A Multilayer Saint-Venant Model

HAL Id: inria-00070749

https://hal.inria.fr/inria-00070749

Submitted on 19 May 2006

HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub-

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non,

A Multilayer Saint-Venant Model

Emmanuel Audusse

To cite this version:

Emmanuel Audusse. A Multilayer Saint-Venant Model. [Research Report] RR-5249, INRIA. 2004,

pp.35. �inria-00070749�

ISRN INRIA/RR--5249--FR+ENG

a p p o r t

d e r e c h e r c h e

Thème NUM

INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE

A Multilayer Saint-Venant Model

Emmanuel Audusse

N° 5249

Juillet 2004

!#"$"&%('*) +-,. ) /0)01213+54

678:9<;=?>A@CBEDGFIHKJL8:9<;3HMGN9<O3PRQTS-N;3H

UVPRWYXK;J?Z[?=]\

^_a`b`(WcPdJeI;#PL;3f7;3PLf7b;gMih0jYkalcmnBEocNQqprpq;J?kascsaltBvucj`b_YwY;3H

xzy {3|:}~€Y|YA‚ ;QqM-JRPLWGeNbf;_]M;:ƒ&„_aPLQq_YMcJ0Wa…(JR7;9NprJRQqpq_†Fc;:P‡DI_aQqM-JdˆŠ‰‹;:Mb_YM-J0HRFIHKJL;:9Œ 67b;f:pq_cHRHRQTf:_ap

D_aQqMcJRˆŽ‰‹;3Mb_aM-JHRFIHKJL;:9QqH _ƒ‘;:pqp’ˆŠ“GMWƒM”_a``bPRW†•IQq9<_aJRQqWYM&Wa…#JL7;–QqMbfWc9<`PR;2HRHRQq—pr;˜=A_†„GQr;3PdˆKD-JRWc“Y;2H

;2S-Nb_JLQrWcMbH‘…™WYPHR7b_apqpqWƒšƒ_JR;3P›bWƒHƒQrJR7 …™PR;3;#9<W„-QqMw<—iWYNMbeb_aPLFYŒ‹œŠJLH;žfQq;:Mbf:FYŸPLWY—bNbHKJLM;3HLH_aM(e

pqWƒfWY9<`NJL_JLQrWcMb_ap fWcHdJ9_a“c;#Q’J]„Y;:PLFf:WY9<9<WYMpqFzN(Hd;2eŒ=;:„c;:PRJR7;3pr;2HRHQrJLH?PL_YMwY;#WY…‡_a``pqQTf:_JLQrWcM

QTH#pqQr9<QrJR;2e5_YMbe5Q’JteW-;2H#MWYJt_apqprWƒ¡JRW˜_Yf3f;2HRHAJRWJL7;„Y;3PdJLQqf3_ap‹`PLWa¢(pr;<Wa…‘JR7b;7WYPLQr£3WYM-JL_Yp‹„Y;3prWIfQrJKFYŒ

¤?;:Mbf:;t_aM(e JR7b_YM“IHJRW_`PL;3fQTHR;_aMb_YprFIHRQqHWY…0JR7;tHR7b_YprpqWƒ¥ƒ_JL;:P?_cHRHRN9<`IJLQrWcM¦ƒ§;`bPRWc`(W-Hd;#7;3PR;_

Mb;:ƒ¥_Y``PLW†•IQr9_JLQrWcMzWY…*JR7b;=?_†„GQr;3PdˆKD-JLWY“Y;2HV;2S-Nb_JLQrWcMbH‘ƒ7Qqf7fWcMbHRQqHdJLHQqM¦_¨HR;JWY… fWcN`pq;3eDI_aQqM-Jdˆ

‰‹;3Mb_aM-JHRFGHdJR;39H:ŸG;•GJL;:MbeH‘JL7;#P_aMwc;?Wa… „_apqQqeIQrJKFz_YMbezwcQr„c;3H_`PR;2fQTHd;©eI;2HRf:PRQq`IJRQqWYM Wa… JR7;©„c;:PRJRQTf:_ap

`bPRWY¢bpr;§Wa…(JR7;‘7WYPLQq£:WYM-J_ap-„Y;:pqWIfQrJKF©ƒ7Qrpq;`PL;3HR;:PL„-QqMwJL7;fWc9¨`bNIJL_aJRQqWYMb_Ypc;:f:Qr;3MbfF#WY…bJL7;‘f:pq_cHRHRQTf:_ap

D_aQqMcJRˆŽ‰‹;3Mb_aM-JHRFGHdJR;39Œ ‚ ;#„_apqQqeb_JR;©JL7;#9<WIeI;:pªJL7PRWcNwY7HRWY9<;©M-Nb9¨;3PRQTf:_Yp;•_a9<`pq;3H3Œ

«¦¬-­ª®L¯t° }± {Y³² PL;:;©HRNPd…´_cf;]=?_†„GQr;3PdˆKD-JLWY“Y;2H‹;3S-Nb_aJRQqWYMbH3ŸGHd7b_YprpqWƒ”ƒ_JR;3P‘_cHRHRN9<`IJLQrWcMµŸc7GFIeIPLWcHdJL_aJRQTf

_Y``PLW†•IQr9_JLQrWcMµŸIDI_aQqM-JdˆŠ‰‹;:Mb_YMcJHRFIHKJL;:9Ÿ9tNbp’JLQrpT_†FY;3P‘9<WIeI;3p¶Ÿ„GQTHRf:WcHRQ’JKF

·V¸(¹´º»¶¼¾½ ¿ªÀ0Á Â

{  =?WYNbH#QqM-JRPLWIeINQTHdWcMbH©NM;MWcN„Y;3prpq;<„_aPLQq_YMcJL;eNbM HdFIHdJR8:9<;eI;zDI_YQrM-JRˆŽ‰‹;3Mb_aM-J9tNbp’JLQ’ˆ

f:WYNbf7b;YŒµ;HRFGHdJR839<;eI;¦DI_aQqM-JdˆŠ‰‹;:Mb_YM-J<f:pq_cHRHRQqS-N;;3HdJ3ŸVe; `(_aP<HdWcM$;žf3_YfQrJRO;JžHL_˜PLWY—NbHdJR;2HRHR;YŸ

HRWYNb„Y;:M-J]NIJRQqprQTHRO`iWYNP©9<WIeIO:pqQTHd;3P]pq;3H]O2fWYNbpr;39¨;3M-JLHAQrMbf:WY9<`PL;3HLHdQq—pq;3HHRNPR…´_Yf:;pqQr—bPR;;:M5;3_aN•`i;:N

`bPRWY…™WYMbeI;2H:Œ‡=?O3_YM9<WYQqMbHHRWYM¦eIWY9_YQrM;#e_a`b`prQTf:_aJRQqWYMPL;3HdJR;#pqQr9<QrJRO;:JQrpµMb;teIWYMbM;_Yf3fO3HS-N<NM;

„GQrJR;2HRHR;9¨WFc;:MMb; eI;¦pO3fWcNpq;:9<;:M-J3Œ \]PYf:;5NMb;_YMb_apqFIHd;¦eIO:JL_YQrpqO:;¦eI;p7GFG`(WYJR782Hd;¦;2_aNI• `i;:N

`bPRWY…™WYMbeI;2H:ŸIMbWYNbH`PLWY`iWcHRWYM(H§NbM;MWcN„Y;3prpq;#_a``PLW†•IQr9_aJRQqWYMe;3HO3S-Nb_aJRQqWYMbHe;=?_†„GQr;3PdˆKD-JLWY“Y;2H:ŸIS-NQ

f:WYMbHRQTHKJL;;:MNMb;AHRO:PLQr;eI;]HdFIHdJR839¨;2HVeI;]DI_aQqM-JdˆŠ‰‹;:Mb_YMcJ‘fWYNb`prO2H:Œ§;JRJR;]_a``PLWIf7;9tNbp’JLQqf:WYNbf7;?`(;3Pdˆ

9<;:JAeOJR;3MbeIPL;#pr;eIWY9_aQqM;eI;t„†_YprQTeIQrJROteN¦9¨WIeI83pr;#;:J…™WYNbPRMQrJ?NMb;teI;3HLfPLQq`IJRQqWYM`PLO3f:QqHR;eIN¦`PLWa¢bp

„c;:PRJRQTf:_Yp0e;¨„GQrJR;2HRHR;<7WYPLQr£3WYM-JL_Ypr;cŸªJRWcNIJ#;:M `PLO3HR;:PL„†_YM-JApT_ HRQq9¨`bprQTfQrJRO;:Jp;žf:_cfQrJROeI;<p_Y``PLWGf7b;

D_aQqMcJRˆŽ‰‹;3Mb_aM-Jtf:pq_cHRHRQqS-N;cŒ >?M;ž„_apqQTe_JLQrWcM eN 9<WGe8:pq;z_aN5JRP_†„Y;3PLH#e;žpOJLNbeI;zeNMb;žPRNb`IJRNPL;zeI;

—(_aPLPL_YwY;?;2HKJ`bPRWc`(W-HdO3;YŒ

¡° |2{ ® € {A"! S-Nb_aJRQqWYMbH eI;V=?_†„GQq;:PRˆ¾DGJRWY“c;3HHRNPR…´_Yf;‹pqQq—PR;cŸ†;3_aN•A`i;:N#`PLWa…™WcMbeI;3H3Ÿ†_a``PLW†•IQr9_aJRQqWYM

7GFIeIPLWcHdJL_aJRQTScNb;YŸIHdFIHdJR839¨;#eI;tD_aQqMcJRˆŽ‰‹;3Mb_aM-J2ŸI9¨WIeI83pr;©9Np’JLQqf:WYNbf7b;YŸ„GQTHRf:WcHRQ’JLO

$ %

*0 V 

œ¾MJR7bQqH0`b_Y`(;3P0ƒ§;_aPL;VQqM-JR;3PR;2HKJL;3eQrM¨9<WGe;:pqQrMwAJR7b;HRWaˆ¾f:_Yprpq;3eHd7(_apqprWƒƒ_JL;:P ›bWƒH3Œ*œŠJ‡fW„c;:PH _A„Y;3PRF

pT_aPLwY;‘P_aMwc;§WY…ª_Y``pqQqf3_JRQqWYM(H0QrM¨JR7b;eIWY9_aQqM¨Wa…iJR7;wc;:WY`b7-FIHRQqf3_ap-›bWƒH ˆµPLQr„c;:PH:ŸpT_a“c;3H3ŸYfW-HKJ_ap_YPR;2_YH3Ÿ

WIf:;3_aM(H:Ÿ_aJR9<WcHR`7;3PR;cŸ_†„†_Ypq_YMbf7;2H:ŒqŒrŒ0œ¾MWYPeI;3P§JLW`PLW„-QTeI;#_¨M;:ƒ 9<WIeI;:pqQr£2_JLQrWcMHdJR;3`—(;:JKƒ§;3;:M JR7;

f:WY9<`pq;•IQrJKFtWa…ªJL7;…™Npqp(=A_†„GQr;3PdˆKD-JRWc“Y;2H0;3S-Nb_JLQrWcMbH0…™WcPVQqMbfWc9¨`bPR;2HRHRQr—bpr;›bWƒH‡ƒQ’JL7<…™PR;3;?HdNbPd…´_cf;_aM(e

JL7;]prW-HRH‘Wa… wY;:Mb;:P_apqQ’JKF<Wa…*JR7;#fpT_YHLHRQqf3_apiD_aQqMcJRˆŽ‰‹;3Mb_aM-JHdFIHdJR;39 ŸGƒ‘;AQqM-JRPLWIeINbf;]QqMzJL7QqH‘`b_Y`(;3P_tM;3ƒ

9NprJRQqpq_†Fc;:P§DI_YQrM-JRˆŽ‰‹;3Mb_aM-JVJKFG`i;?9<WIeI;:piƒ7QTf7žfWcMbHdQTHdJLH‡QqMz_Hd;:J§Wa…*f:WYN`bpr;2ežDI_aQqM-JdˆŠ‰‹;:Mb_YMcJ‘HdFIHdJR;:9H

&(')

&*,+

&(')-)

&. /1032 465

Œ

5"7

&(' ) - )

&* + &

&.

8

')-:9)

+<;

')=?>

@BADC ' @

E F

/

;HG

= >

@IADC

'

@3J

9

E &

&.

' )

=K>

@IADC ' @ +

EL - )M

CON - )

')M

C + ')

N

EL

- ) N - )3P

C

')

+

')3P

CRQ 465

Œk 7

ƒ7b;:PL;TS

' ) 2 ' ) - )U

S* 2 . U

QTH¨JR7; „Y;2f JLWYP¨Wa…JL7;¦fWYM(Hd;3PR„_JLQr„c;ž„_aPLQq_Y—pq;3Htˆ©ƒ_JL;:P¨7;:QqwY7-J_aMbe eIQTHKˆ

f7(_aPLwY;YŸYJR7GNbH -V) S* 2 . U QTHVJL7;]„Y;:pqWIfQrJKF¨ˆ‹_YMbežƒ7b;:PL;JR7b;©HRN—bHLfPLQq`IJRW5QTH‘PR;3pq_aJR;3ežJRWJR7;©fWcMbHRQqeI;3PR;2e

pT_†FY;3Pˆ‘JR7GNbHXWZYK[¨_aPL;PL;:pT_JR;2e JRWzJR7;¨M;:QqwY7G—iWYNPLQqMwžpT_†FY;:PH3Ÿ(_a—iW„Y;_YMbe¦—i;:pqWƒ#Œ:\ QTH?JR7;tJRWaJ_ap

MGN9—(;3PWa…0pq_†Fc;:PH:Œ

L

e;:MWYJR;3H‘JL7;#„GQqHLfW-HdQrJKFžfWG;:f:Qr;3M-J3Œ

67b;#9<_YQrMQqM-JR;:PL;3HdJWa…‹HRNbf7¦_t…™WcPR9NpT_JRQqWYM QTHJRWž_cf:f;2HRH‘JLW<JR7;t„Y;:PRJRQTf:_Yp`PLWa¢bpq;©Wa… JR7;7bWYPLQr£3WYM-JL_Yp

„c;:pqWIfQrJKFzƒ7Qqpr;t`PR;2Hd;3PR„GQqMw<JR7b;tfWc9¨`bNIJL_aJRQqWYMb_Yp*;žf:Qr;3MbfF Wa…0JR7b;DI_aQqM-JdˆŠ‰‹;:Mb_YM-J?9<WGe;:pŽŒ ² NPRJR7;3Pdˆ

9<WcPR;#JL7;t9<WIeI;3p „c;:PLQ’¢b;2HHRWY9<;HdJL_Y—QrpqQrJKF`PLWY`i;:PRJRQq;3Hˆ§;:•GQTHdJR;:M(f;tWY…‡_aM˜;3M;:PLwYFcŸb`iWcHRQ’JLQr„GQrJKF WY…‹JR7;

ƒ_JL;:P7b;:QqwY7-JLHˆ©_YMbeHdWc9<;ž`7GFIHdQTf:_YpV`PLWY`i;:PRJRQq;3Hˆ]_awcPR;3;:9<;:M-JƒQrJR7JL7;f:pq_cHRHRQTf:_apDI_aQqM-JdˆŠ‰‹;:Mb_YM-J

HRWYpqNIJLQrWcM…™WcP§f:WY9<`b_JLQr—bpr;QqMQrJRQT_apifWYM(eIQ’JLQrWcMbH3Ÿc`PL;3HR;:PL„_JLQrWcMWY…HdJL_aJRQqWYMb_YPRF¨HKJ_JL;3H3Œ^]AM;?Wa…ªJR7;AwYW-_apTH

WY…*JL7;#`PL;3HR;:M-J`b_a`i;:PQTH_apTHRWtJLW;•GJR;3MbežJL7;3HR;#`PLWY`i;:PRJRQq;3H‘JLW¨MGN9<;3PRQTf:_YpµHdQq9tNpT_JLQrWcMµŒ

67b;fpT_YHLHRQqf3_ap‘D_aQqMcJRˆŽ‰‹;3Mb_aM-J<HdFIHdJR;:9`_kacbQTH¨_–ƒ§;3prp‘“GMWƒM _a`b`PRW†•IQq9_JRQqWYMWY…JR7;…™PL;:;HRNPR…´_Yf:;

=A_†„GQr;3PdˆKD-JRWc“Y;2H ;2S-Nb_JLQrWcMbH …™WYPVHR7b_YprpqWƒ$ƒ_JL;:P0›bWƒH3ŸYƒ7QTf7<7b_YH‡—(;3;:MƒQTeI;:pqFt„_apqQTe_JL;3e…™WYP‡„_YPRQqWYNbH

wc;:Wc`7GFGHRQTf:_apY›bWƒH3ŸaHdNbf7¨_YH PLQr„c;:PH*WcP f:Wc_YHdJL_YpI_aPL;3_YHV_kYscbŠŸaWYP0;:„c;:MWGf:;3_YM_aM(et_JL9<WcHR`7;:PL;§eF-M(_a9<Qqf

WcP_†„†_Ypq_YMbf7;2HV›(WƒH:_j"b ƒ7;:M¦fWc9<`pr;:JR;2ezƒQrJR7¦_a`b`PRWc`PLQq_aJR;AJR;:PL9H:Œ

67b;e;:PLQr„_JLQrWcM WY…?JL7;–DI_aQqM-JdˆŠ‰‹;:Mb_YM-JHRFIHKJL;:9 …™PLWY9 JR7;˜=A_†„-Qq;:PRˆKD-JRWc“Y;3H;3S-Nb_aJRQqWYMbH…™WcPžHd7b_YprpqWƒ

QqMbf:WY9<`PL;3HLHdQq—pq;›(WƒH<ƒQ’JL7 _5…™PR;3;¦9¨W„GQqMw —(WcNMbe_YPRF QqHžMWƒf:pq_cHRHRQqf3_apƒ7;3M JR7;¦„GQTHRf:WcHRQ’JKF$QqH

Mb;:wYpq;3fJR;2e?_uYscbŠŒ Z§NJ¨JL7QTHžeIWG;3H¨MWYJ_YprpqWƒ JLWXKN(HKJLQ’…™F$JR7b_aJ¨JL7;¦PLQrwc7cJ]XKN9<`bH<ˆ©JL7b_Jz_Y``(;2_aP<QqM

eb_a9³—PL;3_a“IHAWYPA7-FIeIP_aNbprQTfXKN9<`bHAˆ_aPL;#JL7WcHR;Wc—IJL_YQrMb;3e¦ƒ7;3M–NbHRQqMwžJL7;9<Wc9¨;3M-JRN9³ˆ_aMbe–MWaJ

JL7;„Y;3prWIf:Q’JKF–ˆA_YH©JL7;žfWcMbHR;:PL„†_aJRQq„Y;<„_aPLQq_Y—pq;YŒžœ¾Md_

5

j"bŠŸ0\];3PR—i;3_YN_aMbe5U0;:PRJR7b_Y9<;`PRWc`(W-Hd;_ …™Npqp

e;:PLQr„_JLQrWcM—(_YHR;3e¨WYM_aMž_cHdFG9<`IJLWaJRQTf_aMb_YprFIHRQqH0WY…ªJR7;?eQr9<;:M(HdQqWYMpq;3HLH‹=?_†„GQq;:PRˆ¾DGJRWY“c;3H ;3S-Nb_aJRQqWYMbH3Œ œ¾M

`(_aPRJRQTfNpT_aP JR7;3FHKJLNbeIF©JL7;QrMI›(N;:Mbf:;§WY…JR7b;§„GQTHRf:WYNbH JR;3PR9 _YMbeQrJLH PL;:pT_JLQrWcMƒQ’JL7tJL7;§…™PRQTf JLQrWcMJR;3PR9Œ

67b;:MJR7b;tfpT_YHLHdQTf:_Yp*DI_aQqM-JdˆŠ‰‹;:Mb_YM-J?HdFIHdJR;:9 ƒQrJR7˜_¨…™PLQqfJRQqWYMJR;3PR9JRNbPRMbHWYNIJJLWž—i;t_<£:;3PRWYJR7WYPeI;:P

_Y``PLW†•IQr9_JLQrWcM˜Wa…§JR7;ž=A_†„GQr;3PdˆKD-JRWc“Y;2H?;2ScN(_JRQqWYM(H]…™WYP©QrM(fWY9<`PL;3HLHRQr—pq;¨›bWƒH]ƒQ’JL7_z…™PL;:;<9<W„GQrMbw

—iWYNbMbe_aPLF NMbeI;3P¨JR7;¦HR7b_apqprWƒ ƒ_JL;:P<_YHLHdNb9¨`JRQqWYMµŒ 67;3F _apTHdW5eI;:PLQq„Y;_˜„GQqHLfWcNbHDI_aQqM-JdˆŠ‰‹;:Mb_YM-J

HRFIHKJL;:9¡JR7PLWYNwc7_]¢bPLHdJ‡WYPeI;3P‹QrMG„Y;2HKJLQrw-_JLQrWcMµŒ*œ¾Mž_©PL;3f:;:M-J‡ƒ§WcPR“ _5

ucbŽŸIDI_Ypr;3PRQb_YMbe

²

;3PRP_aPLQI;•GJR;3MbeI;2e

JL7QTH¨_Y``PLWc_cf7nJLW–JL7;¦k³f:_cHd;žQqMbf:prNbeQrMw _5HdpqWƒprF5„_aPLFGQrMw–JRWc`(WcwYP_a`7GFn_YMbe$_YM$_JL9<WcHR`7;:PLQTf

`bPR;2HRHRNPL;?JL;:PL9 Œ

AQqPR;2f J…™PL;:;HRNPd…´_cf;#=?_†„GQq;:PRˆ¾DGJRWY“c;3HfWc9<`NIJL_aJRQqWYMbH_YPR;©`iWcHLHdQq—pq;H_k 5

b*—NJA_aPL;#HKJLQrpqp*;:•G`i;:M(HdQq„Y;#_aM(e

PL;3S-NQqPL;<HRWY`b7QqHdJRQTf:_aJR;2en_YprwcWYPLQ’JL79H:Œ<œŠJQTH#M;3f:;3HLHR_YPRFJLW˜_Ye_Y`IJt_u 9<;2Hd75JRWJL7;9<W„Y;39¨;3M-J©WY…

JL7;¨…™PR;3;¨HRNPR…´_Yf:;¨_YMbe˜JLW JR7;<„_aPLQq_aJRQqWYM˜WY…VJL7;<—(WYJdJRWc9 JRWc`(WcwYP_a`7GFQqM5HR;3eIQq9<;:M-JL_aJRQqWYM5f:_YHR;3H3ŒtœŠJ

PL;3S-NQqPL;3H_YpqHRWJRW eI;3_Yp*ƒQrJR7–JR7;QrMbf:WY9<`PL;3HLHdQq—QqprQrJKFfWYM(eIQ’JLQrWcMµŒ?67;<DI_aQqM-JdˆŠ‰‹;:Mb_YM-J]fWc9<`NIJL_aJRQqWYMbH

_YPR;#9Nbf7¦9<WYPL;#;žfQq;:M-JAHRQqMbf;#JL7;:FeI;2_ap*ƒQrJR7¦NbM-„_YPRFGQqMwzk¡9¨;2Hd7b;3H3ŒZ‘NIJAJR7;t—b_Yf“GwYPLWYNbMbezWY…

JL7QTH?fWc9<`NIJL_aJRQqWYMb_Yp*HRQr9<`pqQ’¢if:_JLQrWcMQqHA_apTHdW¨JR7b;9_aQqM¦prQq9<Q’J_JRQqWYM¦WY…0JR7;D_aQqMcJRˆŽ‰‹;3Mb_aM-J?_Y``PLWc_Yf7

QrJNbHd;2H‘WYMpqF<QrM-JR;3wYP_JL;3ežS-Nb_YMcJLQ’JLQr;2H:Œ‹67;©fpT_YHLHRQqf3_apDI_aQqM-JdˆŠ‰‹;:Mb_YMcJHdFIHdJR;39 QqH‘PL;:pT_JR;2e<JRW<_ K9<WaJLQrWcM

—GF¨HdpqQqf:;?_a`b`PRW†•IQq9_JRQqWYM*Ÿ JR7;?7WYPLQr£3WYM-JL_YpI„Y;3prWIfQrJKFeW-;2H‡MWYJ§eI;3`(;3Mbe<N`iWYM<JR7b;?„Y;3PdJLQqf3_apf:W-WcPdˆ

eQrMb_aJR;cŒ‡67;:PL;…™WcPR;©HdWc9<;AQqMI…™WYPL9_JLQrWcMQTHprW-HKJ_YMbeJR7;©`P_YfJRQTf:_apP_aMbwY;]Wa… „_YprQTeIQrJKFžWa… JR7;HRFIHKJL;:9

QTHHRWY9<;JLQr9<;2H„Y;3PRFžpqQq9¨QrJR;2e‘Q’J…´_YQrpTHJLWPR;3`PRWIeIN(f;©JR7b;fWYPLPL;3f J„c;:PRJRQTf:_ap*`PRWY¢bpq;]WY…0JR7;7bWYPLQr£3WYM-JL_Yp

„c;:pqWIfQrJKFAˆµ_YMbet;3„Y;3MJR7b;‘f:WYPLPR;2f J _†„c;:P_awc;3e©7WcPRQq£:WcMcJ_ap-„Y;3prWIfQrJKF]ˆƒ7;:MJR7;‘…™PLQqfJRQqWYMWYMJL7;—(WYJdJLWY9

QTH#MWYJtHd9_Yprp‡;3MWYNbwY7µŸ _apTHdWQrJQqH#MbWaJt_a—bpr;<JLW¦PR;3`PRWIeIN(f;HdN(f75`b7;:MWc9<;:Mb_¦_cH]JR7;žPL;3f:QrPfNbpq_aJRQqWYM

eN;©JRW¨JR7b;HdNPR…´_Yf:;©ƒQrMbe QqM _f:prW-Hd;2ežpT_a“c;YŒ

² WYP‘JR7b;3HR;APL;3_cHdWcMbH:Ÿ-QqMcJL;:PL9<;3eIQT_JL;A9<WIeI;:pTH‘7b_†„Y;A—(;3;:M…™NPRJR7;3P‘N(Hd;2eŒ‡@¦WcHdJWa…µJL7;:9 _YPR;

Om!

" k

H !I

Œ0œ¾MZ_mŸ

5

lŸckYjcbŽŸANb_aPRJR;3PRWcMQŽŸYDI_apq;:PLQ # f:WYMbHRQqe;:P0JR7bQqH§Hd;:J‡Wa…_a``PLW†•IQr9_aˆ

JLQrWcMbH]Wa…‡JL7;<u =?_†„GQq;:PRˆ¾D-JLWY“c;3H;2S-Nb_JLQrWcMbHA_YMbe¦JL7;:FneI;:PLQr„c;eIQª;:PL;:M-J©_Y``PLW†•IQr9_JL;9<WIeI;:pTH:Œtœ¾M

`(_aPRJRQTfNpT_aP2ŸbQqM _5

lŸkYj"bŠŸ(JR7;3F ƒ‘WYPL“zWcM¦JR7b;HRWaˆ¾f:_apqpq;3e 7GFGePRW-HKJ_JRQTf#_a`b`PRW†•IQq9_JRQqWYMQqM–_ƒ_†FzJL7b_J

QTH?_ž¢bPHKJAHdJR;3`¦JRWƒ_aPeJR7;<DI_aQqM-JdˆŠ‰‹;:Mb_YM-J?9<WGe;:p0HdQqMbf;JR7;3Ff:WYMbHRQTeI;:P?JR7;QrM-JR;3wYP_JL;3e¦fWcM-JRQqM-NbQ’JKF

;2S-Nb_JLQrWcMnJLW–PLQqHR;<JR7b;zƒ_JR;3P7;3Qrwc7cJ¨_YHt_YM;•I`pqQqf:Q’J¨„_aPLQq_Y—pr;cŒ¦=;:„c;:PRJR7;3pr;2HRH#JR7;—b_Yf“GwYPLWYNbMbe–WY…

JL7; 9¨WIeI;3prQq£3_aJRQqWYM QqH<HdJRQqprp‘JR7b;f:pq_cHRHRQTf:_ap=?_†„GQq;:PRˆ¾DGJRWY“c;3H©…™WcPR9_YprQTHd9ŸV_aJ¨pq;3_cHKJ¨…™WYPJR7b; 7bWYPLQr£3WYM-JL_Yp

9<Wc9¨;3M-JRN9 ;2S-Nb_JLQrWcM0NM“GMWƒMbH_aPL;©„Y;3prWIfQrJRQq;3H_YMbežJL7;fWcM-JRQqM-NbWYNbH‘`PLWY—pq;:9 QTHfWcMbHRQqeI;3PR;2ežWcM

_#u¥eWY9_aQqMµŒ067;;•I`pqQTfQrJL_JLQrWcMWa…ªJR7b;Ak!

5 …™WYPL9 WY…iJL7;`PLWY—bpr;39 QTH‡`i;:PR…™WYPL9<;3e<QqMž_Hd;2fWcMbe

HdJR;3` JL7PRWcNwY7$JR7;f7WcQqf:;WY…JR7;„c;:PRJRQTf:_apeIQTHRf:PR;:JRQq£3_aJRQqWYMJL7;¦eIWY9_YQrM QqH<eIQq„GQqeI;2e$QqM-JRW Hd;3„Y;3PL_Yp

pT_†FY;3PLH‡WYMƒ7QTf7JR7b;]HL_a9<;Ak ¢bMbQ’JL;A;3pr;39¨;3M-J‘_a`b`PRW†•IQq9_JRQqWYM<QTHVN(Hd;2e<JRW¨eI;3HLfPLQr—i;JL7;A7bWYPLQr£3WYM-JL_Yp

„c;:pqWIfQrJKFYŒ<67;<pq_†Fc;:PH]f3_aM5—(;¨¢•I;2en7bWYPLQr£3WYM-JL_Yp HdJRP_J__cH©QrMd_

5

lŸ*kcj"bVWYP©—b_cHd;2e–WcM–JL7;HdWYˆŠf3_apqpr;2e

HRQqwY9_JLPL_YMbHd…™WYPL9<_aJRQqWYM _YHQqM _k 5 bŽŒ

]ANbP _Y``PLWc_Yf7”f3_aM —i;5HR;:;3Mš_YH _ "H# ! ( ## "!#Ÿ]Q¶Œ;YŒ _ eQqHLfPL;JLQr£2_JRQqWYM”WY…

JL7;u¡=A_†„-Qq;:PRˆKD-JRWc“Y;3H;3S-Nb_JLQrWcMbHƒ7;3PR;JR7;„Y;:PRJRQTf:_Yp*wcPRQTe

4' ) 7

QqHA;:„cWYpq„Y;3e$ pT_zDI_aQqM-JdˆŠ‰‹;:Mb_YM-J%

_YMbeHRWY9<;ž„GQqHLfWcNbH#;&i;2f JLH¨_aPL;ž“c;:`IJ2Œ¦œ¾MJR7QTHQ’J¨wYWG;3H#WcM;zHdJR;3`…™NPRJR7;3PtQqMJR7; _aMb_YprFIHRQqHWa…_

5

j"b

_YMbež_cHV…™WcPVJL7;©fpT_YHLHdQTf:_YpªDI_YQrM-JRˆŽ‰‹;3Mb_aM-J‘HRFIHKJL;:9 ƒ‘;?;:•G7bQr—QrJ_aMz;3M;:PLwYF_YMbeƒ‘;]f7b_YPL_cf JL;:PLQr£3;?HdWc9<;

;2S-NQrpqQq—PRQqN9³HKJ_JL;3H3Œ§[?H?WYM;f3_aM˜HR;:;cŸIJR7;Hd;:J?Wa…VDI_YQrM-JdˆŠ‰‹;:M(_aM-J?HdF-JL;:9H

465

Œ

5c7

ˆ

465

Œk 7

Ÿb—i;:QqMwzktŸ(QqH

_Y—pq;©JRWž`PL;3HR;:PL„Y;©fWc9<`NIJL_aJRQqWYMb_Ypµ;žf:Qr;3MbfFcŒ‘67;f:WYMbHR;:PL„_JLQr„c;A…™WcPR9Wa…0JR7b;;3S-Nb_aJRQqWYMbHWcMƒ‘_aJR;:P

7b;:QqwY7-JLH

4

')

7

pq;3_YebH‡JLW;:•-JL;:Mbež¢bMQrJR;]„YWcprN9<;]HRf7;39<;3HVJRW

465

Œ

5"7

ˆ 4 5 Œk 7

Œ 67QTH‘9<WaJRQq„_JL;3H‘WYNPf7WcQqf:;

WY…VƒPLQ’JLQrMw

465

Œk 7

_cHA_YM–NbHRNb_Yp0f:WYMbHR;:PL„_JLQr„c;tpq;…J#7b_aM(e–HRQqeI;¨`pqNbH]MWYM fWcMbHR;:PL„†_aJRQq„Y;tf:WYPLPR;2f JLQrWcMbHˆ

WcMžJL7;APLQqwY7-J‘7(_aMbeHdQTeI;YŒ0œ¾Mbe;:;3ežJR7;#f7b_Yprpq;:MbwY;?7b;:PL;Aƒ‘WYNpTe—i;]JRW¨`(;3Pd…™WcPR9 _aM _apqwYWcPRQrJR79 _Y—pr;AJRW

“c;:;3`

4Q7

HdJL_Y—QqprQrJKFžˆ

')

0

_aMbe ;:M-JLPRWc`-FzQrM;2S-Nb_apqQ’JKF

4

QqQ

7

f:WYMbHR;:PL„_JLQrWcMWY… 7;3Qrwc7cJH_aM(ezJLWaJ_apµ9<Waˆ

9<;3McJLN9

= ')()

ƒ7;:M

/?0H4 QqQrQ

7

JL7;`b_YPdJLQqf:Npq_YP‡HKJ_JL;3H ')

/

'

\ Ÿ )

/

wcQr„c;3H‹_f:WYMbHRQqHdJR;3M-J

_Y``PLW†•IQr9_JLQrWcMžWa…*JR7;f:pq_cHRHRQqf3_api9<WYMWcpq_†Fc;:P‘D_aQqMcJRˆŽ‰‹;3Mb_aM-JHRFGHdJR;39 ˆ‡ƒ7;:M

/10 4 Qq„

7

JR7;HdJR;2_YeIF

HdJL_aJR;2H0Wa…µ_]pq_Y“Y;_J‡PL;3HdJ‡_aPL;`PR;2Hd;3PR„c;3eŒ^]ANP‹ƒ‘WYPL“`PLW„-QTeI;2H‹_]¢bPLHdJVHdJR;:`<QqM¨JR7QTHVeIQqPL;3f JLQrWcM¨_YMbe<f:_YM

—i;tPL;:pT_JL;3e JLWz_ƒ‘WYPL“zWa…§D_apq;:PLQ _aM(e *_a£3£3_YPRWcMQ^_kck"b09¨WYJRQq„_JR;2e¦—GFzJL7;HL_a9<;t_Y``pqQqf3_JLQrWcMbH:ŒZ‘NIJ

JL7;A_Y``PLWc_cf7;3H‡_YPR;AScNbQ’JL;AeIQi;3PR;3M-J‘HRQrMbf:;_kYk"bª`PL;3HR;:M-JLHVMWYMzf:WYMbHR;:PL„_JLQr„c;?DI_aQqM-JdˆŠ‰‹;:Mb_YMcJ‘HdFIHdJR;:9H

eQqHLfPL;JLQr£3;3e$JR7PLWYNwc7&_n¢bMbQ’JL;¦;:pq;:9<;:M-J…™P_a9<;:ƒ‘WYPL“ªŒ$œ¾M _ bŸ"b?U‹_YPR;2H # `PR;2Hd;3M-J<_ —QrˆŽ›bNQTe

HR7b_YprpqWƒƒ‘_aJR;:PA`PLWY—pq;:9 _aM(eJR7;¨HdFIHdJR;39JL7;:F˜f:WYMbHRQTeI;:PA`PR;2Hd;3M-JLH?_YpqHRWf:WY9<9<WYM¦`b_YPdJH?ƒQrJR7–JR7;

JKƒ‘W¦pT_†FY;:PH#„c;:PHdQqWYM Wa…

4 5 Π5"7

ˆ

465

Œk 7

Œ˜[AHQqM1_kck"bVJL7;:Ff:WYMbHRQqe;:Pt_¦MbWYM$f:WYMbHR;:PL„_JRQq„Y;<„c;:PHdQqWYM_aM(e

JL7;:F ;•I7Qq—QrJJR7(_JJL7;tHRFGHdJR;39 QTHMW9<WcPR;#7GFG`(;3PR—iWYpqQTf©ƒ7;:MJL7;teI;3MbHdQrJRQq;3H?Wa…0JR7b;#›bNQTeH?_YPR;]JLW-W

f:prW-Hd;cŒ ‚ ;`PLW„Y;©QqMJR7;t`b_a`i;:P?JR7b_aJJR7;tpr;:…JA7b_YMbe¦HdQTeI;#WY… JL7;tHRFIHKJL;:9

465

Œ

5"7

ˆ 4 5 Œk 7

QTH7GFG`(;3PR—iWYpqQTf

ƒ7bQqf7 QqH?_aMWYJR7;3P‘9<WYJRQq„†_aJRQqWYM …™WYPWcNP‘…™WYPL9tNpT_JLQrWcMµŒ

67b; WcNIJRpqQqM;¦Wa…AJR7;¦`b_Y`(;3P<QqHž_cH…™WYpqprWƒH3Œ œ¾M DG;2f JLQrWcM knƒ‘;_Y``pqFJR7b;˜Hd7b_YprpqWƒ ƒ_JL;:PHRf3_apqQrMbw

JLWJR7b;¦…™PR;3;–HRNPd…´_cf;˜=A_†„-Qq;:PRˆKD-JRWc“Y;3H<;2S-Nb_JLQrWcMbHž_aM(e ƒ§;¦`PL;3HR;:M-J_YMbe _aMb_YprFIHR;JKƒ§W7GFIeIPLWcHdJL_aJRQTf

9<WIeI;3pqHžˆ¨_YHžQrMbeQqf3_JR;2e QrM ² QqwYNbPR;

5 Œ5

Œ67b;:M”QqM¥DG;3fJRQqWYM u ƒ‘;–eI;3PRQq„Y;JL7;–9Np’JLQrpT_†FY;3PzDI_aQqM-Jdˆ

‰‹;3Mb_aM-J©9<WGe;:p

465

Œ

5c7

ˆ

465

Œk 7

_aMbe–ƒ§;<eIQTHRf:NbHRHAQrJLH];3HLHd;3M-JRQT_ap `PRWc`(;3PdJLQr;2H:Œ©œ¾M DG;3fJRQqWYM˜l ƒ§;;3HdJL_Y—prQTHR7

JL7;?eIQTHLfPL;JR;„c;:PHdQqWYM<WY…iJL7;9<WIeI;:p(NbHRQrMw©JL7;¢(MQ’JL;„YWcprNb9¨;…™P_a9<;3ƒ§WcPR“ªŒ*œ¾MzDI;3f JLQrWcMzj]ƒ§;?`PR;2Hd;3M-J

HRWY9<;M-Nb9¨;3PRQTf:_Yp PL;3HRNprJLH©_aM(e˜fWc9¨`(_aPLQqHRWYMbH?—(;:JKƒ§;3;:M–JR7;<9<WYMWcpq_†Fc;:PAD_aQqMcJRˆŽ‰‹;3Mb_aM-J#_aMbe–=?_†„GQr;3Pdˆ

DGJRWY“c;3H‹HRFIHKJL;:9H0ƒ7QTf7¨„_YprQTe_JL;‘JR7;9<WIeI;:pŽŒ ² QrMb_YprpqFtQqMžDG;3fJRQqWYM ©ƒ§;f:WYMbf:prNbe;_aMbe`PL;3HR;:M-JVHdWc9<;

`i;:PHR`(;2f JRQq„Y;2H:Œ

Viscous Saint-Venant system Saint-Venant system

Hydrostatic model

Saint-Venant system

Classical Multilayer

Shallow water scaling

Viscous hydrostatic model Navier Stokes equations for incompressible free surface flows

² QrwcNPL;

5 Œ5 ² PLWY9 =A_†„-Qq;:PRˆKD-JRWc“Y;3H‘;2ScN(_JRQqWYM(HVJLWzDI_aQqM-JdˆŠ‰‹;:Mb_YM-J‘JKFG`i;©9¨WIeI;3pqH

µ¨  <© ¨z0µ ©  ž ]

 

² WYpqprWƒQqMw _5

j"b?ƒ‘;¦`PL;3HR;:M-J7;:PL; JR7;¢bPHdJHdJR;3`bHWY…AJL7;˜eI;:PLQq„†_aJRQqWYM&Wa…]JR7;–D_aQqMcJRˆŽ‰‹;3Mb_aM-J<JKFG`(;

9<WIeI;3pqHV…™PRWc9 JR7;©=?_†„GQq;:PRˆ¾D-JLWY“c;3H‹;2ScN(_JRQqWYM(H:Ÿ JR7;©eI;3PRQq„_JRQqWYMzWa…JR7b;]HRQr9<`pqQr¢b;3ež=A_†„GQr;3PdˆKD-JRWc“Y;2H

9<WIeI;3pqH‘ˆVHR;:; ² QrwcNPL;

5 Œ5 Œ ‚ ;#QqMcJLPRWIeIN(f;©_Mb;:ƒ¥…™WGf:NbHWYMJR7b;©S-N;2HKJLQrWcMWY…*JL7;eIQTHRHRQr`(_JRQqWYMWa… JR7;

;3M;:PLwYFcŸƒ7bfQTf7tQTH0_aM¨;3HLHd;3M-JRQT_ap-`PRWc`(;3PdJKF#WY…JR7b;=A_†„GQr;3PdˆKD-JRWc“Y;2Hµ;3S-Nb_aJRQqWYMbH3Œ*œ¾M¨`b_YPdJLQqf:Npq_YP QrJ‡_apqprWƒH

N(H]JRW˜eIQTHRf:PRQq9<QrM(_JR;_YM

##" #

…™WcPR9 Wa…VJL7;ž„GQTHRf:WYNbH]7GFIeIPRW-HKJ_JLQqf9<WGe;:p‡QqM-JRPLWGeNbf;2e

QqM&DIN—bHR;3f JLQrWcM kIŒ

5

Œ@¦WcPR;3W„Y;3P#ƒ§;fpq;3_YPRpqF5eQqHdJRQqMwYNbQqHR7 JL7;—(_YHRQqfz; ª;3f JH¨Wa…JL7; HR7b_YprpqWƒ ƒ‘_aJR;:P

_cHRHRN9<`IJLQrWcM$ˆDGNb—bHd;2f JLQrWcM”kŒ

5

ˆ#…™PLWY9 JR7;¦WcM;3H<WY…A…™NPRJR7b;:Pž_YHLHRN9<`IJRQqWYMbH_Y—(WcNIJž„GQqHLfW-HdQrJKF _aM(e

…™PLQTf JRQqWYMˆ‘DGNb—bHd;2f JLQrWcMkŒkŒ

‚ ;fWcMbHRQqeI;3P‘JR7;f:pq_cHRHRQqf3_api…™PL;:;HRNPd…´_cf;©=?_†„GQq;:PRˆ¾D-JLWY“c;3H§;3S-Nb_JLQrWcMbH

&

&.T+

&

& / 032 4

kIŒ

5"7

&

&* + & 9

&. + &

& +

&

&. / EL

& 9

&.

9 + L & 9

&

9 + L & 9

&. & 2 4

kIŒk 7

&

&* + &

&. +

& 9

& +

&

& / N

;:+

L & 9

&.

9 + L & 9

&.Z& + EL

& 9

&

9 2 4

kIŒu 7

ƒQrJR7

*

032 .

2 0

' S* 2 . U 2

ƒ7b;:PL;

S* 2 . 2 U QTHžJL7;57WYPLQr£3WYM-JL_Yp„Y;3prWIf:Q’JKFcŸ

S* 2 . 2 U QqHJL7;–„c;:PRJRQTf:_apA„Y;3prWIfQrJKFYŸ S* 2 . 2 U JR7;

`bPR;2HRHRNPL;§_aM(eƒ7;3PR;

' S* 2 . U e;:MWYJR;3H*JR7b;‘ƒ_JL;:P 7;:QqwY7-J2Ÿ

;

JR7;‘wYP_†„GQ’JKF©_aMbe L JL7;§„GQTHRf:WcHRQ’JKFcŒ =?WaJRQTf;

JL7b_J…™WcPHRQq9¨`bprQTfQrJKFžƒ‘;©f:WYMbHRQTeI;:P§JR7;©›i_J—(WYJdJLWY9 f3_YHR;YŒ œ¾MT_

5

ucb0DI_apq;:PLQ_aMbe ² ;:PLP_aPLQi;:•GJR;:M(eI;3ezJR7;

_YMb_apqFIHdQTHWa…]\];3PR—i;3_YN _aM(e U0;:PRJR7b_Y9<;T_

5

j"bf:WYMbf:;:PLMQrMbw˜JR7b;f:pq_cHRHRQTf:_apDI_YQrM-JdˆŠ‰‹;:M(_aM-J¨9<WIeI;3p§JLW5_

HRpqWƒprF&„_aPLF-QqMw —iWaJRJRWc9 Œ 67;n`PR;2Hd;3M-Jƒ‘WYPL“&eIWG;2HzMWYJQqM-JRPLWGeNbf;nM;:ƒ f:WYM-JRPLQr—bNIJRQqWYM WcM”JR7bQqH

`(_aPRJRQTfNpT_aPJRWc`Qqf5_aM(e”JR7b;5HL_a9<;5;•GJR;3MbHdQqWYM¥f:_aM¥—(;eIWcM;n…™WYPJR7; 9tNbp’JLQrpT_†FY;3P9<WIeI;:p]JL7b_Jƒ§;

QqM-JRPLWIeINbf;zQrM DI;3f JLQrWcM uŒ–=?WaJLQqf:;_apTHRWJR7b_aJƒ‘;z_YHLHdNb9¨;žJR7b_aJtJR7b;ž…™PL;:;zHRNPR…´_Yf;zQqH¨_…™NMbfJRQqWYM WY…

S. 2 * U Œ

]AM<JL7;?—iWaJRJRWc9 ƒ‘;Af:WYMbHRQTeI;:PV_MWt`i;:M;:JRP_JLQrWcMžfWYM(eIQ’JLQrWcMz_aM(e<ƒ§;A;3HdJRQq9<_aJR;JL7;…™PLQTf JRQqWYMžJR7PLWYNwc7

_<f:W-;:žfQq;:M-J¦_aMbe _<=?_†„GQq;:PfWcMbeIQrJRQqWYM

S* 2 . 2 0 U

/102 L &

&

S* 2. 2 0 U / S* 2 . 2 0 U Q 4

kIŒl 7

]AMJR7b;©…™PR;3;©HRNPR…´_Yf:;]ƒ‘;#fWcMbHdQTeI;3P_MbW<HdJRPL;3HLHfWcMbeIQrJRQqWYM

L &

& +

&('

&.

N

EL

&

&.

&('

&. + L

&

&. / 0

WcM / ' S* 2 . U 2 4

kIŒj 7

N

EL

&

& + L &

&

&('

&. + L

&

&.

&('

&. / 0

WcM

/ ' S* 2 . U 2 4

kIŒ

7

_YMbezJR7;#“GQqM;:9_JLQqf©—iWYNbMbe_aPLFžfWcMbeIQrJRQqWYM

&('

&* + S* 2 . 2 / ' S* 2 . U U

&('

&.

N S* 2 . 2 / ' S* 2 . U U / 0 Q 4

kIΠ7

‚

;#PL;3f:_YprpiJL7b_J

4

kIŒ

5"7

Ÿ 4 kIŒl 7

_YMbe

4

kIΠ7

pq;3_YeJLWJL7;©…™WYpqpqWƒQrMw¨9_YHLH;3S-Nb_aJRQqWYM

&('

&*H+

&

&.

/10

Q

‚

;tPL;3f3_apqp*_apTHRW<JR7b_aJ]_<…™NM(e_a9<;:M-J_ap0HKJ_a—QqprQrJKF `PRWc`(;3PdJKFQqH?PR;3pq_aJR;2e JRWžJL7;#…´_cf J?JL7b_JAJR7;=?_†„GQr;3Pdˆ

DGJRWY“c;3H‘;3S-Nb_aJRQqWYMbH_ceI9<Q’J?_YM;3M;:PLwYF

S* 2 . 2 U /

S* 2 . 2 U +

S* 2. 2 U / 9 + 9

E S* 2 . 2 U + ; Q

_YMbezJR7b_aJJR7;©…™WcprpqWƒQrMbw;2ScN(_JRQqWYM 7WcpqeH

&

&*

+ &

&.

S

+ U N L E &

&.T+

&

&+

&

&.

/ N

9

S* 2 . 2 0 U N EL

&

&. 9 + [E &

&Z+

&

&. 9 + &

&

9

Q

²

WYP…™NPRJR7b;:P_aM(_apqFGHRQTH§WY… JL7;©…™Npqp*=?_†„GQq;:PRˆ¾DGJRWY“c;3H§;3S-Nb_aJRQqWYMbHƒ‘;©PR;:…™;:PJRW_klbŠŒ

‚ ;_YPR;zMWƒ QrM-JL;:PL;3HdJR;3e QrMJL7;Hd7b_YprpqWƒ ƒ‘_aJR;3P›bWƒH3Œ–[?H¨NbHRNb_Yp§ƒ‘;žQqMcJLPRWIeIN(f;žJKƒ‘W–f7b_YPL_cf JL;:PRˆ

QTHdJRQTfAeIQq9<;:MbHRQqWYMbHC_YMbe QrMzJR7;]„Y;3PdJLQqf3_ap(_YMbež7bWYPLQr£3WYM-JL_Yp(eIQqPR;2f JLQrWcMbHVPL;3HR`i;3f JLQr„c;:pqFYŒ067;©Hd7b_YprpqWƒ

ƒ_JL;:P§›bWƒH_aPL;?JL7;:Mf7b_YPL_cf JR;3PRQq£:;2e<—GF<JR7;]…´_YfJJR7b_aJ QTH§„c;:PLFHd9_apqpµfWc9<`b_aPL;3eƒQrJR7Œ‹67GNbH

ƒ‘;#f:_YM9_a“c;]JR7;HRWaˆ¾f:_Yprpq;3eHd7b_YprpqWƒ ƒ_JL;:P_YHLHRN9<`IJRQqWYMµŸIQŽŒ;YŒ‹ƒ‘;]QqM-JRPLWIeINbf;#_!RHd9_apqp`b_YPL_Y9¨;:JR;3P%

/

Œ067;:M¨_…JR;3P PL;3HLf:_YprQqMwAƒ§;‘f:_aMƒPRQrJR;

4

kŒ

5c7

ˆ 4 kŒu 7

_YH _]eIQr9<;3MbHdQqWYMbpr;2HRH =?_†„GQq;:PRˆ¾D-JLWY“c;3HµHRFIHKJL;:9

&

&. +

&

& / 032 4

kIŒa 7

&

&*H+

& 9

&. + &

& +

&

&. / E & 9

&.

9 +

9 & 9

&

9 + & 9

&. &2 4

kIŒm 7

9

&

&* + &

&.`+

&

9

& +

&

& / N

;X+

9 & 9

&.

9 + & 9

&.Z& + E & 9

&

9 Q 4

kIŒ

5 s 7

=?WaJLQqf:;©JR7b_aJ / L S- U QqHJR7;eIQq9<;:MbHRQqWYMpq;3HLH…™WYPL9 WY… JL7;#„GQqHLfW-HdQrJKFzf:W-;:žfQq;:M-J?_aMbeJL7b_J7;3PR;

_YMbe<QqMz_apqpbJL7;AeQr9<;:M(HdQqWYMpq;3HLH‡;2S-Nb_JLQrWcMbH

;

eIWG;3H‡MbWaJ‘e;:MWYJR;JL7;?wcPL_†„GQrJKFt—NIJ§JR7; ² PLWYNbe;MGN9t—i;:P

; S - U Œ

67b;©_cHRHRWIfQT_JL;3e—(WcNMbe_YPRFžf:WYMbeIQrJRQqWYM(HWYM JR7;#—iWaJRJRWc9

4

kŒl 7

_aPL;APL;3HLf:_Ypr;2ez_YH

S* 2. 2 0 U

/102 &

&

S* 2. 2 0 U

/

S* 2 . 2 0 U 2 4

kIŒ

55c7

ƒ7b;:PL;

/

I- QTH*JL7;eIQr9<;3MbHdQqWYMbpr;2HRH …™WYPL9Wa…JL7;‘…™PRQTf JLQrWcMf:W-;:žfQq;:M-J3Œ 67;§—iWYNbMbe_aPLFfWcMbeIQrJRQqWYMbH

WcMzJL7;©…™PL;:;#HdNbPd…´_cf;

4

kŒj 7ˆ 4 kIŒ

7

_YPR;]PR;2HRf3_apq;3e_YH

9 &

& +

&'

&.

N E &

&.

&'

&. +

&

&. / 0

WcM / ' S* 2 . U 2 4

kIŒ

5 k 7

N E

&

&

+ &

&

&'

&.

+ 9

&

&.

&('

&.

/ 0

WcM

/ ' S* 2 . U Q 4

kIŒ

5 u 7

67b;]“GQqM;:9_aJRQTf©—(WcNMbe_YPRFzf:WYMbeQ’JLQrWcM

4

kIΠ7

QqHNMbf7b_YMwY;2eŒ

!"$#

‚ ;#HdQq9<`pqQ’…™FzJR7;HRFIHKJL;:9

4

kIŒ

5"7

ˆ 4 kŒu 7

—GFžPL;J_aQqMQqMwtJL7;#£:;3PRW<_aM(ež¢bPHdJWcPLe;:P§JR;:PL9H:Œ ‚ ;©WY—IJ_aQqM _

„GQTHRf:WYNbH‘7GFIeIPLWcHdJL_JLQqfA9<WIeI;:p

&

&. +

&

& / 032 4

kIŒ

5 l 7

&

&* + & 9

&. + &

& +

&

&. / E & 9

&.

9 +

9 & 9

&

9 + & 9

&. & 2 4

kIŒ

5 j 7

&

& / N

;X+

& 9

&. & + E & 9

&

9 + 9 & 9

&.

9 2 4

kIŒ

5 7

ƒ7b;:PL;

*

032 .

2 0

' S* 2 . U Q

67b;–_YHLHdWIf:Qq_aJR;3e —iWYNbMbe_aPLF f:WYMbeQ’JLQrWcMbHz_YPR;¦JL7;nHL_a9<;–_cHJL7;–WcM;3Hžf:WYPLPR;2Hd`iWYM(eIQrMbw JRWJL7;˜…™Npqp

eQr9<;:M(HdQqWYMpq;3HLH=?_†„GQq;:PRˆ¾DGJRWY“c;3H§;3S-Nb_aJRQqWYMbH3Œ

¬ ~I}&%('*),+ - H c /. I " 10

m! !I # 10 k "&0 *03!

!I:2. 303

" k

54#

!: 10*.m m

3. m!k m!

303D

m

k6.

303H##" !

57&0 m! m98

R

#kZ2. 10 " #k

54#

8V k " "I 30 H :4# H#^

k # V

O!I ;7I H

¬ ~I}&%('*)<'>= @?BA 1CEDGFG?HA 1CI:FJ8V 0 V10

#Hk K. . 303X

303"L0

m m

Om! NMD m #

m m

2.:

( # T 10 m!k !

?HA 1COA;FcG?BA 1C9PGF

=?Wƒ ƒ‘;f7;2f“JR7(_J‡JL7;„GQqHLfWcNbH‹7GFIeIPLWcHdJL_JLQqf_Y``PLW†•GQq9_JLQrWcM¨HR7b_aPL;3H0ƒQrJR7<JL7;…™Npqp(=A_†„GQr;3PdˆKD-JRWc“Y;2H

;2S-Nb_JLQrWcMWcM;3M;:PLwYFžHdJRPLNbf JLNPL;YŒ ‚ ;#QrM-JRPLWIeINbf:;AJL7;#7GFIeIPRW-HKJ_JLQqfA;:M;3PRwcF

S* 2. 2 U /

S* 2 . 2 U + S* 2. 2 U / 9

E S* 2. 2 U + ; Q 4

kIŒ

5 7

Q§¬ ~R'*),+TS 03U.98^ " ##"H:4# V03!

.W?HA 1CYX9FcG?BA 1CI5F

&

&*

+ &

&.

S + U N E &

&.<+

&

&+

9

&

&.

/ N 9 S* 2 . 2 0 U N E

&

&. 9 + [ E [ &

&

+

&

&. 9 +

&

&

9

Q

Z . 67b;#`PRWGWY…*NbHR;3Hf:pq_cHRHRQqf3_apfWc9<`NIJL_aJRQqWYMbH3Œ ² WcprpqWƒQrMbwtJL7;#`NPL`(W-Hd;©WY…0^;39_aPL“zkIŒ

5

ŸIpq;JNbH

MbWaJRQTf;©JL7;Qq9<`(WcPdJ_aMbf:;#Wa… JR7;;3`bHRQrpqWYM¦HLScN(_aPL;AJL;:PL9 §Q’J?_YprpqWƒHJLWQrMbf:prN(eI;#JR7b;#JR;3PR9 fWYM-J_aQqMQrMbw

&[ &\

QqM¦_<HRS-Nb_YPR;AJR;3PR9 _aMbezJR7;3M JRW<;•I7Qq—QrJJR7;eI;2fPL;3_cHdQqMw¨Wa… JR7;#;3M;:PLwYFcŒ

67b; f:WY9<`NIJ_JRQqWYM(_ap§f:WY9<`pq;•IQrJKF _YMbe$f:WcHdJWY…JR7QTH¨„GQqHLfWcNbHt7GFIeIPLWcHdJL_aJRQTf9<WGe;:pPR;39_aQqM$HdQq9<QrpT_aP

JLWtJL7;]WYM;2HVWY…µJR7;A…™Npqpµ=?_†„GQq;:PRˆ¾D-JLWY“c;3H‹;2ScN(_JRQqWYM(H:Œ0œ¾Mz`(_aPRJRQTfNpT_aP‘ƒ§;]eIW¨MWaJPR;2fW„c;:P‹JL7;]f:pq_cHRHRQTf:_ap

7GFIeIPLWcHdJL_aJRQTf_a``PLW†•IQr9_aJRQqWYMµŒ ]AM;žƒ_†F˜JLW˜wYW˜…™NPdJL7;:PQrM JR7; HdQq9<`prQr¢(f3_JRQqWYMQTHtJRW˜QqM-JRPLWGeNbf;z_

HLf:_YprQqMwQqMžJL7;]„-QTHLfWcHRQrJKF¨_YMbe<…™PLQqfJRQqWYMzf:W-;:žfQq;:M-JLH3Œ0œŠJHR;:;39<HVJRW—(;]QqM_Yf3fWcPLe_YMbf;ƒQrJR7zJR7;]`7GFIHKˆ

QTf:_Ypª—b_Yf“GwYPLWYNbMbe QrMbe;:;3ezJR7b;]„GQTHRf:WcHRQ’JKFžf:WG;žfQq;:M-Jf:_aM —i;]PL;:pT_JL;3ežJRW¨JR7;]JLNPR—bNpr;3M-J„GQTHRf:WcHRQ’JKFWY…

JL7;]›bWƒ ƒ7QTf7 QqH‘QqM`b_YPdJLQqf:NpT_aP‘PR;3pq_aJR;3ezJRWJR7b;]wc;:Wc9¨;:JRPLQqf]e_aJL_YH_YMbeJL7-N(H‘JRW¨JR7;#HR7b_apqpqWƒ”ƒ‘_aJR;:P

_cHRHRN9<`IJLQrWcMµŒ =?;:„c;:PRJR7;3pr;2HRH JL7;:PL;QTH‹9_aMGFƒ_†FIH JRWeI;:¢bM;JL7QTHVHLf:_apqQqMw  Q’JVQTH‡`iWcHLHRQr—pq;JRWfWYM(HdQTeI;:P

_YMQTHdWYJRPLWY`QTf„GQTHRf:WcHRQ’JKF¦fWG;:f:Qr;3M-JLH©_aMben_apTHdWJRW QqMcJLPRWIeIN(f;eIQi;3PR;3McJ©`iWƒ§;3PLHAWa…

ˆ?Hd;3; _lbŠŒ¤;3PR;

ƒ‘;#Hd7(_apqpµHdN`b`(W-Hd;]JR7(_Jƒ‘;©_YPR;©QqM JR7;©…™WcprpqWƒQrMbwPL;:wcQr9<;

/ 2

_aMbe

/

Q 4

kIŒ

5 a 7

#266# U "$#

‚ ;zf:WYMbHRQqe;:P©JR7b;zHdFIHdJR;39

4

kŒa 7ˆ 4 kŒ

5 s 7

ƒQrJR7 JR7b;ž`b_aPRJRQTfNbpq_YP©…™WYPL9CWa…JL7;„GQqHLfW-HdQrJKFn_YMben…™PRQTf JLQrWcM

f:WG;žfQq;:M-JLH

4

kIŒ

5 a 7

_aMbenWYMb;PL;J_aQqMbH?WcMpqFJR7;<£:;3PRWWYPeI;:PAJR;3PR9H3Œ ‚ ;Wc—IJL_YQrM–JR7;<„c;:PLFf:pq_cHRHRQTf:_ap

7GFIeIPLWcHdJL_aJRQTf?9<WIeI;:p

&

&.<+

&

& / 02 4

kIŒ

5 m 7

&

&* + & 9

&. + &

& +

&

&. /

& 9

&

9 2 4

kIŒkas

7

&

& / N

;2 4

kIŒk

5c7

ƒ7b;:PL;

*

032 .

2 0

' S* 2 . U Q

[…JR;3PtHdQq9<`pqQ’…™FGQqMw¦JR7b;<JR;:PL9H#Wa…fWcPRPL;3HR`iWYMbeIQqMw WYPeI;:P2ŸJL7;ž—iWYNMbeb_aPLF–fWcMbeIQrJRQqWYMbH

4

kIŒ

55c7

ˆ 4 kIŒ

5 u 7

—i;3f:WY9<;

S* 2 . 2 0 U / 032 4

kIŒkYk

7

&

&

S* 2 . 2 0 U / S* 2 . 2 0 U 2 &

&

S* 2. 2 ' S* 2 . UU

/1032 4

kIŒkau

7

S* 2. 2 ' S* 2 . U U / 0 Q 4

kIŒkl

7

67b; HRFIHKJL;:9 QTH¨HdJRQqpqp_YHLHdWIfQT_JL;3eƒQrJR7 JR7b;“GQqM;:9_JLQqf —iWYNM(e_aPLFfWYM(eIQ’JLQrWcM

4

kIŒ 7 Œ 6 _Y“-QqMwnQrM-JRW

_cf:f:WYNM-J‘JR7;#`PL;3HLHRNPR;©—iWYNM(e_aPLFžfWcMbeIQrJRQqWYMWYMJR7;©…™PL;:;#HRNPR…´_Yf;

4

kIŒkl

7 Ÿ 4 kŒk

5"7

QqH;2S-NQr„_Ypr;3McJJLW

S* 2 . 2 U / ; S' S* 2 . U N U Q 4

kIŒkYj

7

œŠJAQqH?MWƒ¥`iWcHLHRQr—pq;#JLWzf7;3f“zJR7b_aJJR7b;t7GFIeIPRW-HKJ_JLQqf©_a`b`PRW†•IQq9_JRQqWYMHR7b_aPL;3HƒQrJR7JL7;…™Nbprp =?_†„GQr;3Pdˆ

DGJRWY“c;3H<;2ScN(_JRQqWYM&WYM ;3M;:PLwYF$HdJRPLNbfJRNPL;YŒ =A_a9<;:pqF f:WYMbHRQqe;:PLQrMwnJR7b;¦7-FIeIPLWcHdJL_aJRQTf ;:M;3PRwcF

4

kŒ

5 7 Ÿ

5

Q§¬

~R'*)<' S 03U.98^ " :4# 03!

. ?BA 1C FG?HA A X9F

&

&*

+ &

&.

S + U

/ N 9 S* 2. 2 0 U N

&

& 9

Q

Z . 67b;#`PRWGWY… NbHR;3HWcM f:pq_cHRHRQqf3_apfWc9<`NIJL_aJRQqWYMbH3Œ

67bQqH‘7GFIeIPRW-HKJ_JLQqf9<WIeI;3p(ˆ‹WcP‘HRWY9<;?„_YPRQT_aM-JLH§ƒQrJR7z7bWYPLQr£3WYM-JL_Yp(„GQTHRf:WcHRQ’JKF¨WYP‘WaJL7;:PHR`(;2fQr¢(fJL;:PL9H‡ˆ

QTH„Y;3PRFzNbHd;2ezQqMwY;3WY`7GFIHdQTf:_Ypb›(WƒHHKJLNbeIQq;3H3Œ ² WYP…™NbPdJL7;:PPL;…™;3PR;3Mbf;2HHd;3;A…™WcP;•_a9<`pq; _l(Ÿ

5

sIŸ

5

mcbŠŒ

 : <  

‚ ;<f:_aMnMWƒ eI;3PRQq„Y;tJR7;žDI_YQrM-JdˆŠ‰‹;:M(_aM-J?JKFG`i;9<WIeI;:pTHAˆHd;3; ² QqwYNPL;

5 Œ5

Œ©6 WeIWzJL7QTHAƒ‘;¨`(;3Pd…™WcPR9

_YM _cHdFG9<`IJLWaJRQTf©_aMb_YprFIHRQqH‘WY… JL7;#7GFGePRW-HKJ_JRQTfA9¨WIeI;3pqHWc—IJL_YQrMb;3e QrMJR7b;#`PR;3„GQrWcNbHHd;2f JLQrWcMµŒ

œ¾M _5

j"b0JR7;<_aNJR7WcPLHA`PL;3HR;:M-JA7WƒJLW`bPRWc`(;3PRpqF¦eI;:PLQr„c;t9<WYMbWYpT_†FY;:P]DI_aQqM-JdˆŠ‰‹;:Mb_YMcJ]9<WGe;:pTH:Œ#œ¾M-JRPLWaˆ

eNbfQqMw_aM¦_†„Y;3PL_YwY;2e<„Y;:pqWIfQrJKF

- S* 2. U / [

' S* 2 . U

S* 2 . 2 U

Q

_YMbe eI;:`(_aPRJRQqMwPR;2Hd`i;3fJRQq„Y;:pqF˜…™PLWY9 JR7;ž7GFIeIPLWcHdJL_aJRQTf9<WIeI;3p

4

kŒ

5 m 7ˆ 4 kŒk

5"7

_aMbe5…™PLWY9 JR7;ž„GQTHRf:WYNbH

7GFIeIPLWcHdJL_aJRQTf¦9<WGe;:p

4

kIŒ

5 l 7ˆ 4 kŒ

5 7

ˆ<NMbeI;3PJL7;–7GFG`iWaJR7b;3HRQqH

4

kŒ

5 a 7

ŸJR7;3F e;3eINbf:;–JL7;nf:pq_cHRHRQTf:_ap

D_aQqMcJRˆŽ‰‹;3Mb_aM-JHRFGHdJR;39 ƒQrJR7 …™PRQTf JLQrWcM

&('

&*

+

&('-

&.

/ 02 4

uŒ

5"7

&('-

&* + &

&.

'-:9

+ ; ' 9

E / N - Q 4

uŒk 7

_YMbe _„GQTHRf:WYNbHD_aQqMcJRˆŽ‰‹;3Mb_aM-JHRFGHdJR;39 ƒQrJR7 …™PRQTf JLQrWcM

&('

&* +

&'-

&. / 02 4

uŒu 7

&('-

&* + &

&.

'- 9 + ; ' 9

E / N

[

+

-

+ L &

&.

' & -

&. Q 4

uŒl 7

_cH…™WYPL9<_Yp_YHRFG9¨`JRWaJLQqf_Y``PLW†•GQq9_JLQrWcMbHtQqM S U

ˆPL;3HR`(;2f JRQq„Y;3prF S 9 U

ˆtWY…?JR7b;=A_†„GQr;3PdˆKD-JRWc“Y;2H

;2S-Nb_JLQrWcMbH:Œ‹@¦WcPR;©eI;:JL_aQqpTH_a—iWYNIJJL7;3HR;eI;:PLQq„†_aJRQqWYMbHf3_aM—(;]…™WcNMbe QrM<_

5

uIŸ

5

j"bŠŒ

67b;3HR;JKƒ‘W˜DI_aQqM-JdˆŠ‰‹;:Mb_YM-J©9<WIeI;3pqH#_YPR;<_YHLHdWIf:Qq_aJR;3enƒQrJR7 _¦eIQqHLHRQr`b_aJRQqWYMnWa…;:M;3PRwcFYŒ¨œ¾MbeI;:;2e5QrM-JRPLWaˆ

eNbfQqMw

'

/

'

+ '

/ '- 9

E + ; ' 9 E 2

5I5

ƒ‘;©;3HdJL_Y—prQTHR7z…™WcPJR7;f:pq_cHRHRQqf3_apµDI_YQrM-JdˆŠ‰‹;:M(_aM-J9¨WIeI;3p

4

ubŒ

5c7

ˆ 4 ubŒk 7

&('

&* + &

&.

-

'

+ ; ' 9

E ?/

N - 9 2

_YMbez…™WYPJR7b;#„-QTHLfWYN(HWYM;2H

4

uŒu 7ˆ 4 uŒl 7

&('

&* + &

&.

-

'

+ ; ' 9

E N L &

&.

'-

& -

&. / N

[ +

- 9 N L ' & -

&. 9 Q

¬ ~I}&%$),+ m303 10

R

? P 1C:FG? PAYF " !

?7

303 0 !

& ?BA 1C FG?BA AC:F@803# 303#

.mk # 1 10 m

/ 0 = !mm! 8V m :03"

S* 2. 2 U / - S* 2 . U 2 S* 2 . 2 U / N

&(-

&. 2 S* 2 . 2 U /d;

S' S* 2. U N U 2

80 S' 2 - U S* 2. U

K.10 #

m ( #

& ?P 1C:FG?PAYF =

"B## 303 m

J.T10 "

m

H !I

= !Im!

S

'

2

- U S* 2 . U . 303

"

m

?P<PGFG?PX9FK8^ 10 6.mk # !I

.R10 "

0! & ?BA 1C;XOFG?BA 1C I5F . ! H . #k O

H303U.O8^ "Z:4#

&

&.

'

& -

&. ?/

'

& 9 -

&.

9 Q

‚ ;©MWƒ eI;3PRQq„Y;©_t9<WcPR;A`PR;2fQTHd;©_a`b`PRW†•IQq9_JRQqWYM*Œ ! HR`(;2fQT_apqprFzƒ§;]ƒQqHR7žJLW“c;:;3`HRWY9<;AQqMI…™WcPR9_JLQrWcM

WcMzJL7;#„Y;3PdJLQqf3_apHdJRPLNbf JLNPR;©WY… JL7;#7WYPLQq£:WYM-J_apª„Y;3prWIfQrJKFž_YH9<WYJRQq„†_aJR;2eQqM JR7;#QqM-JRPLWIeINbf JLQrWcMµŒ

" ## "* 6UR" U !"$#

‚ ;f:WYMbHRQTeI;:PJL7;t7GFIeIPLWcHdJL_aJRQTf©9¨WIeI;3p

4

kIŒ

5 m 7ˆ 4 kIŒk

5c7

Œ ‚ ;¢bPHKJAQrM-JRPLWIeINbf:;t_žeIQTHRf:PR;:JRQq£3_aJRQqWYM¦QqM¦JR7;

„_YPRQT_a—pq; ˆHd;3; ² QrwcNPR;ubŒ

5

Œ67;:M…™WcPAHRWY9<; \ ƒ‘;teI;¢(M; \ QrM-JL;:PL9¨;2eIQT_JR;#ƒ_JL;:P7b;:QqwY7-JLH

) S* 2 . U HRNbf7zJL7b_J

0/

S* 2. U C S* 2 . U 9 S* 2 . U QQQ

> P C S* 2. U > S* 2 . U / ' S* 2 . U Q

‚ ;f7b_YPL_cf JR;3PRQq£:;:\ pT_†FY;3PLH§JR7bPRWcNwY7JR7;e;¢bMQrJRQqWYM Wa… JR7b;#QrMbeQqf3_JRWcP…™NMbfJRQqWYMbH

) S* 2 . 2 U

W [ 2

\ ) S* 2 . 2 U /

[ "! )P

C S* 2 . U ) S* 2. U

0$#&%('*),+(-/.102)2

JL7b_J?_YPR;©_Ye„Y;3fJR;2ez—GFJL7;©›bWƒ

& )

&* + & )

&. + & )

& /?0

Q 4

uŒj 7

5

River bottom Free surface

' 9 S* 2 . U

' C S* 2 . U '

S* 2 . U

' S* 2 . U

S* 2 . U

S* 2 . U

9 S* 2 . U

C S* 2 . U S* 2 . U

S. U

² QrwcNPL;©ubŒ

5

‹[9tNprJRQqpq_†Fc;:P_a`b`PRW-_Yf7

67b;:M…™WYP;2_Yf7zpT_†FY;3Pƒ§;#eI;:¢bM;#QrJLHƒ_JL;:P7;:QqwY7-J ') S* 2 . U —GF

W [ 2

\ ' ) S* 2 . U / ) S* 2 . U N )3P

C S* 2 . U 2

_YMbe _aM_†„Y;:P_awc;3e„c;:pqWIfQrJRQqF -) S* 2 . U

W [ 2

\

- ) S* 2 . U / [

') S* 2 . U

S* 2 . 2 U

Q

‚ ;fpT_aQq9 JL7b_J

¬G° } ¬ $),+TS 03 #

&T8^ 30 .mk # !IVm! #

&('

C

&* +

&('

C - C

&. / 032 4

uŒ

7

&('

C - C

&* + &

&.

' C -:9

C

+<;

' C &

&.

>

@IADC ' @ /

EL

- 9 N - C

' 9 + ' C N - C 2 4

uΠ7

&(' )

&*+

&(' ) - )

&. / 032 4

uŒa 7

&(')-)

&* + &

&.

')-:9

) + ; ')

&

&.

>

@IA C ' @ / EL

-)IM

C N -)

' )IM

C + ' ) N EL

-V)

N

-V)P

C

' ) + ' )3P

C 2 4

uŒm 7

! W / E

QQQ\ N [

5

&('

>

&* +

&'

> - >

&. /?032 4

uŒ

5 s 7

&('

> - >

&*

+ &

&.

' >

-:9

> + ; ' > &

&.

>

@IADC ' @ / N

EL

- > N - > P C

' > + ' > P C 2 4

uŒ

55c7

.m .H

X ;7I H S U m ! 8^ 10 !#k m!

"

.:10/0! H !I ! 10 ,. 2.:303 " #kI

:4#

Z . ‚ ;Wc—IJL_YQrM<—GFQqM-JR;3wYP_JRQqWYM¨Wa…

4

uŒj 7

QqM<JR7;„c;:PRJRQTf:_YpbeIQqPR;2f JRQqWYM \ 9_YHLH‹—b_apT_aM(f;;3S-Nb_aJRQqWYMbH

&(' )

&*+

&

&.

/ 032 4

uŒ

5 k 7

_YMbe\ “GQrMb;:9_JLQqf©—iWYNM(e_aPLFzfWcMbeIQrJRQqWYMbH_aJJR7;#QqM-JR;:PR…´_Yf:;3H

& )

&* + S* 2 . 2 / ) S* 2 . U U & )

&.

N S* 2 . 2 / ) S* 2 . U U

/10

Q 4

uŒ

5 u 7

>AHdQqMwJR7;<_†„Y;3PL_YwY;2e„Y;:pqWIfQrJRQq;3H -) ƒ§;¨ƒPLQ’JL;tJL7;<9<_cHRHA—b_Ypq_YMbf;¨;3S-Nb_aJRQqWYMbH

4

ubŒ

5 k 7

QqMnJR7;DI_aQqM-Jdˆ

‰‹;3Mb_aM-J…™WYPL9_apqQqHR9

&(')

&*+

&

&.

')-)

/?0

Q 4

uŒ

5 l 7

‚ ;¨MWƒgQrM-JL;:wYP_JL;tWcM˜;3_cf7˜pq_†Fc;:P?JR7;¨;3S-Nb_aJRQqWYM

4

kIŒkas

7

Œ#67;t…™WcPR9³WY…‡JR7;¨7GFIeIPRW-HKJ_JLQqf#`bPR;2HRHRNPL;

4

kIŒkYj

7

pq;3_ceH‘JRW

&

&.

/

;

&'

&.

/ ;

&'

&.

/ ; ')

&('

&.X2

_YMbeJL7GNbH¨ƒ§;WY—IJ_aQqM ˆ©JL_Y“-QqMwnQrM-JRW _Yf3fWcNM-JtJR7b;“GQqM;:9_JLQqf—iWYNMbeb_aPLF f:WYMbeQ’JLQrWcMbH

4

uŒ

5 u 7

ˆ\

9<Wc9¨;3M-JRN9;3S-Nb_JLQrWcMbH

&

&*

+ &

&.

9

+<;

')

&('

&. /

&

&

S ) S* 2. U U N

&

&

S )P

C S* 2. U U Q 4 uŒ

5 j 7

=?Wƒš;2ScN(_JRQqWYM(H

4

kIŒkas

7

_YMbe

4

kŒkYu

7

Qr9<`pqQr;

& 9

&

9 / S U 2 &

&

[ A / S U 2 &

&

[ A

/102

_YMbezJR7;3PR;:…™WYPL;

S* 2 . 2 U / - S* 2 . U + S U

0

' S* 2 . U 2 4

uŒ

5 7

ƒ7(_JQr9<`pqQq;3H

[

') S* 2 . U 9 S* 2 . 2 U

/ -

9) S* 2 . U + S U Q

5

¤?;:Mbf:;©ƒ§;f3_aM ƒPRQrJR;]JR7b;#9¨Wc9<;:M-JRN9;3S-Nb_aJRQqWYMbH

4

uŒ

5 j 7

QqMJL7;tDI_YQrM-JdˆŠ‰‹;:M(_aM-J…™WYPL9<_YprQTHR9

&

&*

')-V)

+ &

&.

')-:9

) + ; ')

&('

&. /

&

&

S ) S* 2 . U U N

&

&

S )P

C S* 2 . U U + S U Q 4 uŒ

5 7

‚ ;eIPLWY`nJR7; S U _YMben9tNprJRQq`pqF

4

ubŒ

5 l 7 _aMbe

4

ubŒ

5 7

—GF - 9 ¥QrM5WYPeI;3PAJLW¦PR;2fW„c;:PJL7;„_aPLQ’ˆ

_Y—pq;3HAƒQ’JL7neIQq9<;:MbHRQrWcMµŒ ² QqMb_YprpqF¦ƒ§;¨_a``bprF¦_¢(MQ’JL;¨eQª;:PL;:Mbf:;9<;:JR7WIe¦QqM˜JL7;„c;:PRJRQTf:_Yp eIQqPR;2f JLQrWcM

JLW]JR7;PLQrwc7-J 7b_YMbe<HdQTeI;Wa…(JR7;9<Wc9¨;3M-JRN9 ;3S-Nb_JLQrWcMbH ƒ7;:M<QrJ‡QqH‹f:WYMbf:;:PLM;3etƒQ’JL7_aM¨QrM-JR;3Pd…´_cf;cŒ

"*"$@W" # #2 "$ 6U " UU "$

67b;]9NprJRQqpq_†Fc;:P?DI_YQrM-JdˆŠ‰‹;:M(_aM-JHdFIHdJR;:9H`PL;3HR;:M-JHVJKƒ‘W¨…™NMbe_Y9<;:M-JL_Yp*HKJ_a—QqpqQ’JKFž`bPRWc`(;3PdJLQr;2H:Œ

} °V° {´| ° *) + S

03 # #

& ?P APGFG? PAGF

303 3

"

. 10 8V# 03# "&0 : m :0 #& =

m!R8^ 10 ##"Z :4

H

!I " ' /

=1>

@IADC ' @

!H "! # " 303 "

/ ) /

')- 9)

E ' + ;

')

E 2

8V #

0 10.O8^ "Z54#

&

&*

'

+ &

&.

' - ) ) + ' -

)^;

')

E / N

-:9

C N EL

>

9 S ()

N

-)3P

C U9

')

+

')P

C Q

Z .

67;#eI;3„Y;3prWc`9<;:M-J‘WY…*JL7;©`PRWGWY… QTH§„c;:PLFžfpT_YHLHdQTf:_YpiHRQqMbf;©QrJ9<Qq9<Qqf“IHƒ7b_aJQTHeIWcM;?…™WcPJR7;

f:pq_cHRHRQTf:_ap‡DI_YQrM-JRˆŽ‰‹;3Mb_aM-JHdFIHdJR;39 Œž67b;M;:ƒ eIQTHRHRQr`(_JRQq„Y;¨JL;:PL9CQqMnJL7;;:Mb;:PLwYF¦;3S-Nb_aJRQqWYM5QqHteIN;<JRW

JL7;`PL;3HR;:Mbf:;<Wa…VJL7;žHL_a9<;<„-QTHLfWYN(H]…™PRQTf JLQrWcMnJL;:PL9C_J#JL7;QqMcJL;:PR…´_Yf:; W N [ E QqM5JR7;9<WY9<;3McJLN9

;2S-Nb_JLQrWcMbH§…™WYPJR7b;#pq_†Fc;:PH W$_aMbe W N [YŒ

67b;ž9tNprJRQqpT_†FY;:PtD_aQqMcJRˆŽ‰‹;3Mb_aM-JtHRFIHKJL;:9H;3MbHdNbPR;2H_apTHdW˜HRWY9<;PL;:pT_JLQrWcMbH©ƒQ’JL7JR7b;zfpT_YHLHdQTf:_YpVDI_aQqM-Jdˆ

‰‹;3Mb_aM-JHRFGHdJR;39

} °V° {´| ° *)1' S

03 #

?BP APYFcG? P AGF

#c

10

mm!

I O :4# k

') S

02

. U / 2

-) S

032 . U

/10

W / [ QQQ\ 4

uŒ

5 a 7

80 10 H

.mk d 303 m 303

m^

?BP 1C:FG?PA;F

" !

. 303 #

= !mm! . 8V :03"

:303X m!

- ) S* /

02

. U / - S. U ! ' ) S* / 02 . U

/

) S. U ? :0<303 = ) / [ F #

"

W / [ QQQ\ ' ) S* 2 . U

/

) ' S* 2 . U 2

-V) S* 2 . U / - S* 2 . U 2

80 /?30 F?7;F

H2.O303

B

& 8^ 30

S 2 - U S* 2 . U

5

Z .

U‡PRWGWY…´HA_YPR;Wc—G„-QqWYN(H:Œ]=WaJLQqf:;tJL7b_JAJR7;¨`PL;3HR;:PL„_JRQqWYM¦WY…VHdWc9<;HdJR;3_ceIF¦HKJ_JL;3HAQqH]_…™NM(e_ˆ

9<;3McJ_ap Hd`i;3f:Q’¢(f`PRWc`(;3PdJKF Wa…0JL7;¨DI_YQrM-JRˆŽ‰‹;3Mb_aM-JJKFG`(;9<WGe;:pTH_uŸ

5

acbŠŒ?67;tHR;3f:WYMbe`iWYQqM-J?QqH?QqHAQqM

_YwYPL;:;39¨;3M-J?ƒQ’JL7n^;39_aPL“ubŒ

5

HdQqMbf:;tJL7;9NprJRQqpq_†Fc;:PAHRFIHKJL;:9 QTHA_YM–QqM-JR;:PL9<;3eIQT_JL;t9<WIeI;:p0—i;JKƒ‘;:;3M

JL7;#7GFIeIPRW-HKJ_JLQqf]HdFIHdJR;39 _aMbeJL7;#fpT_YHLHdQTf:_Yp*DI_aQqM-JdˆŠ‰‹;:Mb_YMcJHRFIHKJL;:9Œ

> "*K229 U "$ # 9 " #2#2*"*

!6UR" U "

67b;A9Np’JLQrpT_†FY;3PD_aQqMcJRˆŽ‰‹;3Mb_aM-JHdFIHdJR;39

4

uŒ

7ˆ 4 uŒ

55c7

7b_YH§JKƒ‘W9_YQrM eIPL_†ƒ—(_Yf“IH:Œ ² QrPHKJ_YH‘Wc``(W-Hd;2e

JLW©JR7;9<WcMWYpT_†FY;3P‹D_aQqMcJRˆŽ‰‹;3Mb_aM-J‹9<WIeI;:pTH0JR7b;`PL;3HLHdNPL;§JR;:PL9H‡_aPL;MWaJ‡QqMž_©fWcMbHd;3PR„_aJRQq„Y;§…™WYPL9 _aM(e

JL7GNbH‘JR7;3QrPe;¢bMQrJRQqWYM QTHMWaJWY—G„GQrWcNbH‘ƒ7;:MHR7WIf“GH‘WIf:f:NP2Œ0=?;:„c;:PRJR7;3pr;2HRH§9_aMGFžƒ‘WYPL“GH§7b_†„c;A—i;:;3M

e;:„YWYJR;2eJRW¨JL7QqH?ScNb;3HdJRQqWYM_aMbeHdWc9¨;©PL;3f:;:M-Jƒ§WcPR“IH§`PRWc`(W-Hd;]MGN9<;:PLQqf3_apªƒ_†FGH§JLW<f7bW-W-Hd;©_ KPLQrwc7-J%

HRWYpqNIJLQrWcMzˆVHR;:; _

55

b*WcP _ab*QrM JL7;fWcMcJL;•GJWa… JR7;tDI_YQrM-JRˆŽ‰‹;3Mb_aM-JfWc9¨`bNIJL_aJRQqWYMbH3Œ

67b;‘HR;3f:WYMbe#`iWYQqM-J‹QqH 9tN(f7t9<WYPL;V`PLWY—bpr;39<_aJRQTfaŒ ² WYP JL7;‘HRQq9¨`bprQTfQrJKF#Wa…JL7;`NPL`(W-Hd;§pr;:J‹NbH0fWYM(HdQTeI;:P

JL7;§JKƒ§WApq_†Fc;:PHµ„Y;3PLHRQqWYMWY…JR7;HRFGHdJR;39

4

ubŒ

7 ˆ 4 uŒ

55c7

Œ ² WYpqprWƒQqMw _ cbIƒ‘;§ƒPLQrJR;§JR7;§JKƒ§WApq_†Fc;:PH*HRFIHKJL;:9

NbMbeI;:PJL7;fWc9<`b_YfJ‘…™WcPR9

&

&* + S U

&

&. /

S U 2 4

uŒ

5 m 7

ƒ7b;:PL;

/ ' C

' C - C

' 9

' 9 - 9

2 S U / 0

EL

- 9 N - C

' 9 + ' C N - C

0

EL

- C N - 9

' 9 + ' C

2

S U / 0 [ 0 0

N - 9

C

+<;

' C E - C ; ' C 0

0 0 0 [

; ' 9 0 N - 9

9 + ; ' 9 E - 9

Q

[AH‘JR7;#„c;:PRJRQTf:_Yp`PRWY¢bpq;]WY…*JL7;#„Y;3prWIfQrJKFžQTHwYQq„Y;3M—GFJL7;#PR;3pq_aJRQqWYM

4

ubŒ

5 7

ƒ§;#7b_†„c;

-V)

S* 2 . U / - S* 2 . U + S U W / [ 2 E Q 4

uŒkas

7

} °V° {´| ° *) !#303 ?PAGF 303

#c 8V

( #

?P1C F

0 3# m < c# 80 - C S* 2 . U

/ - 9 S* 2 . U 10 "B c

2.H303

Hk 7 S U 03c c

0 " " k k

Z . 67b;#;:QqwY;:MG„_apqN;2H§WY…*JL7;#9_JRPLQr•

S U _aPL;#HdWcprNIJLQrWcMbH‘Wa…

9 N E - C + -:9

C N ; ' C

9 N E - 9 + -:9

9 N ; ' 9 /d;

9 ' C ' 9 Q

5

67b;:PL;…™WYPL;NM(eI;:PAJR7;¨_YHLHdNb9¨`JRQqWYM

4

uŒkas

7

¢bPLHdJAWcPLe;:PA_Y``PLW†•IQr9_JLQrWcMbHQqM

- C N - 9

Wa…‹JL7;;3Qrwc;:MIˆ

„_YprN;2H_aPL;

\

/

- Y

; S' C + ' 9 U 2 4

uŒk

5c7

/ - Y - C N - 9

E [ N ' C N ' 9

' C + ' 9 9 2 4

uŒkYk

7

ƒ7b;:PL;

- / ' C - C + ' 9 - 9

' C + ' 9 2 - / ' C - 9 + ' 9 - C

' C + ' 9 Q

=?WaJLQqf:;?JL7b_JQqMžJR7b;©f3_YHR;

- C / - 9 / -

JR7;©;:QqwY;3MG„†_YprNb;3H§—(;2fWY9<;]PR;2_api—NIJJR7b;©HRFIHKJL;:9 QTHHKJLQrpqpMWaJ

7GFG`i;:PL—(WcprQTf©HdQqMbf; - QTH_¨eWYN—pq;#;:QqwY;3MG„†_YprNb;]_cHRHRWIfQT_JR;2ežƒQrJR7¦_<WYMb;©eQr9<;:M(HdQqWYMb_Yp;:QqwY;3MbHd`(_Yf;cŒ

67b;–;:QqwY;3MG„†_YprNb;3H

f7b_aP_YfJR;:PLQq£:;JR7b;–—b_YPRWIfpqQqMQqf`(_aPRJWa…©JL7;–›(Wƒ³_YMbe”_aPL;˜PL;:pT_JL;3e&JRW$JR7;

f:;:pq;:PLQ’JKF–Wa…_¦HdQqwYM(_ap‡_JtJR7;QqM-JR;:PR…´_Yf:;—(;:JKƒ§;3;:M5JR7b;JKƒ§W¦pT_†FY;3PLH _ Ÿ

5

cbŠŒ 67;3QrPtQr9_awcQrM(_aPLF˜`b_aPRJLH

_YPR;#PL;:pT_JL;3eJRWzJR7;t;•IQqHdJR;3Mbf;tWa…‹QrM(HKJ_a—QqprQrJRQq;3HA_JJL7;tQqM-JR;3Pd…´_cf;#ƒ7QTf7¦pr;2_Ye JLWz_9<Q’•IQqMwž—i;JKƒ‘;:;3M

JL7;#JKƒ‘WžpT_†FY;:PH3Œ‘¤?;:Mbf:;Q’J]QqH?Mb_JLNP_apµJL7b_J?JR7;—b_aPLWIfpqQrMQTf©;:QqwY;3MG„†_YprNb;3HWY…‹JR7;9NprJRQqpq_†Fc;:P?HRFIHKJL;:9

7(_†„Y;_ MWcM–„_aMbQqHR7QqMwQq9_awcQrMb_YPRF`b_YPdJ#HRQrMbf:;¨JR7;<7GFG`(WYJR7;2HdQTH

4

uŒj 7

_a—iWYNJ]JL7;<Qr9<9<QTHRf:Qr—QqpqQ’JKF˜WY…

JL7;pT_†FY;3PLH0QTH‡MWYJ‡prQqM“c;3e¨JRWJR7;`b7-FIHRQqf3_ap—(_Yf“GwYPLWYNM(eŒ Z‘NIJVQrJ‡…™WcprpqWƒH‡_apTHRW©JR7b_aJ‡JR7;?`PR;2fQTHd;„_YprN;

WY…

QTH§MbWaJPR;3pr;3„_aM-JHRQqMbf;]ƒ§;#_YPR;]MWaJQqMcJL;:PL;3HdJR;2ezQqMzJL7;©;:„cWYpqNIJRQqWYMWa…µJL7;3HR;K„GQqPdJLNb_apAQqMcJL;:PLMb_ap

QqM-JR;3Pd…´_cf;3H3Œ

67b;©JKƒ§WWYJR7;3P;:QqwY;3M-„_YprN;2H

\

f7b_YPL_cf JR;3PRQq£:;]JL7;—b_YPRWYJRPLWY`QTf]`b_aPRJWa… JR7;›bWƒ _YMbe¦_aPL;#PL;:pT_JR;2e

JLW©JR7;]f;3pr;3PRQrJKFWa…µ_#HRQrwcMb_ap(_J‡JL7;…™PL;:;HRNPd…´_cf;cŒ 67;3PR;:…™WYPL;JR7;?9tNprJRQqpq_†Fc;:PVDI_YQrM-JRˆŽ‰‹;3Mb_aM-J‡9<WIeI;:p(QqH

QqM¦_awcPR;3;:9<;:M-JƒQ’JL7JL7;fpT_YHLHRQqf3_apPL;3HRNp’JH_a—iWYNIJJL7;#9<WYMWcpq_†Fc;:PDI_aQqM-JdˆŠ‰‹;:Mb_YM-J9<WIeI;3pqH…™WcPƒ7QTf7

JL7;#;:QqwY;3MG„†_YprNb;3H§ˆV_aMbezJR7GNbHJR7;f:;:pq;:PLQ’JLQr;2HWa… _<HRQrwcMb_apµ_aJJR7;©…™PL;:;#HRNPd…´_cf;]ˆV_aPL;

-

Y

; '

ŒVDGQqMbf;

JL7;3HR;f;3pr;3PRQrJRQq;3H‹_YPR;ƒ‘;:pqp’ˆŠ“GMWƒMJLW]—i;QqM¨wcWGWGe¨_Yf3fWcPLe_YMbf;‘ƒQrJR7¨JR7;;:•I`(;3PRQq;:Mbf:;3H3Ÿ†JR7;_YwYPL;:;39¨;3M-J

—i;JKƒ‘;:;3MJR7b;?JKƒ‘Wt9<WIeI;:pTH‘_aMbe<JL7GNbH§JR7;]„_apqN;?WY…

\

QTH§_YMž;3HLHR;:M-JRQT_api`PLWY`i;:PRJKFWY…µJR7;A9tNprJRQqpT_†FY;:P

HRFIHKJL;:9Œ

"*K " " #2#2*"*!6UR" "$#

6 W W„Y;3PLf:WY9<;zJL7;˜eIQržfNprJRQq;3HeIN;JRW JL7;3HR;¦MWcM&f:WYMbHR;:PL„_JRQq„Y;¦HdJRPLNbf JLNPL;_YMbe&MWYM 7GFG`(;3PR—iWYpqQTf

M(_JRNbPR;©Wa… JR7;#9NprJRQqpq_†Fc;:PDI_aQqM-JdˆŠ‰‹;:Mb_YM-JHRFIHKJL;:9

4

uŒ

7 ˆ 4 uŒ

55"7

_aM(ežJL7;:PL;…™WcPR;]JRW_†„cWYQTez_a`b`b_aPLQ’JLQrWcM

WY…QrMbHdJL_Y—QqprQrJRQq;3HtQrM JL7;žMGN9<;:PLQqf3_apVHRQq9tNpT_JLQrWcMbHH_ cb§ƒ‘;ž`PLWY`iWcHR;<JRW˜f:WYMbHRQqe;:Pt_˜HRpqQrwc7cJLprF5eIQi;3PR;3McJ

HRFIHKJL;:9

&('

C

&* +

&('

C - C

&. /?032 4

uŒkau

7

&('

C - C

&* + &

&.

' C - 9

C

+<;

' C G

=1>

@IADC ' @ J

E

5

/ ; G = >

@IADC ' @ J 9

E &

&.

8 ' C

= >

@IADC ' @ F +

EL

- 9 N - C

' 9 + ' C N - C 2 4

uŒkl

7

&(' )

&*

+

&(' ) - )

&.

/?032 4

uŒkYj

7

&(')-)

&* + &

&.

')- 9

) + ; ')

G

=?>

@BADC ' @ J

E

/ ; G = >

@IADC ' @ J 9

E &

&.

8

')

= >

@IADC ' @ F +

EL

-)IM

C N -)

'

)IM

C + ' ) N -)

N

-)3P

C

' ) + '

)3P

C 4

uŒk 7

! W / E

QQQ\ N [ 2

&('

>

&* +

&('

> - >

&. /1032 4

uŒk 7

&('

> - >

&* + &

&.

' > - 9

> + ; ' > G = >

@IADC ' @ J

E

/ ; G

=K>

@IADC ' @ J 9

E &

&.

8 ' >

= >

@IADC ' @ F N

EL

- > N - > P C

' > + ' > P C Q 4

uŒka

7

} °V° {´| ° *) !# 303

? P1CI:F 30

;8

#<2.:303

H

& 0

303

.O8^ " 3#k

?Z CGF S 0

# m! H # " 10k "&0 03!

! "#H

0 3# m

?Z A;F S 03 m # "I#

\

? PAC:F

!# ;7I

-

CRN - 9

.:8V 2.

"B c

?Z PYF

8V# 0 "&0 V!I 10 m

#c X

303 k "&0 $03!

!

H R10$

0Z T10 ( #O;7I '

) / ' \ - ) / -

?Z X9F

S

0

3303R

.O303 54#

303D!

k m 10 &c 2. 303 8V 03# "&0

! 2.R303X!

50"B m :0 #

!# 7I . 10#

m

,.m !Hk "&0 03!

!I

!# m!

3 #

Z .

[?H…™WYP©JR7b;žMWYMf:WYMbHR;:PL„_JRQq„Y;<9<WIeI;:pVƒ§;žf:WYMbHRQqe;:P©…™WYPHdQq9<`prQTfQrJKF˜JL7;JKƒ‘W¦pq_†Fc;:PH©f:_YHR;YŒ

¤?;:Mbf:;]JR7;#;3Qrwc;:MG„_apqN;3HWY…*JR7b;#9<_aJRPLQ’•WYMJR7;#pq;…J?7(_aMbe HdQTeI;_YPR;©HdWcprNJRQqWYMbHWY…

9 N E - C + - 9

C N ; ' CON

; ' 9

E 9 N E - 9 + - 9

9 N ; ' 9 N ; ' C

E / ; 9 ' C ' 9

Q

5

67b;:PL;…™WYPL;]NMbeI;3PJR7;_cHRHRN9<`IJRQqWYM

4

ubŒkYs

7

¢bPHKJWcPLeI;3P_a``bPRW†•IQq9<_aJRQqWYMbH§Wa… JR7b;];3Qrwc;:MG„_apqN;3H_YPR;

\

/ - Y ; S' C + ' 9 U 2

/ - Y ; S ' C + ' 9 U

E 2

ƒ7b;:PL;

- / ' C - C + ' 9 - 9

' C + ' 9 2 - / ' C - 9 + ' 9 - C

' C + ' 9 Q

UVPRWc`(;3PdJLQr;2H

4U 5"7

_aMbe

4

U§k

7

…™WcprpqWƒ”Qr9<9<;3eQq_aJR;:pqFYŒ‹UVPRWc`(;3PdJLQr;2H

4

U§u

7

_aMbe

4

U‡l

7

_YPR;]fWYM(Hd;2ScNb;:Mbf:;3H‡WY…

Wc—G„-QqWYN(H f:WY9<`NIJ_JLQrWcMbH0ˆ*MWYJRQTf;‘JR7(_J RHdNbQ’J_a—pq;QrMQrJRQT_ap(ƒ‘_aJR;3P‹7;:QqwY7-J‡e_J_ 9¨;2_aMbH‹JR7b_aJ‹JR7;QqMQrJRQT_ap

MbWYM¦fWcMbHd;3PR„_aJRQq„Y;©`PL;3HLHdNPL;#HdWcNPf;©JR;3PR9QqHAHd9_YprpŽŒ§67;Hd;2fWYM(e`(_aPRJWa…

4

UVu

7

9<;3_YMbHJR7(_J?QqMJR7;

£3;:PLW…™PLQqfJRQqWYM f:_YHR;YŸG_aM(e…™WYPHRNQrJL_a—bpr;]QrMbQ’JLQq_Ypªeb_JL_bŸ-JR7;]MWYM fWcMbHd;3PR„_aJRQq„Y;`PL;3HLHRNPR;]HdWcNPf;JL;:PL9

_YMbezJR7;#„GQTHRf:WYNbHHRWYNPf;AJR;3PR9 „†_YMQTHd7…™WYP;2_Yf7JRQq9¨;cŒ

/ ;E >

@BADC

' @ 9 & \ 8 '

=K>

@IADC

' @ F 2 /

EL S-)IM

C N -) U

S' )IM

C + ' )U N S-)

N

-)3P

C U

S' ) + ' )3P

C

U

[ f:WYMbHR;3S-N;3Mbf; Wa…

4

UVl

7

QTH<JR7b_aJJR7;…™PL;:;¦HRNPR…´_Yf;¦_YMbe JL7;¦JLWaJL_YpeIQTHRf7b_YPRwc;WY…?JL7;¦9tNprJRQqpT_†FY;:P

HRFIHKJL;:9 _YPR;t¢bPLHdJ©WYPeI;:P©_a``bPRW†•IQq9<_aJRQqWYM¦WY…‡JR7;f:pq_cHRHRQqf3_ap0DI_aQqM-JdˆŠ‰‹;:Mb_YMcJ©PL;3HRNp’JH:Ÿ`bprNbH©f:WYPLPR;2f JRQqWYM(H

eN;©JRW¨JR7b;#9¨WcPR;#HRWY`7bQqHdJRQTf:_aJR;3e„Y;3PdJLQqf3_apª`PLWa¢bpq;©Wa… JR7;©›bWƒ¥„Y;3prWIfQrJKFYŒ

¬

~I}&%$)<' -

! .m `303 !k m

303 m

# c k 2.Z303 H8V

k #V0 3# m &c#k303#

Z303 m

803# 8Vm

!I #

8^ 30 30 Z H

m 10 <. 303Z# "I#

:4#

303 m.: 3. m

.303 m

X-V)

/ - S 0

&g

0 3# m

m

303 ! H

2.X303

# m! # "I#

m

:4( 303 # 2.X 3.m

m

(k BA

2.#

2COA

. !

¬ ~I}&%$)1 S 0 H

. . m

# "!#(

10 80 :0

57 # 303 E71. 303 #

m:( #

& ?P1CGFcG?P AYF "! # "

&\

; ' 9) m

#c $

#$ S 0Z m

# c$3k

10

; ' ) &\ =

@A ) ' @

! m m # # ! " 0

m $ S 03

#c #k

0 m

10 # "B c

. 50Z# !m! mm! .m g10

.:10 #

m

$

10

k : " 8^ 10 10 #

mV

&

m m c#

*. m

k303#

30

H303"!

. !I,. #

!I

#V .X10:10 m 3#k ?Z

AYFG?

Z

XOFH!

H10 "B#

.X100 m

2. 303 .m60!

! U.U.m g10 m "B c

\

?PA CGF

¬ ~I}&%$) 10

#

.R10 # "!#

#c 10 D m!

.8V 03c: ! 10X Hk m m #^2. 10

:4 k

k ^ m

303 .!. 303

#c $H !IV!Z 10 :4 k m30 E7

5

!

m

0^ 303 m k . 303 m

c! "!#! 8V

^10XI

V54# #k E7B

!

X #

:4# #V! # # .m "# = .98

303 10

. 10

#c m

X

c#k

X303

#c 2. 303 #

m

?HBF ^I 54# #k Z303

2. " H3 " 0

# !

8V98^ 303 10

2. 10

:4 #k 10 !

# &c#

. .m m# " &

P C

V0‘   :

‚ ;˜MWƒ³`PL;3HR;:M-JžJL7;–eIQTHLfPL;JR;–„Y;:PHRQrWcM WY…#JR7;–f:WYMbHR;:PL„_JLQr„c;¦9tNprJRQqpq_†Fc;:PD_aQqMcJRˆŽ‰‹;3Mb_aM-JzHRFIHKJL;:9

4

uŒkau

7ˆ 4 uŒka

7

Œ‡67;fWYM(Hd;3PR„_JLQr„c;A9NprJRQqpq_†Fc;:P?DI_YQrM-JdˆŠ‰‹;:M(_aM-J?HdFIHdJR;39

4

uŒkau

7 ˆ 4 uŒka

7

`PL;3HR;:M-JHHdWc9<;

Mb;:ƒ JR;3PR9H§ƒ7;:MfWY9<`b_YPR;2e<ƒQ’JL7JR7b;]f:pq_cHRHRQqf3_ap(D_aQqMcJRˆŽ‰‹;3Mb_aM-JHdFIHdJR;:9

4

uŒ

5"7

ˆ 4 ubŒk 7

ˆ QqMžJL7;AHRWYNbPLf:;

JL;:PL9H_YMbe;:„c;:MQqM JL7;f:WYMbHR;:PL„_JRQq„Y;`(_aPRJ3Œ5DI;:„Y;3PL_Yp§HdJRP_JL;:wYQq;3H_aPL;ž`iWcHLHdQq—pq;_aM(e …™WYPQrMbHdJL_YMbf;

U‹_YPR;2H0;J§_Yp¶ŒR_ cb(`PL;…™;3P‹JRWfWYM(HdQTeI;:P0JL7;…™NpqpªHdFIHdJR;:9 _aMbe¨—NbQrprJ‘_#HR`(;2fQr¢(fHRWYpq„Y;3P0…™WYP‹JL7;JKƒ‘W#pq_†Fc;:PH

f3_YHR;YŒ$DGQqMbf:; 7b;:PL;ƒ‘;ƒQTHR7 JRW5JLPR;2_J¨_cH¨9<_YMGFpq_†Fc;:PHt_YH¨`iWcHLHdQq—pq;ƒ‘;…™WYpqprWƒ _YMWaJL7;:P<HdJRP_JR;3wYF

JL7b_J¨f:_aM—i;HR;:;3M _YH_˜9¨WIeIQr¢b;2e pq_†Fc;:PRˆŽ—GF-ˆŽpT_†FY;3P]_Y``PLWc_cf7–ˆ]ƒQ’JL7 PR;2Hd`i;3fJJRW˜JR7;JL;:PL9<QrMWcprWcwYF

QqM-JRPLWIeINbf;2e¦QqM _ cbŽŒ©œ¾Mbe;:;3e–ƒ§;¨fWcMbHdQTeI;3P?JL7;9NprJRQqpq_†Fc;:PAHRFIHKJL;:9 _YH:\ fWcN`pq;3e–9¨WIeIQr¢b;2e–DI_aQqM-Jdˆ

‰‹;3Mb_aM-JHdFIHdJR;:9H§_aMbe<ƒ‘;Af7WGW-Hd;JLW_Yeb_a`IJ§_YMž;•IQTHKJLQrMbwDI_aQqM-JdˆŠ‰‹;:Mb_YM-J‡¢bMQrJR;]„YWcprN9<;?HRWYpq„Y;3P‹JRWtJR7;

9NprJRQqpq_†Fc;:P#f:_cHd;ˆ…™WYP_ wY;3M;:P_ap QqMcJLPRWIeIN(f JRQqWYMnJRW JL7;<¢bMQrJR;„cWYpqN9<;9<;JL7WIeH#PL;…™;3PAJLW<_

5

-Ÿ kaubŽŒ

! Hd`i;3f:Qq_YprpqF¨JR7;©eIQTHRf:PR;:JRQq£3_JLQrWcMJR;2f7MQTf:H§7b_†„Y;JRWt`bPR;2Hd;3PR„c;JR7;]`PLWY`i;:PRJRQq;3HVWa…µJL7;]f:WYM-JRQqMGNWYN(H‡9tNbp’ˆ

JLQrpT_†FY;3P9¨WIeI;3p*HdJL_JL;3eQrM JL7;#U‡PLWY`iWcHRQ’JLQrWcMbHuŒ

5

_YMbeuŒkIŒ

6 W¦_a``bPRW†•IQq9<_aJR;JR7;žHRWYpqNIJLQrWcM S ')

S* 2 . U 2 -)

S* 2 . U U 2W / [

QQ

\ WY…§JL7;9tNprJRQrˆŠpq_†Fc;:PDI_aQqM-JdˆŠ‰‹;:Mb_YM-J

HRFIHKJL;:9—-F eIQTHRf:PR;:JR;#„_apqN;3H S'

)

2 - )

U 2W / [ QQ\

2 2

ƒ§;#QqM-JRPLWGeNbf;_cHNbHRNb_ap*_HR`b_cf;ˆ

JLQr9<;©eIQTHRf:PR;:JRQq£3_JLQrWcM—b_YHR;3e<WcMz_wcPRQTeWY…µ`(WcQrM-JLH . M

C

9

ƒQ’JL7HR`b_Yf:;?HdJR;:`(H . / . M

C

9 N . P C

9

_YMbe WcM$_nwYPLQqeWY…?`iWYQqMcJH

*

eI;¢(M;3e —GF

* /

=

*

ƒ7;:PL;žJL7; JRQq9¨;HdJR;:`(H

*

ƒQrpqp—(;

`bPR;2fQTHd;2ezpT_JL;:PJR7bPRWcNwY7 _ ² f:WYMbeIQrJRQqWYM*Œ‡67;:M ƒ§;#N(Hd;]JR7b;©¢bMQrJR;#„YWcprNb9¨;]…™P_a9<;:ƒ‘WYPL“ªŒ

‚ ;f7bW-W-Hd;©JLWžQqMbfpqNbeI;;•I`pqQqf:Q’JL;:pqFzJL7;t`PL;3HLHRNPR;HRWYNbPLf:;©JR;:PL9 QrM¦JL7;¢(MQ’JL;t„cWYpqN9<;HdWcpr„c;:P2Œ67bQqH

…™WcprpqWƒHNbHRNb_apQTeI;2_YH‘…™WYP?—(WYJdJRWc9 JLWY`iWYwYP_a`b7Qr;2H§ˆVPR;:…™;:PJLWZ_

5

bŠŒV67;#„GQTHRf:WcHRQ’JKFHdWcNPLf:;?JL;:PL9f3_aM—(;

QqM-JR;3PR`PL;JL;3e¨_YH‹_A…™PLQqfJRQqWYM¨JR;3PR9 —i;JKƒ‘;:;3M¨JR7;pT_†FY;3P ƒ‘;_aPL;fWYM(HdQTeI;:PLQqMw]_YMbetJR7;JKƒ‘W©M;:QqwY7G—iWYNPLQrMbw

pT_†FY;3PLH3Œ0[?HNbHRNb_apƒ‘;©JRPL;3_aJJR7QTH…™PLQqfJRQqWYMJR;:PL9 Qq9<`pqQqf:Q’JLprFcŒ‡67QqHpr;2_YeH‘JLW<HRWYpq„Y;©_<pqQrM;2_aPHRFGHdJR;39Œ

² WYP0JL7;HdQq9<`pqQqf:Q’JKFWa…iJR7;`bNPR`iWcHR;ƒ§;MWƒ eIQq„GQqeI;JL7;fWc9<`NIJL_aJRQqWYM<QqMJKƒ‘W#HKJL;:`bH*;•I`pqQTfQrJ§fWc9ˆ

`bNIJL_aJRQqWYM¨Wa…bJL7;‘›bNI•I;3H J_a“GQqMwAQqM-JRW#_Yf3fWcNM-J JL7;`PR;2HRHRNPL;§HRWYNPf;§JR;3PR9 _aMbeQr9<`pqQTfQrJ‹f:WY9<`NIJ_JLQrWcM

WY…*JL7;©…™PRQTf JLQrWcM¦HdWcNPLf:;?JL;:PL9H:Œ

" " # " #2"*

6 Wz`i;:PR…™WYPL9 JR7;¨;•I`pqQTfQrJ]HdJR;:`nƒ§;NbHR;_¢(MQ’JL;„cWYpqN9<;t“GQqM;JLQqfHRf7;39<;YŒ©67;wc;:M;3PL_Ypµ…™WcPR9³Wa…V_

¢(MQ’JL;#„YWcprN9<;©9<;JL7WIeQTH

M

)

N

)

+

) M N

)P

N *

)

/102

ƒ7b;:PL;

)

/ ' )

2

)

/ ' )

-

)

QTH‹JL7;?„c;3f JLWYP‹WY…JR7;NbM“GMWƒMbH3Ÿ

/ *

.

QqH‡JR7;APL_aJRQqW

—i;JKƒ‘;:;3M5HR`b_cf;_aMbenJLQr9<;žHdJR;:`(H:Ÿ _aMbenƒ7;:PL;¨JR7;žeQqHLfPL;JL;›(NI•

) M

QqHt_aM _a``PLW†•IQr9_aJRQqWYM–WY…

JL7;#;•_cf J›bNI•z;2HKJLQr9_aJR;3e_aJ`iWYQqM-J . M

Œ

67b;#“-QqM;:JRQTfHRf7b;:9<;#QqH?_<`b_aPRJRQTfNpT_aPƒ_†FJLWzfWcMbHKJLPRN(f JJR7;tM-Nb9¨;3PRQTf:_Yp›bNI•Œ

²

WYPJR7;tf3_YHR;©Wa…0JR7;

D_aQqMcJRˆŽ‰‹;3Mb_aM-JHRFIHKJL;:9 QrJ`PL;3HR;:M-JLH_¦„Y;3PRF5wYWGWIefWc9<`PRWc9<QqHR;ž—i;JKƒ‘;:;:M HdJL_Y—QqprQrJKF_aMbe _cf:f:NPL_cfFcŒ

67(_aM“IHJLWz_9<Qqf:PRW-HRf:WY`QTf]QrM-JL;:PL`PR;:JL_aJRQqWYMWa…0JR7b;;3S-Nb_aJRQqWYMbH?Q’JA_YprpqWƒHJLWzfWcMbHKJLPRN(f J9_Yf:PRW-HRf:WY`QTf

HLf7;39¨;2HJL7b_J#`PL;3HR;:PL„Y;HdWc9¨;;3HLHd;3M-JRQT_ap0`PLWY`i;:PRJRQq;3HAWa…‡JL7;¨f:WYM-JRQqMGNWYN(H?9<WIeI;:p ]`(W-HdQrJRQq„GQ’JKFWa…§JR7;

ƒ_JL;:P7;3Qrwc7-J3Ÿ;:M-JRPLWY`GF QqM;2ScN(_apqQ’JKFcŸ_Y—QqprQrJKF JLW JRPL;3_Jž„_cfNN9 _aPL;3_cH:ŒqŒqŒ ‚ ;˜PL;…™;3PJRW _k b…™WYPz_

wc;:M;3PL_YpµHdNbPR„c;:FžWY… JL7;“GQrMb;JRQTf©JL7;:WcPRF _aMbeJRWZ_kIŸik b …™WYP…™NPRJR7b;:P?eI;:JL_YQrpTH?_Y—(WcNIJ“GQrM;:JRQTfHLf7;:9<;3H

…™WcPJR7;tD_aQqMcJRˆŽ‰‹;3Mb_aM-JHRFGHdJR;39Œ

67b;¨9NprJRQqpq_†Fc;:P#fWcMbHd;3PR„_aJRQq„Y;t`bPR;2HRHRNPL;tJL;:PL9 eIWG;3H©MbWaJ©¢J;•_cf JRpqF¦QqMnJL7;<NbHRNb_ap0…™P_a9<;<ƒ§WcPR“¦WY…

JL7; D_aQqMcJRˆŽ‰‹;3Mb_aM-JJKFG`i;9<WIeI;3pqH3Œ–67GNbHQ’J¨QqHMWaJ¨WY—G„GQrWcNbH#JL7b_J<_apqpVJL7;ž¢(MQ’JL;z„cWYpqN9<;HLf7;:9<;3H

JL7b_JA7b_†„Y;©—i;:;:MneI;:„c;:pqWY``i;3eJRWHdWcpr„c;AJL7;tf:pq_cHRHRQqf3_ap D_aQqMcJRˆŽ‰‹;3Mb_aM-J?HRFIHKJL;:9f3_aM¦—i;;•GJL;:Mbe JLWJR7;

9NprJRQqpq_†Fc;:P#f:_cHd;cŒ ‚ ;žfpT_aQq9 JR7(_J#JR7b;¨;:•GJR;:M(HdQqWYM5Wa…‘JR7;<“GQqM;JLQqfHLf7;:9<;¨JRW JL7;9tNprJRQqpT_†FY;:P#f3_YHR;

QTH]„Y;3PRFMb_aJRNP_apŽŒœŠJ#QqH#_YMWaJL7;:P©_aPLwYNb9¨;3M-J?…™WYP©QrJLH#f7WcQqf:;YŒœ¾Mn…´_YfJAJL7;<WYMpqF˜eIQi;3PR;3Mbf;<ƒQrJR7nJR7;

f:pq_cHRHRQTf:_ap-f:_cHd;_aPLQqHR;3H*QrMJR7b;‘eI;:¢bMQrJRQqWYM¨Wa…JL7;\]Qr—b—bH ;3S-NQqprQq—PLQrN9JL7b_J‡_Y``(;2_aPH*QqMJL7;9<Qqf:PRW-HRf:WY`QTf

QqM-JR;3PR`PL;J_JLQrWcM WY…0JR7;D_aQqMcJRˆŽ‰‹;3Mb_aM-JAHRFIHKJL;:9Œœ¾MbeI;3;3e JR7b;t9<WIeIQ’¢(;3e˜\]Qq——bHA;3S-NQqprQq—PLQrN9 QqM-„cWYpq„Y;2H

_aJJR7;#HL_a9<;]JRQq9¨;]JL7;#ƒ‘_aJR;3P‘7b;:QqwY7-JWa…*JL7;#pT_†FY;:P_aM(ežJL7;]JRWaJ_apƒ_JL;:P7;:QqwY7-JWa… JR7;©›(Wƒ#Œ0@˜WYPL;

`bPR;2fQTHd;3prF˜_YMbe5ƒQ’JL75PL;3HR`i;3f J©JLWJL7;MWYJL_JLQrWcMbH©QqM _k"b‡JL7;z\]Qq——bH#;2ScNbQrpqQr—bPRQqN9 …™WcP©JR7;pT_†FY;3P:W QqH

MbWƒ

\ ) S* 2 .

2 U /

') S* 2 . U

S* 2 . U S N

-) S* 2 . U

S* 2. U U 2

ƒ7b;:PL;

S* 2. U / ;

=1>

@IA C ' @ S* 2 . U

E Q

67b;M;:ƒ MWcM˜fWYM(Hd;3PR„_JLQr„c;©`PR;2HRHRNPL;#HdWcNPLf:;]JR;:PL9 QqHAeIQqHLfPL;JLQr£3;3eJR7bPRWcNwY7¦_<9<QrMb9¨WIepqQr9<QrJR;3PJRW

;3MbHRNPR;]JL7;#PRWc—NbHdJRM;2HRH‘WY… JL7;HRf7b;:9<;

)

/ 0

S

U

9

.

#

N

2

N

Q

4

lbŒ

5"7

" /# U / 2

67bQqHQq9<`prQTfQrJAHdJR;3` eIWG;2HMWaJ_i;2f JJR7b;eIQqHLfPL;JL;©ƒ‘_aJR;:P7;3Qrwc7-JLH3Œ‡67;:PL;…™WcPR;

' M C

)

/ ' M

)

Q 4

lbŒk 7