!!! #'$'+$!(!+G6!!*&-%-$'!+!%!,#$+-!!%9%)%(!*'#'$!$-!!:%-! β P$'(R+Q!'##!-!9')! $(!#'*!:!(!*'#'$!+'&%'$!:%-! α P$'(R+Q! '##!-!9')! $('-% '% 1 " PLQ O = γ = β α 6 %!(!*'##)'%*!*# *%-$!**'%!*(!'9$+ '%!%#'$'+$!*!99!)9*!*!,#$!**%*+#!( %+$!(%(!" 6 PJQ O = ω µε 6 %#! '$*!,#$+!$!*9')! $*α! β (!'+'%$!* 7'%!" PMQ α ω 2 εx '% ( δ ) µε Y 2 2 = ω µε Y ε Y + : 2 '% ( δ ) = P3Q β ω µε Y 2 εx 2 + ε Y + 2 = ω µε Y = 2 2 '% ( δ ) 6!*#**!(!,$'$!!*9')! $*(!#'*!!('-% '%( %*&%':!*'$* #**! (! )')!$ ' #!$+7- (-!)$!!99!)7! ( +! )$$!*#%('%!% *'%!*$!'%*%7!$*!** 7'%!*" β α PINQ ε Y= : µω αβ PI2Q ε X= 6 µω!*$*) $'%(!,#$+!$!*#-%+%!*(!#$#'&'%!%!$+!(!7!**! (! 9')! $(! '-6 '7!**!P+R*Q!*!%!=#'$$( 9')! $(!#'*!" PIQ ω = 6 β 1!9')! $('-% '%#! -&'!+!%?$!!,#$+-!%P#R+Q )%7!$!%P(R+Q6!*)!! (!$%$! %- *!$'$!!%!# *!'$&!+!% *-!# $)'$')-$*!$!*'-% '%*6

35 :(&#%$5$,,-"(.'+$9#%)(-"'#=.$,$"+$.-')+)-=.$!!9')! $(! '-;P*'%*(+!%*%Q:$*'$&!+!% *-!%**+!:*!% ($!)!+!%=#'$$(!%7!$*!(!'%&!(!#!$!6!(!$%!$#'$'+$!:(-A=%$(!% P2LQ:(-9%!$'##$!%$!-%!$&!(**#-!!-%!$&!*)8-!#'$).)!6%'" PIIQ 2 Y ; = ε '% δ = ε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µ 6 '#!$+-'-+'&%-!( %! $)!!*(%))%$U-!#'$*'!%! $!%* *'%)!*#$-*!%'%)!!(!$%$!#$#$--6$: %!,*!!#! (!+'-$',9!$$+'&%-!*('%*!*+!,&-&!*!! $!%! $!*!%&-%-$'$*9'!6%(!$*(!!!*- (!*$!%('%)+#!(!+!* $!* (!#!$+-'-$-'*-!** $(!*$)!*=$*9$!*!%! $*!%9!$,.(!(!9!$;!9! '#$% 2331<: ' +'A$- (!* +'-$', &-&!* #$-*!%! %! #!$+-'- +'&%-!#$'!+!%-&'!=)!!( 7(!" PI1Q µ = µ N6

36 4 "#$%&%'#(#)"*$+(*,#)%,)"*$,,-"(.-')%(*(% '%*! )'($! ( $'7' #$-*!%- ): % * #*!$%* (%) )!!.#*! *+#9)'$)!('%*!%*!+!(!*9$+!*6 9' )!#!%('%$!+'$!$!('%*(!%+$! *!*'##)'%*(!'+-(!$'('$:!*A!*!%)!$)!=)'*!$%!*%#'*(!*+'-$',&-&!*+'**%(!* --+!%*(!%' $!+-'!;9[*:)'%'*'%*:)_!*:!)6<6'%*)!)'*!!%! $&! $:.#*! µ = µ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permittivité diélectrique polarisation interfaciale interfaciale polarisation dipolaire dipolaire atomique électronique fréquences géoradar polarisation atomique polarisation électronique fréquence [Hz] /& $!26" 7 % (! ' #!$+7- (-!)$! * $!%*!+! ( *#!)$! -!)$+'&%-!6$#$--('((7-(!** *)!#-*-!)$!*;('#$* -&!%! ')' *8'*233<6

38 7 "#$%&%'#(#)"*$+(*,#)%,)"*$,,-"(.-')%(*(% % *'%'%%(!* *)!#--!)$!:%'" PIHQ ε ( χ + χ + χ + χ ) ε = N $ ' ' +666 : Z χ$!* ' * *)!#- -!)$! -! ', #-%+%!* (! #'$*'% %!$9')'!P*'%*(+!%*%Q: χ :'* *)!#--!)$!-!',#-%+%!*(! #'$*'%(#'$!:!)6!*9$-!%)!*(!$!','%(!*#'$*'%*-!)$%!!%!;(!$($!(! 2N 2H! 2N 2I G $!*#!)7!+!%< *% $* * #-$! $!* ', 9$-!%)!* $'('$6!* #-%+%!*)%)!$%!%!%*!+!(!**(!*6,9$-!%)!*$'('$!!%'*!%)!(! #'$)!* (#'$!*: )! *% )!* #-%+%!* (+%!%! )!* ' )+#*% +%-$'&!(!'$)! (-!$+%!'7'! $(!'#!$+7-;ε O1:L# $! '$G] ε OJ:H# $')')!< H 6!* +-)'%*+!* (! #'$*'% (#'$! #$-*!%!% (!* 9$-!%)!* (! $!','% #$)!*(!*9$-!%)!*$'('$"!!**%( %!&$'%(!+#$'%)!('%*'+!* $!Z' $)!)%!%(!* +-)!*(#'$!*6 '#$#$%(!' #$-*!%!('%* %! $)!!* (-!$+%'%!6 %!99!!' #$-*!%! %! $* 9$!#!$+7-;ε OM2<:(! = '*#!) (**.+-$! (! *' +-)!6 ' 9$-!%)! (! $!','% '**)-!!* (! $($! (! NG6!* #'$*'%* (! )'$&!* (!*#')! *% )'$')-$*-!* #'$ (!* 9$-!%)!* (! $!','%'$&!+!%# *'**!*;(!$($!(!2=2NN8G<!%!#'$)#!%#'*=' #!$+7-('%*''%(!(!*9$-!%)!*$'('$6 ' #!$+7- (-!)$! )%*(-$-! * $!%*!+! ( *#!)$! (!* %(!* -!)$+'&%-!*7'$!(%))%*(-$'!+!%!%9%)%(!'9$-!%)!6 (&$%###'*'+$9#%=.$$<<$9#$(.<%'=.$"9$,%(*(%!#'$'&$'#!#$-)-(!%'#$-*!%-!*+-)'%*+!*-*',#'$'+$!*)%* 9*(! -!)$+'&%-*+!('%*!*+!,&-&!*6 *'7%*# )%*'!$!:( %! #'$:'##$,+'% µ = µ N#!?$!#*-!!!(' $!#'$:)%( )7--!)$!! #!$+7-(-!)$!;' *!%*(!*- '%*(!',F!<#! 7!%*-)$$!* *'9$+! ( %#'$'+$! %!"'#!$+7-(-!)$!!99!)7!6 *'%*+'%!%'%%* %-$!**!$',$($!*(!&$'%(! $(!)!#'$'+$!('%*!(+'%!# *$!*$!%(!* 9$-!%)!*$'('$6 H '%*)!+-+$!:# $# *(!)'$-:!*!*7'! $*% +-$!*(!#!$+7-(-!)$!*!$%!,#$+-!*(!+'%$!$!'7!#'$$'##$='#!$+7-( 7(!69%(!%!#'*' $($!*%'%*:% * %%$( *%* #'* (! 7'$'! * ##-+!%'$!6 **: -)$ $! $& $! *! B ε ε N O1:LC *!$' *.*-+'!+!%$!+#')-!#'$B ε O1:LC6

39 :(&#%$5$,,-"(.'+$9#%)(-"'#=.$,$"+$.-')+)-=.$ 8 <>></' ++!% * '7%*(-A= +!%%%-!% 26266I: %!*#'*#**!(! +!* $!$(! +'%$! (*%)! ' )%( )7- -!)$! (! ' #!$+7- (-!)$! ;' *!%* #-%+-%&!(!)!*(!,!$+!*<6!!'+!* $!(!#'$'+$!*!99!)9*;'%'.*! (!'$-#%*!!%#'*!!!% '($' $!(!#'*!( %+! = %!!,)'%-!)$!<!*!%7*'&!'!6 *'7%*(%)$'**!+- %'$&!!%*!+!(!+!* $!*(!#!$+7-(-!)$!!99!)7!; ε <:)%)!$%'% %!&$'%(!7'$--(!+!,&-&!*6 %!%*!+!(!+!* $!*$-'*-!*('%*!)'($!(!)!+-+$!%--'A -!*(!* 7'! $***!*(!$'7',#$-)-(!%*; '%)!' 23M3:'$(2332: $%!$233I<6!* (99-$!%*(*#*9*%*$ +!%', *-*# $$-'*!$)!*'%'.*!** 7!%!+?+! #$%)#!(!+!* $!" %'%'.*! $(!$-*!' +!* $!'#'*!!'+# (!(!*%(!* $-9-)!*! $'%*+*!* * $ % -)'%% (! $)!: #')- = %-$! $ ( %! )!! )','!6 ' )!! *-! ('%*! )'($! (!* $'7', #$-*!%-* ('%* )! +-+$! ;9& $!26I< *%*#$! (! )!! (-7!##-! = %* /$'%`'* ( -$! ; '%)!' 23M3<6%)') %7!$*!#!$+!(!%$!*7'! $*(!*#'$!*$-!!*!+'&%'$!*(! ε!(!'#!$+-'-+'&%-!6! *!*$-* '*!% ** $'#!$+7-*!$% )++ %-* )6!* +!* $!* ( #'$'+$! µ *%!% &-%-$' $* $ -!*! %! #!$+!!%#'*(!(*%&!$'#!$+-'-+'&%-!(!*-)'%%*(!)!!( 7(!6 % )+#'$'*% (!* +!* $!* (! 7!**!* ') *!* $-'*-!* * $ -)'%%: ' $!#$-*!%'7-(!*+!* $!*(-!)$!*!*!,)!!%!6 conducteur externe échantillon de roche conducteur interne /& $!26I"!! )','! *-! # $ $-'*!$!* +!* $!* (! #!$+7- (-!)$!!99!)7!6 -)'%%=+!* $!$!*9'`%%-!%).%($!)$!,:(%!('+$!!,-$! $!* (!%7$%1N++6!!)!!*%*#$!(!)!!(-7!##-!#'$%* /$'%`'*( -$! ; '%)!' 23M3<6

40 ? "#$%&%'#(#)"*$+(*,#)%,)"*$,,-"(.-')%(*(% ' 9& $!261#$-*!%!!!*) $!*(!#!$+ 7-*(-!)$!*!99!)7!*!% 9%)%(!'9$-!%)!6!'!' 262$!&$ #!!*7'! $*!%!** $!*#'$!*$-!!!+'&%'$!(! ε :* $'#'$!$-!!(!')%( )7--!)$!!99!)7! σ ;$!'% P2IQ<:* $'%&!(!#!$!!* $*%%7!$*!:!9')! $(! '-;$!'%*P2LQ!PIIQ<! * $'7!**!(!#$#'&'%;$!'%PIQ<6!*7'! $**%9 $%!*# $$*9$-!%)!* (99-$!%!*6!*$!+'$!* #! 7!%?$!9$+ -!*=#'$$(!)!*(%%-!**%%+$! *!*" \! #'$'+$! ε 7'$!!% 9%)% (! ' 9$-!%)!: )!! 7'$'% %!* #'* %-&&!'!6'%*'#$!*!'-(!*)'*: εy! ε X*%(-)$**'%!*6!!' #'$! +'&%'$!(! '#!$+ 7-(-!)$!!99!)7!(!!' (-+%-$'*-! ' &+!%!=)' *!( #-%+%!(!$!','%(#'$!(!'+-)!(!' *! +'%9!*!=!%7$%NG6!!99!$!*!)!#!%('%+'$&%'6!*7'$'%*(! ' #!$+ 7- (-!)$!!99!)7! % (! $* 9$!* +#)'%* * $! )+#$!+!% ( %! %(! -!)$+'&%-! ('%*! +! &-&!6 * $!7!%($%** $)!*!99!*!%26I6 \ %9'*'%'*$')%(!*+!* $!*B!,$?+!*C:$-'*-!** $(!!' :%)%*'! =LN G:!*7'! $*( #'$'+$! ε Y7'$!%!%$!H!HN6!*)%$'*!*(! #!$+ 7- #! 7!% (%)?$! +#$'%* ('%*!* +!, &-&!*6 ' 7'$'-(!)!#'$'+$!) 7$! %$($!(!&$'%(! $6 \!*7'$'%*( #'$'+$! σy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

41 :(&#%$5$,,-"(.'+$9#%)(-"'#=.$,$"+$.-')+)-=.$ 7 a. Calcaires saturés 8 b. Granitoïdes ε e eau douce,1 g/l NaCl,5 g/l NaCl 2, g/l NaCl ε e tonalite saine (Vienne) granite mésocrate (Grimsel) granite leucocrate (Grimsel) granite rose ε e 8 6 ε e fréquence [MHz] 1 1 fréquence [MHz] 9 c. eaux 45 d. roches argilisées 85 eau déminéralisée eau de mer marne (Gard) marne (Mont Terri) tonalite argilisée (Vienne) ε e 8 ε e ε e 6 ε e fréquence [MHz] fréquence [MHz] /& $!261"!* $!(!'#!$+ 7-(-!)$!!99!)7!* $(7!$*-)'% %*(!$)!*6!* +!* $!*(;)')'$!**' $-*'7!)(99-$!%!*)+#*%*(!' < *$!%!99!(!'*'%- (!!' *' $'%!*-)'% %*6!*+!* $!*/#$-*!%!%!*7'! $*!%!** $(99-$!%*.#!*(!&$'%!*:+!,9'7$'!*= *'%( &-$'('$6!*+!* $!*9#$-*!%!%!* )'$')-$*!*(!#!$+ 7-(!!' (!+!$!(!!' (-+%-$'*-!6!*+!* $!** #$-*!%!%!*$-* '*!% ** $(!*$)!*'$&*-!*:#! 9'7$'!*=!+#( &-$'('$6!,)!#% (! ' #'$! +'&%'$! (! ' #!$+ 7- (-!)$!!99!)7! (!!' (-+%-$'*-!:!*)!*) $!**%(-)$**'%!*!%9$-!%)!6

42 "#$%&%'#(#)"*$+(*,#)%,)"*$,,-"(.-')%(*(%

43 :(&#%$5$,,-"(.'+$9#%)(-"'#=.$,$"+$.-')+)-=.$!%+$!,+(!*!+#$!*!,*!%# $(-)$$!'(-#!%('%)!!%9$-!%)!(! '#!$+7-(-!)$!!99!)7!69%(!#$#*!$ %9$+'*+!+&%!!%$!)!* (99-$!%*+(!*:% *)++!%`%*#'$!,#$+!$ε* *'9$+!" PILQ ( ω ) ε χ ( ω ) σ + ε ω / ε = N \ ε P/R+Q(-)$')%$ %(!*+-)'%*+!*(!#'$*'%)'$')-$*-*#'$(!* 9$-!%)!*(!$!','%'$&!+!%* #-$! $!*='&'++!(!9$-!%)!*='!! %*%-$!**!:! $-#%(!%(%)%*'%'%-+!%!% '($' $!(!#'*!' )'+#-!)$!6ε!*(%) %#'$'+$!$-!!)%*'%6 \ σ/ PR+Q(-)$ ')%$ %(! ')%( )7--!)$!(-9%! = 9$-!%)! %!;a($!)) $$!%<6!#'$'+$!!*-&'!+!%$-!!)%*'%6 \ χ P*'%* (+!%*%Q!* ' * *)!#- -!)$!!99!)7!: )'$')-$*'%!* +-)'%*+!*(!#'$*'%%%)+#$*('%*!!$+!ε 6!*(99-$!%*+(!* %--'$(-**!(*%&!%* $'+'%$!(!)'$')-$*!$ )!(!$%!$#'$'+$!:!*)+#!,!!7'$'!!%9$-!%)!" PIJQ χ ( ω ) χ Y( ω ) χ X( ω ) = 6! +(!(!!.!;231H<!,#$+! '* *)!#--!)$!!% 9%)%(! ε ;(-9%# *' <:( %!7'! $+!(!#!$+7-='**!*9$-!%)!*ε P/R+Q:!( %!+#*(!$!','%τp*q:;!+#*%-)!**'$!# $-'$ %!#'$*'%(%%-!<" PIMQ χ ( ω )!*+(!*(!!>!;2312<" PI3Q χ ( ω )!(!!>'7(*%;23H2<" P1NQ χ ( ω ) 2 ε ε = 6 ε 2 + ωτ N 2 ε ε = : 2 ε + N 2 2 = ε N ( ωτ ) α ε ε ( 2 + ωτ ) β *%(!*7'$'%!*( +(! (!!.!:*%$( *!%(!*9')! $*('+$**!+!% ;$!*#!)7!+!%α! β P*'%*(+!%*%Q<6 *#$#*%* *'%( %!' $!'##$)!:'!+!%(99-$!%!:* ##$+'%' %%(!9$-!%)!(!$!','%6!!(!$%$!*%*#$!( +(!(!5%*)!$;23JJ<: '##!- B$-#%*!(-!)$! %7!$*!!C6 * $!9$+ %*)! +(!(! ' +'%$! * 7'%!;!%(!$!'$(233M<" ω 2 P12Q χ ( ω ) = χ 2 ) ω : π 6

44 2 "#$%&%'#(#)"*$+(*,#)%,)"*$,,-"(.-')%(*(%!!!,#$!**% *! %!9$-!%)!(!$-9-$!%)!ω #!?$!(-9%!(!+'%$! '* +!%'$$'$!;%!*'&#'*( %#'$'+$!('A *!+!%( +(!)++!!* 9$-!%)!* (! $!','% (!* +(!* #$-)-(!%*<6'%*)!+-+$!:'9$-!%)!(! $-9-$!%)! ω *!$' A $*9,-!=2NNG6!#'$'+$! χ!*'#'$!$-!!(!' * *)!#-='9$-!%)!(!$-9-$!%)!]P*'%*(+!%*%Q: %9')! $)'$')-$*'% %!%*-(! ' 7'$'%!% 9$-!%)!; 7'$!!%$! N! 2<6 '%* ' 9$+ '%(! 5%*)!$:! $'##$!%$!#'$! $-!!! +'&%'$!(! '* *)!#-!*)%*'%!' 7'! $(!)!$'##$!*9%)%( #'$'+$!6 9%(!)'$')-$*!$ '#!$+7-(-!)$!!99!)7!:!%+$!(!#'$'+$!* )%*'%*!$-!*%-)!**'$!*=')'$')-$*'%( %+! (%%-!*!* 7'%" \ '$!# $!+(!(!!.!; σ / :ε : ε :τ <: \ )%# $!+(!(!!>!; σ / :ε : ε :τ:α <: \ )%# $!+(!(!!>'7(*%; σ / \ '$!# $!+(!(!5%*)!$; σ / : ε : ε :τ:β<: : ε : χ :<6 '#$*!* +(!* (!!.!:!>!!!>'7(*%: ' * *)!#- -!)$!!99!)7! +'&%'$!)$@'7!) ' 9$-!%)! A * = ' 7'! $ ω = π τ :# * (-)$@' (!=6 '* *)!#-$-!!!**$)!+!%(-)$**'%!+'*)!!(-)$**'%)!!*+',+'!='9$-!%)! ω = π τ 6!*+(!**%* $ *-*# $)'$')-$*!$!*+-)'%*+!*(!#'$*'%(!.#!(#'$!=#$,+-(!*9$-!%)!*(!$!','%6 '#$*!+(!(!5%*)!$:#'$!*$-!!!+'&%'$!*%*$)!+!%(-)$**'%!*:!!!*'9$-!%)!6 '9& $!26H#$-*!%!' $!(!7'$'%(!'* *)!#- -!)$!!99!)7!!(!'#!$+7-(-!)$!!99!)7!# $)!* '$!+(!*6!$'%!* *+#9)'%* #! 7!%?$! '##$-!*6!* #'$!,!+#! #**! (! )%*(-$!$! ' )%$ % #.*! (! ' )%( )7- = 9$-!%)! %!!* +#)!+!%%) *!('%*'+(-*'%(!'#'$!+'&%'$!(!'* *)!#- -!)$!6!*'$*#**!(!%!#'*#$!%($!!%)+#!')%$ %;+'-+'!< (!$+!σ/ ('%*!)') (!ε6 '$!'%PILQ!*'$*$!+#')-!#'$" ω = ε χ ω ε P1Q ( ) ( ) + ε N 6 '%*)!)'*:!*+(!*(!!>!!!>'7(*%%!%-)!**!%'$*# *!+#!(! '$!#'$'+$!*:!*+(!*(!5%*)!$!!.!:(!$*#'$'+$!*6!$'%* $'7',;$%!$!'6 233I: '% 233L< $!%(!%)+#!(!!+#(! +(!*%*#$-*#'$'(!5%*)!$('%*!*!*')%$ %'!9$-!%)!(!' #!$+7-;ε <%!*#'*#$*!!%)+#!6'%*)!)'*:*! *(!,#'$'+$!**% *-*!'$!'%PILQ!*$!+#')-!#'$"

45 :(&#%$5$,,-"(.'+$9#%)(-"'#=.$,$"+$.-')+)-=.$! 15 susceptibilité électrique 22 permittivité diélectrique 1 Jonscher Debye Cole Cole Cole Davisdon contribution de ε χ e ε e contribution de σ DC /ω 8 14 χ e ε e fréquence [MHz] fréquence [MHz] /& $!26H" $!*(! 7'$'%!% 9$-!%)!(! '* *)!#--!)$!!99!)7!!(! ' #!$+ 7-(-!)$!!99!)7!# $(99-$!%* +(!*;!.!:!>!:!>'7(*%!5%*)!$<6!+(!(!!.!:!>!!!>'7(*%#$-*!%!%(!*#'$!*$-!!* (!'* *)!#--!)$!(-)$**'%!*!%9$-!%)!:!(!*#'$!*+'&%'$!*'99)'% %! 7'$'% B!%))!C6! +(!(! 5%*)!$#$-*!%!(!*) $!*(!* *)!#- *$)!+!%(-)$**'%!*6 $!%$'#'$!$-!!(!'#!$+ 7-(-!)$!!99!)7!: ')%$ %;)%*'%!<( #'$'+$! ε!*'a -!='#'$!$-!!(!'* *)!#- -!)$!]# $!%$'#'$!+'&%'$!:')%$ %;%7!$*!+!%#$#$%%!!=' 9$-!%)!< ( #'$'+$! σ/ Rω!* 'A -! = ' #'$! +'&%'$! (! ' **)!#- -!)$!6!* %-*#$-!** $!*',!**%)'$$'$!*6 P1IQ ε ( ω ) ε χ ( ω ) = 6 N!%*!+!(!*+(!*#$-*!%-*)>(!** *;'%('%*! $9$+!)+#!!('%*! $* 9$+!* *+#9-!*< ' --!*- * $!* 7'! $*!,#-$+!%'!* #$-*!%-!* ' #'$'&$'#! '9& $!26L#$-*!%!!!*$-* '*.#!*6!*#'$'+$!*(!* +(!*%--!% *#'$+%+*'%(!-)'$ '($'!+.!%!%$!!*) $!* +(-*-!*!!*7'! $*$-!!*6 * $!A!%*!+#(!* +(!*(!!.!;9$+!*)+#!!*+#9-!<'%*!! +(!(! 5%*)!$('%**' 9$+!*+#9-! =(!,#'$'+$!*;$!'%* P12Q! P1IQ<6

46 4 "#$%&%'#(#)"*$+(*,#)%,)"*$,,-"(.-')%(*(% 12 a. tonalite altérée 22 b. calcaire saturé ε e ε 19.5 e ε e ε e fréquence [MHz] 1 1 fréquence [MHz] Jonscher 2 paramètres Jonscher 3 paramètres Jonscher 4 paramètres Debye 3 paramètres Debye 4 paramètres Cole Cole 4 paramètres Cole Cole 5 paramètres Cole Davidson 4 paramètres Cole Davidson 5 paramètres données réelles /& $!26L" +#'$'*%!%$!!* (99-$!%* +(!* (! 7'$'%!% 9$-!%)! (! ' #!$+7- (-!)$!!99!)7! # $ (!,!%*!+!* (! +!* $!* $-!!* $-'*-!*!% '$'$!6!*+(!*(!!.!'%*!!+(!(!5%*)!$('%**'9$+!*+#9-! =(!,#'$'+$!*%!#$( *!%#'*(!%*$-* '*6'$)%$!:!*' $!*+(!*;!>!:!>'7(*%:5%*)!$=$*! '$!#'$'+$!*<'A *!%!*(%%-!*$-!!*(! +'%$!*'*9'*'%!!- 7'!%!;) $!*)%9%(!*<6!*' $!*+(!*#$( *!%(!*$-* '**'*9'*'%*6 *$!!%%*9%'!+!%! +(! %-)!**!!+%*(!#'$'+$!*:=*'7$!+(!(!5%*)!$('%**'9$+! *+#9-!=$*#'$'+$!*;$!'%*P12Q!P1Q<6 '#!$+7-(-!)$!!99!)7! *!9$+!'$*" ω ω 2 π P11Q ε ( ω ) = εnχ 2 ) + ε!!$!'%*!$''$&!+!% *-!('%*'*!(!*(-7!##!+!%*6 6

47 :(&#%$5$,,-"(.'+$9#%)(-"'#=.$,$"+$.-')+)-=.$ ' $'--%-$!**'%( *!$(!*+(!*#!$+!'%('))-(!$=(!*7'! $*!% $'##$'7!)(!*#-%+%!*#.*!*(!%9-*;9$-!%)!*(!$!','%:)%( )7-= '**!9$-!%)!:!)6<:!)!:+'&$- %%+$!* #-$! $(!#'$'+$!*6%9':('%*' '%(! (!* 9$-!%)!* $'('$: )!* +(!* %! #$-*!%!% % ' $ *!**! % %)- %-)!**'$!*=)!.#!(!(-!$+%'%6'$!,!+#!:('%*!)'*( +(!!>!: % *'7%*# 7-$9!$ %!%*!+!(!#'$'+$!*; σ / : ε : ε : τ:α<:)$$!*#%('% =(!*7'! $*'!$$'%!*( %#%(!7!#-%+-%&!:#! )%( $!:* $''%(! (!* 9$-!%)!* $'('$: = (!* ) $!* '!+!% *'*9'*'%!* 7* = 7* (!* (%%-!*!,#-$+!%'!*6!+?+!:(!,!%*!+!*(!#'$'+$!*; σ / : ε : ε : τ:α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granite (Grismel) tonalite saine (Vienne) tonalite altérée (Vienne) vitesse [m/ns] facteur de qualité Q fréquence [MHz] 1 1 fréquence [MHz] /& $!26J" K'$'%!%9$-!%)!(!'7!**!(!#$#'&'%!( 9')! $(! '-; # $(99-$!%* +!,"&$'%!*'%;*!(! $+*!<: %'!*'%!! %'!'-$-!;*! (!'K!%%!<6!*) $!*#$#*-!*)%--)') -!*'7!)!+(!(!5%*)!$ =$*#'$'+$!*:!% *=#'$$(!(%%-!*(!#!$+7-(-!)$!!99!)7!+!* $-!*!%'$'$!6 '7!**!!'9')! $(! '-*%)$**'%*'7!)'9$-!%)!6

48 7 "#$%&%'#(#)"*$+(*,#)%,)"*$,,-"(.-')%(*(% ' 7'$'% (! ' 7!**!!* ' )' *! (! ' (*#!$*% (!* %(!* $'('$6!! )'$')-$*!9%('+!%'!'%&!+#*--%-&&-!:#'$'%'&!'7!)'**+!:('%*!*$'7',$''%(!**&%',$'('$6 '(*#!$*%(!*%(!*$'('$)++!%)!)!#!%('%=?$!#$*!!%)+#!! =?$! $!)%%!#'$ ')++ %' -(!* $'('$*!*6 ' +'%$! ('$(!$'(*#!$*%;!# *&-%-$'!+!%(!+(-*!$'7'$'%!%9$-!%)!(!' #!$+7-(-!)$!!99!)7!<' ('$()%**-= *!$(!*+(!*(*=; )%*'%; $%!$! &&%* 2331: '% 233L: $! $'%(A!'% 233M<6?+!*)!* $'7',$!#$-*!%!% %!#$!+$!-'#!('%*')+#$-!%*%(!*)'$')-$*!*(!* *&%', $'('$: * $!*!%!%)$! $* #$)!* (!* 9$+ '%* *-!*!% **+!: %'++!%# $+(-*!$'(*#!$*%(!*%(!***+!*(!7 +!('%*(!*+!, 7*)-'*!* ;ba'$'%*%23j3:!$923m1<6!* +!* $!* (! #!$+7- (% *% **!*!*) $!*(!7'$'%( #'$'+$!;#$ 7!%!)!*+(!*%!#! 7!%?$! *'*9'*'%*6'! $*:'7'$'%( #'$'+$!;!*!' ) ## *+#$'%!!' 7'$'%(!'7!**!* $ %!#'&!9$-!%!!(%%-!6 $!,!+#!#$#*-9& $!26J ;9$-!%)! 7'$'% (! 2N = 2NNG<! ('%*! )'* ( &$'%! (! $+*! ;7$ '%%!,!26I<:'7!**!)$@(!N:2I=N:2H+R%*:* %!' &+!%'%(!%7$%c '$*!!9')! $(! '-)$@(!23=HM:* %!' &+!%'%(!# *(!NNcd '%*!)'*(!'%'-'-$-!( *!(!'K!%%!:'7!**!' &+!%!(!%7$%2Jc:!9')! $(! '-:(!JNc6!*$'7',(!+(-*'%(!**&%',$'('$# *$-)!%* *!%(!*9$+ '%* (!'(-#!%('%)!!%9$-!%)!( #'$'+$! ε- 7'!%!*='#$*!!%)+#!( % 9')! $;7'$'!6F!$*!!9;2331<:'!$'%!'6;233N< *!% %+(!(!!>!;%-)!**'%!+#(!)%#'$'+$!*<6'$)%!;233L'< *! %+(!(!!.!&-%-$'*-:) #-= %+(!(!b!7%>k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

49 :(&#%$5$,,-"(.'+$9#%)(-"'#=.$,$"+$.-')+)-=.$ 8!(!,+!#$9!*)'$')-$*!( % +! (!#$#'&'%(*#!$*9: =; )%*'%6! +(!(! 5%*)!$ =(!,#'$'+$!*'-- *-# $&-%-$!$)!#$9 ;'##$,+'% σ / = ε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ω ) ] \ )') ( %+$!(%(!O ;$!'%PJQ<!% *'%'(-9%%(!'#!$+7- (-!)$!!99!)7! ε :#$#$!=)'!+(!;7$9& $!263<" #$92"εY OL:2;)%*'%<:ε XON:2;)%*'%<] #$9"+(!(!5%*)!$=(!,#'$'+$!*; ON:3M: χ OL:NM<] #$9I"+(!(!5%*)!$=$*#'$'+$!*; ON:H: χ ON:2:ε OL<] ω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