Chapter 5
Prediction ofFloods by Weibull Ploning, lnd cunbel Analyticll Method:
Th. Elfetr ol B.rng.j
51
A diskibution lor AM ooods c5lnot bc chos slcly on
the basis
or{
priori th@reiical arSunents Th. chsacteristics of obsercd flood dala nust
d€l..nined io 6 sitabl€ fdhion Md irk.n into rcmunt vhcn
b€irg
cnoF.
spl6
Som. distributions
from rhcm do not hrrc
6
b€ .xclud.d
ifit
i
be
dislribulion h
b knoM
rha1
n
doo
cl@ctdistiq in @nmor with ob$ned n@d
Until 1970 lhc suitabihy of dy pfliol& diltribution fo. nood
frequency dalysis wa often jud8€d on th. bsis of physiql insp.crion of the
(bla
or.
prcbabilily plot On
scl !
plot th.
wpl.
vslus of lh. htdrologi.al
r@rd apps.r r $ri.s of ploncd poi.c while dE cninAr.d disrdbu{on ofa
prlialr fm, whos slihbility is b.ing dmined, muld app.d a a lin€ ot
curye The fom ofdistribution wh$€ lin€ or tuwe show€d
best
aere€n€.t with
Cwbel [21], in d.monslraling the us
of EV disrribudonq n.de us of @nfrdcn@ i.tads a!o!t thc liic o. aRe of
rhe plott.d poinrs would then b€ chos6.
detuBnatu h. snitabiliry
Il
i
or6.t \o help jndg.
of th.y ditxibutiMJu 14M eri.t, ddta".
thc tuIed distdbutioi otr rh. pobalility plor
t
is now undcr$ood how€vd, bot[ fron th@rericrl s1.ris1i6 and from
lplit smple t*ts on nydroloajc.l drta, that &y
@
giv.n d'st.ibuton
dGplay
r
rMid
sinele
of dlla fiom
a
plofi.d bch.vior which is quit. difmnt fron
pm eT rchtioisn p. fils a $npL frcn a $6igh line popularion @uld
dhplay mrtcd d liurc on r prcba[ility plot ma.kably clos 10 a stRishr line
rh.
on a probabihy
pbr
Ther€for., rn
r. is.
choosing a disribution on the basis of
nol the
full ofih.
ploi,
p*
se. bur
disrinct polribility
i.sp{tion
std3 fioh
ofr
of@or
when
probability plor. This is
the hjghly
unenrin .itur. ofthc
$
fdilid
obsvld d.rr d ! prob.litiiy p.pa
uing r eitrblc plordng poritid aortuh |Id tla thc dd. n tu d by . nnighl lie.
Thi! mthod b mr r@nftrd.d rcw ! d.y.. Hoy cv!. iri tfituriE felturc b ihar om
c{r G how rdl rhc dn. fir th. $$i|ed dirribqrion.
Thc cfteB otfou plorriig -plofiig polirion fomts d €$iirurin8 rhc
plobllilny wi8lr.d nloictn! ofLNO), LP (3), GEv, rnd W.t by dhnbudonr wrc
inv.srisrr.d by H.l'uiir ud BozdlM [2sbl.
In thjs ch.ptd w.ilull ptoning dd cu$.1 nlrlEd @ dies.d ed ts.d for
Findin8 lh..tr6ts of@rurrudion ofbr.6sd or pr.diqion ofnood. Co.lidcne
iitdd3 dc ctlql .d ..d e dm in srlph.
Prub.biliry Pbq Pto(i.! Po{ ioru, Prob.bitt9 PrFr .nd W.ibul nolti.g
A
rcchniqu. B ro plor th.
5.2.1 Pob.Dili9 p.pc.
.!d
Prcbrbiliry
not
In 8.n.rd *hgl thc ondadE di!fiiburion frft{on Pr (x) is plorcd on
&nhnujc p.pcr vru rh. vrluc of X r i6ishr lirc dc mr eh. 10 sd. nddr
lirc oi aithndic p.pq, P. (x) wolld h.vc to bc Sive! by thc qFdion P, (x) =u + b
ftus if Ihc orul.tirc di{riburion ol ! rcr of d.o plon n . nnisl tinc d dirhiEdc
piFr, th. d.ta follo*s I qifom diltdbotion. Prcholiliry p!p.r crD bc dcvdopcd s
th$ uy c!ruhtivc disdburion
d
h. plottcd B
r !r6itl|.
li@.
Cli.r.lty . *pslrc
typ. ofprob.bility plpq is r.qlircd for c$r, oflhc ditrcrcar p.obrhility dturiblrios ro
plot a a llr|j8ht linc. TIE *rling of fi. prcblbility FFr m.y do h.v. ro ch.rg. s
thc
p.6@rd ofr pdiold
dindhodon .n6gc.
Codru.iing pob.bility p.pa ir r proca ofrotr{oming tlE probalil'ry r.dc
so th.t rhc relling @nulltirc odc b . nniaht lie. Thc rtu3lomrion tEhniqu.
u.b. illudnt d bytlsub lMFi'lt16l
A prcblbility plot ir . plor of mSnirld.6s or probrlililt trcmiii.S rtr
prob.bahy ro eign ! d.h poi ir @dro y cLd.d ro u d.r.mining rh. ploui.s
position. lf orc it daling wirh . poplLlio4 ddmi.i.a th. plorri'rg posirioD ir @rcty
. nlt r of d.|miring th. ftrdion of rlE d.r. !du6 lcs! (gora) rhd or c{ud to rhc
v!lu. in quenio.. Thur ih. sdl6r (u861) topddion v.luc wqld ptol r 0.0 sd rhc
l$Ben Gnlll€0 popul.tion v!lu. sould plor !r .1.0. Asignins plonins loririo.s to
emplc data is not
a
sk.ighr foruard b@ue on,
on ndo b.
ture rhar a $mpl€
smllqr a..l l.tAdt v:bq of th. untnoM popuhtion Thus plotting
positions of 0 ed I shNld b. rvoid.d for ernpl. d.rl! unls orc ha rddilion,l
infomation on tn popubtion limit3
Frequdct dalysis i md.ly a poc.dur. for Gtiding rh. ftqu€.cy of
occuf.nc. or probability of occundc. of past dd / or tulur€ d€nr. Ths probabihy
@nhins the
plorting wnh or {irhout any dislriburion.l ssunpliotu is a nelhod of fiequency
aulyes,
vilh ot wiihout
The pro@durc io b. followcd in €ith.r qs.
Hydrologic frequenct analysb can bc mad.
disribltional assunptions.
maling my
is much the
sme. If no distriburional Gunptions de m.d€, the inv€rigalor n€rely plots the
obsfred data on any kjnd of papd ( noi o@swily prcbability p3 er ) dd ues rhis
he. besl judg@dt ro
s
fom dd ro
tbc rugritudc of pasr or tuture d€nts lbr vdious
dalltic{l tehniqu. h uscd, ir is r@mmnded that tne data slill
rhat onc @ gcr cn ids of now vdl rhe dat! fii rhc asdcd &alytical
r€tum p€riods.
b. plotl€d
If
ddmin
an
spor poEntial p.oblcms.
pals do not o@r with lny lix.d p.ftd in imc or rDglDtude Timc
inltuls b.t*a 0od. vrry. L.rg. nood! n tuolly h.w la.gc rdum pdiod! od vi@
vere Tne d.ffnidor of th. Gluo Fiod my not involvc ey r.f€d@ to probabih,
Eo{a€r, a relario$hip belwn lh. prob.bility of o@tu@ of a {lmd tud its r.1un
p.riod betftd cd in jusifi€d A givd nood 9 with . r.lum pdiod T nay br *.eded
one in T yem H6ce th. probabilily of dc..dcn@ i! P (0r > 4) = l|I.
Flood
The cunulative probability
ofnoi-exe.d€ie,
F
(Qr) h
givfl
by
.
F(O,\=P(O"<n\=t:f
Abov. eqution
is the basis
for e*imating thc rognnudc of r
pdiod T subsriturins F (Q.) =r
-
floa, Qr, givo
i13
t€tum
l/T in . k.oM si.tisti@l di$dbunon tuicdon, oie
.n. nognitud. of Qt, Oi.n, th. drita d. plotl.d on prchability psp.r to
cnek wn€rner th.y follov a paniol& di31nbldo4 ro dder ms, ed b cndk fo.
an $lvc
ao.
5.2,2 Plo(i|!glolirions
Prolrbilitt plots rcqlir.6 inirilt .slimite ofrhe problbilily of mnqcc.dq@
F= (F (Qr). rticb ir c.ttcd t ..plolling posirim. ptouing p6itioN rc .ts u$d
to
.siro1e p.rmEr.d by usins rh6 9rcb.liliry wisnl.d f,o@nr, (p\W0 nEdjod
Son. Connonly Uxd nortir!
p6
io! Fotuur.
Ploriing
/V+l
/v
+
l- ln
F=
T
'-:
0.12
-
i
o.44
i-05
,v
t
N + 0.25
0.175
l/+ 025
N +O.2
t-t
M
i
HoskinS
0t5
tv
nnk i. asdding ddq = N- n+r
rot
N. No
in d€@oding o.d.r
- N- i +l
,E
Plolring posnion fornula is civ€n by:
P(r>xq)=n (N+l) ''--------(5.1)
wb@ n= ordr runber or nnk offi€ .vent,
N= Tolal nunrh.. of r@s,
x=Q (c{0 @sc4)
(i)
(ii)
The data are amnged in decreasins oder ofnagnhude to IiDd m.
Tn€ Fobabilily P of.ach erenl being .qun.d to or
is
*Fd.d
calulaled by fornula (5.1) The W.ibull fomula is a @nprcn!. wnh morc
sllrislial ju$iliodon. Ifrh. N vrtua dc distnbu&d ujlomly
100 p€r@nl probability,
th..
th€re must be N+ I intwals,
rhe data points .nd 2 int€rv0js
(ii
)
(Ploltins positiod)
Th.
Euiiene iii€rva|,
a1
bdvq
Nlint
rysls
0
md
belwn
tbe cnds
T (the retun peiod or 6equ.ncr, h c.lculated
r= (N+l!m
T=I/P
s
(5.2)
o3)
dd lanted, ed tbc w.ibull PlottiiS
points (Q,T) colculrled, a shph of masritud. (Q).vs p'obabilily' t(P( pxl
P( x<x ) o. T )l cu b. plotted on probabilily pap$ wilh speoial s.tle or oi
Gumbel piobabilny p.r€. with rcduc€d 5.3l., i€ QVs.Yr to fit a disldbuion
On€
Graphically)
rhc
d.r! ssies
diesed i. stion
Trbl6 5 I I ro 5.1.3
Ctrmb.b
5.1.
show cdolariotu fo. thc gnpns by W.ibull Plot for
I tul d.t!
Atrrlttiql M.rhod
Tnc
fidely
is id€ntili.d
Evl dindhtiion, rl$ krcwn
used di$dbutioo in n@d ft.qucncy
s
Cumb.l disdbutior, is tne
rnalylis
CDL Fx(x)- axp I-€xp [-(x-O/dl, ..(J4),
wnere F(x) b rhe probdbilily of.on-dceddce for
th€ valu. x,
n6t
t
is
th. l@tioi
p!$*td,
P(X>rr=t-e-"
M6r |rr-q,+ €q,
...(5.5)
€=0.57721
vdilnc: dl = 1--=
= 1.645
4I
C@ffi.i6t ofSlcsrEss: C.= 1.1396
C@troi.mofKonosb: G=5.4002
r,=q-oht-lnp]...............(5.6)
rr =d-a
tn
f
(By
iNqtbs
CDF)
r'l
0-;)
l-h
L
1...o.?)
'I
We hl@
u dt.rndiw fom:
rl-k+Kror.
.....(5.8),vh@
l...rs.sr
' 'L t ( -Ill
r./tl
r- = -91,*r'l-r.{
r
NoL: Kr ir th. rtrdudiz.<l OorlbL
Vri.r. (r-
o, o
= l).
x n .l!o elLd Fr.q@icy F.dor.
5.3.2 Grntd"
X, =
t
X, = vrluc
+
Eq! io!
Kro-r b rh.
ofx
with
choe'3 sc|@l cqu.lion of b/dolosicd fr.qlc'cy
t .ct!m p6iod T,
7= nai ofx,
Kf
fr.qudct f.ctor, wiich d.tod!
upon T
ed rI| fr.quaq dkdbudon.
.. (5.r0)
. o.t t)
l ]"** ,.
.,,-=.l^^
T
II
L
It,
tS"
-+c'577
=Dulet's Conta\t\
r1.282s,610,@ ,
valus of ," &d S" arc 8iv.n in labh ?.1 md T.ble
sia, io Eigiffiiog Hydrologt by K. Sub.d.y [ 36 ]
53J
...(5.12),of
T
r"*,,0t
lo
Bar.drced\tiate
? 4, for ditrsern
smpl.
PROCEDT'R.E
The
alor
flood nagnitude
Flood
e
equalioN
us.d und.r thc followiog prcc.dur. to
mtonding
10 a
givo rcluo pdiod
dimr€ fte
bas.d on
Amnal
sdiet
Anege the diehdge data in d.@ndin8 ordq &d
N
l{€re the amudl flood value is lh. vartuc X
and
C.lqlate T
o. for
L
by
rhe
no1€
tlrc
.
givd dau.
WeibullPlo nsPolnioi Fomula
D€rmire,-
vi.
&
rnd S. lppropd.l. ro
siv6
N.usins
t!bl6
[ 36
].
spl€
size
vii.
Dct mirc th. rcquircd
dii.
Chdk wh.ilE rh.
dr@.
rr.
ote rd d!l. fl
vrluc dirdbqrion.
Nor.: Cd@Ltiotu for $c Gr.thr
in Tdl.r 5l I to t.j.3.
t,1
by
CunU Prcblbility Pipd
&
slDM
VEruflCATION
{i)
(ii)
CNlculate
vdu*
of
?lot Xt.vs.T on
Xl for
Sdi
somc T< N,
is
By qtrspolsrioi,
vdw
of
Ning Cunbelt iomulae
Log or Gumb.l Probability Plot.
. slr.idt lirc scn $d point
thd Gamble's dktabltio. k . ,ro!.r fn.
lfthe plor
tIs
wd! wirh lhdrctical Gmbcl's
Xl
for
(2 33,
X ) li6 on it (wno N is 14.).
r>N @ b. d.t@nEd
6ily.
GT'MDEL PROBADILITY IATXR
n
w
is
r.qunc
(l)
o
lbri$a
tlr
gnphic.l
!?tsa|ll!on ofcumbd ilisiribulior Bde
sp€idly ftr*ed for vaioG vdue. orT.
Forr $!b on 6si&.. w.6[l.ud
( sdy
(ii)
(iii)
.id for
lim
eihm.ti! ldle
of t'r
vdtr*
-2 to 5)
lor sl.cr.d v.b4 orT ( Ssy 2. 10, 50, . .) Iind tnc vtlu6 of yr
For xr or Q vdud on ordi rq ue ather dith'rdic or log!'ithmi€
Ar Xr is a lincar tunction of yr, the poinls ( xn yr) fo' Gunbel
dislribution wil plol a! . ltoight lin. on clttbcl p.ob.bilily plot. This lioe an
be used for gr.phicd interpohtion snd ampol.tion. ThG Equaton of the
snanelt liffi, ctlLd "nod.l Jd W.lie tl@ b! GMb.l wthod," ht en d^a *t
is
obt
ie!
ud 3hoM inTabLNo.6.l.t 10TablcNo.6.l.l
in
Chaplq6.
Confid..c. Ljnir. Fo. Pr.dictior of noodr by Crnb.l A..tyljc.t M.rhod
SiM lhe v.lue of tlE vdi.l. f.r . siEn Etum p.riod, xn d.t€min d ty
Guobcls nefiod m hav. .do.s du. to linir.d 6ple d.rl wed, rherefo.€,
6nido@ limils &e deiralrl€.
Th6 @nfiden@ interyal indi€tc thc limit' lboul the calqld.d v.lue b€rwed
shich lh. lrue value @ b€ eid to li. with epeiiic prcb.lity on sanpling €106 or y
For a confid€n@ probalilily C, thc 6nllden@ ioretu
bounded by th€ !€lues X, af,d X1 givei by
x,a
C
=Xr +
f(c)
sq
.
l of lh€ variare Xr
is
(5.D)
wnerc f(c) is. tunction ofth. @indcn@ problbiliry
Bddemined by us,ns rheLsbleoanomar
'dids(rcbdr$
C
in%
t0
68
0.674 1.00
{c)
80
90
95
99
1282
1.645
1.96
2.54
"** =bh
Kr= ft€quocf fadorgMaby.qu|lioi(s
o".' = sr.ndrd ddiaioi of thc
Tabl€s 5 3.6 1 io 5.3.6
ll
.
(5
14)
tl)
mpl.
sboy th. @lculatcd valu* of C I's ar
8OpZ
md 95%,
Gnphs 5 r I to 5. l. 13 show rh. f..quency cuN.! by W€ibull plofiing position,
L S m.tbod and Confide.e inl.ryrls by Cunbel hcthod, L.S. nelhods in disNsed in
Tn.
E|Tectl or Co.rtruction
At
ol B.rn36 on Pi.dicrion ot pak Di!.rrrg.
S!Lt$ B.ngG
Mei dd q
&<x0
&>xro
yd
< xr , i..,.5 flood, for .tl
for tne dara b.tor. rhc
for rhe dau aftd bi@s6.
bs!8.t
d.i! s.t3
by
% ditrcr6.€, for dif.rcnt T-y.es nood{
coisttuclior ofthe ba!8.r vui6 frcn
2
43o/o
2q/ob 45Yr
i . the ef€cts on Dcdidion
26./e to
bdw
the
d.ti els, du'
to thc
for Gudtlu (B)
for Sukku(B)
of [@dt rr Suktur (B) ttuc to rnc @nrtrucrion of
spprcndr.lv €qlal
Gudd! (B)-and of Sukkw (B)
e
l:1.27:
.].
Bcf@ Glddu @)
And Gddu @)
1.38
lr l.lt:1.50
l: | 25:1.29
l:1.32 145
ic.
4.
rhe etrects
ofcrddu (B) > Thc ctrecls of Su*lu (B) and
Mdinum
0ood i!
(i)
y@s
50
(v) 301ts
dpai.d
to
*b0>xs>xoii
r.turn.n r
sins tul da13 of sukru @)
ulinS d!l. bcrorc Grddu (B)
Nirg d.rh .tq cflddu (B)
using tu! beforc Sutlor (B)
uri.g datt .nq Suthr (B)
Hd@ th. nuitun tl@d it dpd€rt ro tctum .t Sukku (B) in lhc shorl'sl
p'dod
D€riod (i.e. 15 y€s) uling th. d;tl ,nd Guddu (B) rnd in rhe longdt
i lo0
1a.2
Ar
yd*) $ins
th. d|t. b.forc Sukh. (B).
Koti B.mg.
M6
dd Qr < xr for dl &ta cn (a d Sutlotr (B))
<
D, xo for ihe d.t. b.rorc .ll b.rfl.sa (M .r sul&ur (B)
Dr > x,o rn.r baras6, d@pt for rhc d.L .ild Su*lur (B)
2.
% ditreioe, for ditr r.nt T-y.$ ll@ds, bet*€d th. do$ scG du. lo
co.$rucrion of barlg€3 vdia fron
tbc
't94/,b 4e/"
ior Sukku (B)
3lY"ro6e/'
41/oro22o/o
i.o. the ef€cts on prcdiclion of loodr
lnd de Brotet due to sukrrr (B).
l: L26: I
L
.t Koid(B)
are the l€ast due ro Gudd!(B)
16
r:1.19:1.55
1ll9:ln
l:1.15:1.50
11128:1.41
I
x rm> x
r>x
The efrects
m
io €&h
is
(i) 100'ts
(ii) l0oy4
(iii)
(iv)
(r)
(t)
(ii)
a3
it SuEor (B)
dpat€d to rctud
lnr
tul dltr ofKotd @)
uins dda bdore Kolri (B)
usiDs
30y€s
20yaE
50y4
l00r!d
20 yds
e
cs,
I t6
(B) >The etr c!3 ofcuddu (B) > Th. .f@ts of sukkur (B)
ofKoti
Mdinum llood
1.39r
Bin8
dlll lnd
Kot i (B)
using dar! b.foc sukru (B)
using drlt ift6 Sukhr (B)
using dal! b.forc Goddu (B)
urins d.tA rn* c!dd! @)
llood to retum at Kotri (B) i. lle shon.d
pcdod (i . 20y6) usins rh€ d a sct .nd Guddu (B) ed in fi. loi861 p.nod
(100 ymhsing dra before Guddu(B) .d bdo€ Kotri (t r.
H.ncq w€
dp€crins
Minum
5.4.5
(ii)D'
2
>rr
for all rh.
b.d.s6.
% difer€.@, for dif.rcnt T-y.es floods,
vuia fton
b€tw4n Suklor (B) dd cuddu(B)
,
6l oel. ;
35.?70
b.t{*nKorn (B) &d S!kku(B)
bdwn Koti (B) dd cuddu(B)
di€ad@ k h.tw*n ih. predictons
at
Kori (B)
md
cuddu(B).
xrm
=
I134:l4a
l: l3lr 142
I
I
i e. lhe 6riG de l.Eali for Glddu (B) ald
Th. ririos e sle itEd3in8 wi$ tift
4
Muinun
For Kotri (B)
l2: 146
0@d it ciP.ct dto
lad
for sukru(B).
r.tnrid
usiis tult d.1s for clddu (B)
$ins tuI dat. for for sukk$ (B)
usii8 tull dala for Kotri (B)
(t
(iD
(iii)
H€n@, we ee dparing m.nnun fl@d to rctun
(
20 y.ds ) and at Kotd (B) i, tnc bngrst pcnod
C!dd!
(B) in th. snon6i pldod
i.
q
L
Md
2
(B)
The .treds on pcdiclion of floodt ar Koti (B) @ rh€ 165l duc to Ouddu
rid rr€atet due io Sukklt (B).
itre'a;eas on prcoiaion i:ittoo<lt at Sutkur (B) doe to th' @hsltuction or
Guddu (B) md Suklur (B) ar lPProxitut.lv equal
ud
<x
i, fd ![
th. drtr
s.a
D! < x o fd allth. dd! !.ls bcforc bu 86
D; > x h ro, all iJE drla *tt ns b.E$e sd sle fo' ftll dar' er3
t for 3ll datt 3.t3 itd b!.Bg*
(B)
'o
m"
fs) .r Kotd < Th. .trock of sul(ku (B) r Kord
"r..li "rctaa'
4. (t) MqinM [o.d to .aum .t Sul&i (B) .GquiB CDnd psiod ror rh'
d!!i .nd cuddu (B) .d lonscsr Fiod ror the dd! b'forc sul&u (B) (i '
3.
'
x
n
>x$>
(b) Maxinun fldd lo r.lur. at Kotri (B) r€qund shortdl P.dod ror rhc drt! !R.'
Crtldu (B) ed lon8.6l Pdiod for th. da1t befor€ rori (B) (i. fot 1wo
5
Th. difcr€ncB belw.ei
b.two
6
rhe
prcdid.d R@ds !t Kord (B) dd at Guddu (B) >
Kotri (B) and sukkur(B) > bctwcd sd&ur (B)
Maximm fl@d
.t Kotd (B)
in the
is
dd
Guddu (B)
lo r€rum rr Guddu (B) i. tlE shond pqiod (20
lo.s6t rcriod ( l0o y4).
rlt
ud
Corclusion
Tft..f6r3
of @mtrudion of
distane. | 00 -
ye6
flood is largd rhe 50 -
Mdinum 0ood is to rerun
in the
la€e
bd.g6 o. a !h. dom itr€d
dep6d upon th.
y.r4 flood (a dpdted) .lc
al cuddu (B) in thc ahonet p6iod and at Kord (B)
p.dod.
33
bl.5.l.l C.lcul.lior for Gtrnb.l Prob,bility P.p.r.nd w.ibull Plor Cudd! B.rrng.
1962-1999
t.t6l
l1
t.la5
Lrl
t.t31
l.
33
a
1.294
C.LrLlio! Aq GlDh|. Pnb.bilrr Prpd Ald W.lbo|r llol, SlltLlt (E)
TrU.No:5.1.2
''
(l9lll - t910)
t.56)
tr
l5l5
Lltt
td.ajj
C.lol.ti..
ror a[.
Cr4f, O. Cmb|.
tu.liliv
o9i,r-lt
POs.!rl wdhdl tlo! K.ti aE
)
0.1'
1.661
0.21t
t.557
0.211
5t
0.21
,1)
2t
t.21t
| 233
23
ItD
0.v2
t03l
0.11
ltl
Lt26
t.t?l
0.?t3
33
0.511
,.nt
n
tE
1.533
n
Lll0
ul
0lt7
t.5x
2311
I
F
q a E
:
E
n
a
a a
E
E
I
E
F
t
*
R
a
E
.t
8
a
F
E
IA
E
3a
a
q
B
I
it
s
€t
&I
E
.ii
a
t
$
J
i
E
q
a
F a
d
d
g
t
€
I
a
a
F
F 8
r
I
e
c
E
I
4
€
s
$
,5i
t^
e *
E
e
F a
e
F
:E
5
l{
F
s
!q
€
re
G
E
$
a
ti
q
D
i9
rl
s
?>
$
!
i
{
E
a
F
6
I
s
E
G
a
;
R
a
3
.i
R
a
I
t
e
a
4
F
I
a
I
E
I
E
&
F
F
E
s
I
ie
$
3
id
"3e
F
€ F
.E
t
f
T
ai
F $
R
t
:l
a
a
g
I
I
a
e
a *
P
E
e a
!
G
a
o
a
R
6
s
E
I
8
q
8
I
t
s
IE
a
6
e
3
i!
q
z
s
a
€ s
;
5
c
a F
s
g
I
I c Is
a
E
g a
€{
€
€
ll
a
5 a
4
!e
.F;
lg
F
a
t1
€
:I
E!
E
3
5 ;
s
!
!
a
a
R
a
a
s
F
a
ts
c g
E
F
a F a
I
\
a F a
c
€
F
$
:
Ie
t
a F
g
I
!
E
s
j
I
:
q
F a a
$
a
I
a
Es
dd
a
la
39
lq
:a
q a
E9
;i
il
a
6
€
t
a
e
a F
F
s
E
I
a
E
a
a
:
g
F
F a a
F
fl
s
I
€
q
a
E
!i
a
t
3
e a
a
a
3 a
s
s
8
c
E
t
fl
c
j
s
I
E
F
E
e
E
a
a
P
g
d^
lvi
E
€a
:R
R
*
g
I
6
B
a
ii
E
3
R
a
d
s a 3
R
R
5
s
g
t
E
€
IE
r9
f
R
de
s
q
q
le
lr
I
a
\
g 3
E
R F
€
p
a $
a
a
J
{
o
6
t
3
i
a s
a
s
a
F F
c
a
E
c a F
s
5
s
a
F
!
I
a
a
P
c
a e e
€
?
I
f
a F
e
s
l.
q
o
R
a
a
!0
st
b
-€$
g
c
a
s
$
6
s
c
a
3
"taa
0
!s
d
I
e
-
s
q
E6
2
h
E6
a
g
s!q
6
.5
E
6
i
i
€g3*
+
6
E
6
F
898
6
F
',18 6
56
g
8
E
;qe
3
aa
e!
ed
d xg
>g
;
s
f I
,6
3
JEEE
!;i
:En
s
6 c
8
B
X
3i
g
Ee
c
E
s
F
t
q
$3
a
,
E
s€
q
F
lEg
3
I
a,t 5
g6
f
H
F
FF
s
R
€i>
6nn
iF
5ii
R
P
F
t
il
€
>A
d
e3r
E"'
$
{
E
I
?
?
H
a
s;
F
g
i5
:
x
a
o
a
sE
q
?
3
g
c
c
g
d
E
g
{
t
3
R
Ii
€$
FA,E
daab
a
R
a
53
si a i€ $
n
I
I
'
I
:
r
I
si
6
E
I
E
T
I
i
|!
6
.,i
z
n
.n
a
,
t,,
-lI
I
,
a
n
i
E;
'E
!F
I
.I,
l{l
t1
t-:
dn
"i
i€
I
a
6
,ll
It
l?
il
T
;si
+tl
;se
il+
6
I
I
8i
TI
l!
?c
d
!E
i;
4\
i€
!5
6
8{
IT
!
-!;
I
I
i9
I
I.
IF
t+
!
o
n
I
I
I
-!t
ES
;!
I
EB
I
li
i{
t2
Nt
.I
n
,l
{l
tl
T1
a
+
i
I
I
IE
ls
i.42
rd
t
I
:
E
6
a
I
+
I
!
5
l6
a
E
t
I
+
t
til!
It
lt
ll
li
llI
l3
I5
tl
t
il
ll
E
ti
l!
t:
E>
E€
EI
,1
ti
i,!
a
tg
:l
II
E
ta
T
t{
t
8l
I
t
+l
t
tl
fl
d
a
rl
I
il
!/
:l
{l
?l
'il
I
t
5
E
!
I
t
a=
+
IE
:i
!!
E;
-!{
ilt
il
d
I
T
i
F
a
r3
il
e
i
+
a
€
a
E
-F
i
i
t+
tf
/!
IT
E:
:s
:!
!
iG
E
la
ir
7
EI
t
!i
!e
E
6
!
E
!l
tl
!l
t
.51
II
el
?l
'l!
XI
:E
6
:e
l6
:tl.
I'r
!-r
t!
3(,
,
dt
llal
il
ll
6
i
|l
tp
IT
It
lr
t4
IY
Ir
li
B
z
t?
t?
lr
ti
tf
it
I
Ig
t
I
ll
li
F
E
t?
ll
*"t
ttl
lsi
iE:
a
e
I
i
e
I
:T
i!
l{
3
a
;!
tt
I
i
r
E
+
!t,t
I
i
!
,
6
t
.l
+
a
I
I
z
i
oitlqaqO
|]
'tl
.!l
:l
a
+
€l
vn
E
a
::
e
:t!a
t
i
!I t
t 0
;E
+
3
i
I
I
r
E
+
I
!
I
I
|l
I
tr
lr
I
3
I
I
I
a
E
T
6
T
ti
{i
..!
c
E
I
rE
a!
!B
F
F
I
I tl
.it
EI
rl
3
tl
tl
T
I
+
I
I
I
t
'tl