düşey manyetik dipol frekans sondajı model eğrilerinin lineer

i ) u s E Y l r A N y E T l K I ) 1 F { . ) ; ,F I i i _ i i , / _ l i 5
SONDAJI Ii'()DEL ECIi 1 ].,ir;I(i ii 1 Ii
L1}iEER SUZGE(: IE(rjiLI.-Ii:iLE
H E S A P I , A I i I ' J I :1: :
S a r nr
ZLrlil-,i.ri_
Y r i I , : s e k i , i s a r : r , ,I ' c l i
.leoflzil:
l ' 1 i i h . A t i t r t ; i 1i m r l h l r
i QQt'r
ANKARA UNiVERSiTESi
FEN Bi LiMLERi ENSTiTtJSiJ
D U $ E Y M A N Y E T I K D r P O L F R E K A N SS O N D A J I M O D E L E G R i L E R i N i N
L r i i E E R S U Z G E CT E O R E M i i L E H E S A P L A N M A S I
sami ziiNeUr
Y U K S E K L i S A N S T E Zi
J E O F i Z T K M L I H E N D i S L i C i ANAB i L J.MDAL I
Bu Te:
)iot
Pr ,: t . - r
3 . 1 -.,
Takclri
i'J'J(j
:
:ECFLT
'l'arihrnde
Alagrdaki
Juri Taraf rndan 7rl(yetmi g)
,,!r oybiriiqi/Oyqoklu!u
D,lC . Dr
Ahrnet
T
i Ie
BASKIJR
Katiul
Edilmiptir.
Y, Doq. Dr
& n r r r H l N D E V A T - iL,
\/
://,i
"i
1
AZET
DOSEY }',1ANYET1K
DIPoL FREKANS SONDAJI MoDEL EG'RTLERTNTN
Lt NEER SzGEq
rEoREMl
Saml
t LE HESAPLANMAST
zuNB0L
Itnka.ra Unlwersltesl
Fen BillmIeri
EnstltOsO
Jeoflzlk
Anabllimdalr
M0hendisIlQl
Danr$nnn
:
Doq.Dr.
Ahmet TuQrul
1SSO, Sayfa
JOrl:
BAPKUR
: 88
Prof.Dn.
All KEqELt
Doq.Dn.
AhrreL Tugrul
BA,SOKUR
Y. Doq. Dn . Oznur I'lt NDEVALLI
D0gey manyeLlk
dlpol
frekans
sondaJlnln
CDI.IDFS) trporlsl
frekans
orLam:-ndakl
MaxwelL
denklemlerl-ne
dayanrr.
Bu
metotda
m.anyetlk
alan
bilegenlerl
elektromanyeLik
CEM-)
wekf6r poLanslyell
kullanriarak
hesaplan:.r.
ManyeLlk rrekt6r
potanslyell
MaxwelI
denklemlerlnden
L0retiIen
dalga
denkl eml nl saSl ar . Bu dal ga denkl eml nl n
qd5z0mtJ sl L j ndl r 1k
koordlnatlarda
gekllnde
verlllr.
Blrer
inLegral
dekleml
lfade
edlLebllen
alan bllegenlerl
l"lneer
sCrzgeg kuram:.ndan
yara,rlanrlarak
sayr sal oI arak hesapl anabl ll r.
Hesapl amal ar
(1946) rdan
Anderson C1S7S): Korkealaakso
ve
Saksa
alrnan
bllglsayar
prograrnr
yardrmr
yaptlm.t.gtrr.
lIe
FORTRAN
dlltnde
yazrlmrg
program
bu
oldukqa
hrzlr
olarak
alan
bi I egenl er I ni n
genl 1 k
or an:. nl
hesapl ar .
Dl'{D FS
model
q-/o- ve, R.o oranrna
eQnllerl
hazrrlanmrgtrr.
Model
96re
2L
eQrll,eri
Rzfo, e karpr
Bu
clz|lmigLir.
+rtllnde
lr=n"l
metoLda
arazi
egri.Ierinln
grafik
yaklagrmla
yorumu
yaprlmrgtrr.
Melodun
ayrrmlrllElnln,
Ost
Labakanrn
lleLkenllglne(ar),
(o./or),
lletkenllk
kontrasLrna
al:-cr -verlcl
arasrndakl
uzakrrgrn
labaka
CR,/D) we alr.cr-verLcl
arastndakl
uzaklrga
Dl'tD FS
nr n
Schl umberger
96r OI mO5tor .
kargr I agLr r:. I ma.sr yapllmr gLrr.
karrnlrgrna
(R) ba{lr
sondal r
oranr
oldueu
j 1e
der
ANAHTAR KELTMELER: Dogey
m.anyetik
dlpol
frekans
sondaJr
CDMDFS), Maxwell
denklemlerl,
manyetik
alan bllegenlerl
el"ekLromanyet"ik
CEld)
,
wekL()r poLanslyell,
lineer
stlzgeg kuramr
ii
ABSTRACT
CAI,CULATION OF THE VERTICAL HAGNETIC DIPOLE F'REOUENCY
SOUNDING HODEL CURVES
BY USING THE LINEAR F'IITER THEORY
Sami ZUNBUL
Ankara University
Graduate School of Natural and Applied Sciences
Department of Geophyslcal Engineering
Supervisor
: Assoc. Prof . Dr
Ahmet Tugrul
BASOKUR
1990, Page: BB
Jury
:Prof.Dr.
AIi KEQELT
Assoc.Prof.Dr.
Ahmet T. BA$OKUR
Assoc. Prof . Dr . 6znur MiNDEvAtLI
The
theory
of
the
Vertical
Hagnetlc
Dipole
Frequency
Sounding (VHD FS) ls based
the
in
on
Maxwell
equatlons
(EM)
frequency domain. In this
method; the Electrornagnetic
potential,
vector
satisf ies
the
is
EM rdave equation,
derived
from
Maxwell
equations.
Solutlon
this
wave
of
equation
can
be
solved
in
the
cylindrical
coordinate
system. The EM field
components
can
be
represented
as
integral
equations.
The sample values of
components
these
are calculated
by
us ing
the
I inear
theory.
f i lter
The
computations
are performed by using
program
the
computer
(1985).
written
by Anderson (1979):
Korkealaakso
and Saksa
The amplitude
ratio
of the field
components can be computed
program in FORTRAN programming
by this
language.
The VMD FS
model curves are prepared
according
conductivity
to
the
(o /o )
ratios
and
the
ratio
of
transmitter-rece iver
zf
distance
with the
to thickness
amplitude
of
plotted
The model curves were
tR/D).
Hr/H, on the
and
ordinate
RTfo,
on
the absclssa.
In this
method, the graphical
approach to the
evaLuation
of
the
f ield
curves
was
carr ied
out.
The
(o /q l, R/D ratio
conductivity
contrast
and R are found
as
2t
parameters
inf luencing
the resolution
of
the
method.
The
results
the
of
VMD FS are
compared
with
Schlumberger
sounding resulLs
obtai ned using the same parameters
as
of
the V|'lD FS.
K E Y W O R D S: V e r t i c a l
Magnetic
Dipole
Frequency Sounding
(VMD FS ),
Maxwell
equations,
Electromagnetlc
(EM) vector
i
a
1
,
L
i
e
a
r
f
i
l
t
e
r
theory.
Potent
tit
IE$EKKUR
Bu galrsmanrn
1le
destek
Prof. Dn.
tglen
g1zlml
lqln
H.
JoloJl
ve Jcloflzlk
blr
werl
AKINr
AYVACI ve A.
hazrrlanmasrnda
glrtslnde
a,
M0hendlsllgt
Ba5kanlrgr
borg
bl1111.m..
Btlglsayar
t rr&,
.
A. O.
personellne'
bana
JeoloJl
yardrmcr
BOIOnO
GOLI e,
bllglsayarlarrndan
Odasr
yararlandrQrm
t na rra deferll
gerekll
UOur
Lezln
Legekk0r0
ve
a
BASI(|'JRr
Tu$rul
sisterrLcrlndcn
Bagkanlrgr
meslektagrm
Mohendlslerl
g6roglerl
ve
KAYIRANT a,
Dalre
6grencllerlnden
bana ftklrlerl
Ahmet
DoC.Dn.
hesaplanmasrnda
egrllerln
olan
oIan,
Turan
verllerln
Bllgi
gerqekle'gnreslnda
yararlandrgrm
Kolu
ve Jeoflzlk
l CI NDAKI i.ER
1
G1R15.
1
2.
MiiTorxJN nio{?lst..
?1.7
Fr-ekans Or-lamrnda
?
'C.
?'-"2.
E ve H lqln
4.3.
EM Vek1-6r Potanslyell
Maxwel I
Dalga
t)enkl emler.l.
. .
Denklr:mler'l
-l
ve Dalga
Denklerni.
4
e.3.1. PeneLrasyon derlnllgl
3.
HO}''OJEN VE TAEAKAT-I
DUSY
3. 1.
MANYET1K
HomoJen
BIR
OnLam.
3.2. 1.1k1
Labakalr
orLam
1q1n
OrLam
Manyetlk
lqln
Alan
4.
MANYET1K
4.7.
1k1
4.2.
T_ we T
lntegralLerlnln
otYardrmr
IIe
Sayrsal
4 . 3.
ma.nyel-lk
.....a5
Vekt-6r
PoLanslyell
Bllegenlerl-
EM geklrdek
..-.."?6
Foknslyonu.
ALAN
Halka
. . .3O
BILESENLERIN1N
SondaJr
SAYISAL
we Kargrlrklr
KuplaJ
Llneer
Olarak
HESABT
. . .34
Oranlarr.
S0zgeq
.36
Kuramr
HesaplanmasL
.....-.4e
kurulmast.
....A2
ZX{ANKS AJ-tprogram:lntegrallcrlnln
4. 3. l.Hesaplama
Yardrrm
Sayrsa.l
ll-e
Olarak
To
we
T,
Hesaplanm:rsl
algorlLm^asr
5.
D T . { DF R E A K A N S S O I I D A J I
5. 1.
lkl
Tabaka
Model
PararneLrelerlne
16
HesapJ.anmasl .
hesabr
3.4.
4. 2.1.S0zge9
(Hz,,alr)
1a
bllegenlerlnln
n-Tabakalr
ve
s
s
A-Lan 81 legenlerlnln
Tabakalr
5.5-
OZERINDE
DIPOI-.
1kl
aLan
ORTAM
OrLam.
3. 1. 1. M.anyretlk
3. e.
z
..
MODEL EGRTLER1
Egrllerlnln
G6re
...
d.B
..4t}
.,....54
S.,ndaJ
lnceLennresl
....5S
6
T I ( t T A B A K A } . ' O D E LE G R T L E R T
.-....64
7.
DI{DFSVERTLERTNINYORUMU
......67
7.1 .
lkl
Tabakalr
Graflk
YoLla
Ortama
.Alt DMD FS Edrllerinln
DeSerlendlrllnresl
7.2.
Ornek Blr
a.
D T ' { DF R E K A N S$ i . { D A J I N I N A Y R I M L I L I G I . . .
8.1.
Da5Ok
l4etodun
S.
D{-{D FS
I t elkenl
16e
Sahl p
......6A
..,.77
O r L a m _ al r d a
PeneLrasyonu.
MODEL
}.,OD€L EG{TlLERl
1 O.
......62
Derperlendlrnx..
EG{?ILERIN'N
...73
SCHLUMBERG{iR
l LE KARSI LA.5|I RI LM.A.SI
SOI.{DAJI
75
SONUCLAR
u|6
KAYNAKLAR.,....
B7
EK-A
SiHGELER
: l ndoksi. yon
B
D
sa)'t sr ,
kalrnlrdr
Labakanrn
:i.
L
E
alan
: Elektrik
F
siddt=i,i
vek Ltir
: Manyet-i k
(vekt'cJr)
yel r
poLansr
Ci =O
havadak
L
-
i=1 .,2.,3-,
f
: Frekans
H
: MarryeLj.k alan
H=
: ManyeLi.k
i-{
: MarryeLik
.
1
(
-t
,
SrCdel-r
(vekt6r)
alantrr
dUsev
brleseni
aLanrn
;rai'a}' (radial)
'
br IeSenr,
, - l/2
: j".
k
),
otrla.nidaki
I
ail
orLama
vayrllm
daiga
sabit-i'
L
RC\) : EM geklrdek
R
: A1:.cr -verici
€
:Elekfriki
0)
: ?nf
(r
:i.
fonksiyonu,
uzaklrk,
arastndakipermiLtivile,
aqr sa1
frekans
,
ileLkenlrdi
orfamrn
L
6
: PeneLrasyon
derj.nlrQr
\
: l nt-eqras)/orr
paranrei,resi
O
:Ke},fr
.i-t
: Manyel- j, k
skaler
CSkin
deplh)
'
,
forrksj.r'on,
gegi r genJ. r l: ,
FUt urr br ri ml erde
mI:s si stenri
esas
a I r n m r 5 t - . rr .
i ,
1. GIRIS
Elektromanlctlk
olan
D0gey
l.lanyetlk
6lg0len
nlcellk
genllk
oranrdrr
Max1-probe
Dlpol
frekansa
(
alanrn
d0gey
rrekt6r
potansl)rell
CHz)
CH"
CDMD
FS)
bilegenlerlnln
adr
olarak,
radlal)
nanyet.lk
bilegenlerl
hesaplanabll
gartlarr
btrl
pratlktekl
Teorlk
denklemlerlnden
slnrr
alan
nretodun
kullanrlarak
uygun
SondaJrnda
manyetlk
Bu
yatay
Maxurell
denkrerntnln
kargr
SondaJr t drr
vc
sondaJlarrndan
Frekans
lH_/H.l>.
Frekans
potansl)reIl,
CEI.D derlnllk
elde
ir.
EM
.rrekt6r
EM
edllen
El-l
dalga
g6zorrcstyle
altrnda
alda
edlleblllr.
Vekt-6r
potansl)rell
H, blrer
lntegral
lntegral
denkremlerl
EM
geklrdek
Lernslr
eden
sonucunda
hesapl anarak
uygun
f onksl )rcnu
konrrolOslrcnu
hesaplannug
denklc'mi
gekl lne
bl llnen
blrer
olarak
blr
elde
lfade
deglgken
1l e
fonkslyonunun
karmagrk
genl l k <>ranr
sa.yr
l*.n.
Daha
1le
ve
Bu
fonkslyonunun
6ncedan
istenen
sayrsal
oran
Hz
yaprlarak
d6nogomo
sOzgeg
fonksl)rcnu
cdllen
edlleblllr.
getlrlleblllr.
sozgeg
geklrdek
kullanrlarak
aran
I eI de edi I l r .
yncdefl
konvolosyonu
bilegenlerl
2 . H E T O D U NT E O R I S i
2. l.Frekans
Ortamrnda
A
o-
genl lk
Haxwell
we ort faz
denklemleri:
ozere,
olmak
A=A.,ri
slntlsoldal
Sekllnde
olarak
deftgen
lzotrop
ortam
asagrdakl
dalga
blClnlne
elektromanlrctlk
lgln
glbl
frekans
tanrmlanrr
sahlp
zamanla
CEID
alanlara
ortamrnda
Maxwell
cward
alt
homoJen
denklemlerl"
1966):
VxE+l c,rp'H=O
I
\b<l{-Co+1 a<^r)'E=O
II
V' E=O
III
V'H=O
rv
Burada;
E = elekLr1k
alan
gtddetl
H = rnanyetlk
alan
Slddqtl
a = ortamrn
al = ?nf
lr=
cttektedl
manyetlk
r .
lletkenllQl
acrsal
e= elektrlkl
1f ade
harmonlk
CVolL/nD,
(A/m),
Cmho,zm),
frekans,
geglrgenl
permlttlvlteyl
1k,
t.l.
E ve
H
I .
.r I I rlal
al.
Iriln
Oalge
wr+ I I -
Dsnf'riemlerl
Maxrr,+l .l
t
eml *rr j lri:r
clenkl
t-r-ri:lsyonel
,
VxVx[
+1 t^lp, Vxl I =O
(1J
( ')'\
VxVxl'l-C c'+ i e<l) VxL. =O
.lenk.Lelrnleri
elde
edilir.
rlenkl<lmlerinden
i:
ve
vazt
II.
Max-well
(1)
kargrI:.klarr
I1 r sa ,
L J-l
VxVxl I +1 ap( o+ i e<o) ' H =O
c4)
eI de
egiLliginden
denkl
gClre
emf eri
Vlt. *ktt
VxVxA=W'
edl I I r .
yararLanarak
denklernlerine
V'E=O
rre
vre
III.
we V'H=O
E ve
H
wek t.()r eI
a-t'n
IV.
oldugundan
Marwe}J
dolayl
nde"rt
(5)
=o
(6)
rlH*pt11=9
sek-lrnde
(;l)
ve
VxVxL, +r c.+r( cr'+l cor)' E =O
denk I eml er 1
C4)
ve
L
litrn
'.lenk I enrl er i ncl,e Y*r-.i n{:
ve
i tlt
iqin
dalga
denklemler.l
elde
edllir
C3)
,
l'l.tl .rrl.r
I
r)l ar-Al
acllarrsllt
2.3.8M
Vektor
EM Leortde
al anl ardan
yasyonl
di f eransi
kullapm:rk
Mallick
III.|,laxwell
19BO).
bir
baska
edrleblIir
Denklemr
qc)zmek rCih
a
oldukqa
1964).
il
lcl
TI
t
<lertrei.Llrkler FIM
LureLl
I ebi. L ern
t r x - - a: tn s i y e l
trygun
oJacakLrr
(PaLra
denklemine
F:in
vekLor
C P h j . J .I j . p s
DaIga
prol>,iemlerr
I c>nslyonlarrnr
E
ve
Potaneiyeli
vr}
bi -l i n l r l * l
cla
ol ar ak
i
I t .1
AYI
',al)1 !l
y,,ryi lt
tl:i i ,l.r
ti
lJr: <terrk I em,l er
rlrr
denk I eml er i
l l t r l r r x > lt . z
( _ ) r1 a n l l
,;l rtlr
i'rlrrv..)
tlDlA
g6re
r()tasyonel1
Bu dururnda
V'E=O
ve
olciuQunclatr
ifade
5ekl1nde
manyeLik
kaynaklar
rCin,
('7)
E =-i o/.rvxF
oi ar ak
Bu
seCr I i r
vekL6r
Bur ada
potansiyelinl
F manyetlk
I I .
vekLc}r
M.axvrel -I
PoLansiyelidlr-
denk 1 eml nde
yer 1 ne
yazarsak,
Vxl l=-i
'.lr;,( o+t
v x ( H+ k
2 f- ) = o
c.r€) Vxi'
(8)
(9)
-
denk ] emJ ni
rot-asyonel
bir
6te
yazar
e'der 1 z .
eI de
we
deglldlr
fonksiyondan
bundan
dolayr
gi bl
a$agrdaki
sk al er
il-l bi
I
(ll+kzl
gdre
t ureLl
lebl
l i r:
(10)
H = - k 2 F- v . C
c11)
(7)
ve
II.
denklemlnl
VxVxA=tZV'
a-fa
M:axwell
eSl t,I 1 Ql nl
yer.i.ne
denkIe,mlnde
de
)
k eryf i
H + k z F= - V - O
yandan
ak
denk.I emi. ner
C S)
kuL lanarak.
- I <,rprH
= -i co;.rVxVxFH=VxVxF
H=V V -r-€r
denkleminl
elde
CT?')
( 11)
ederiz.
( 1-?>
denk I eml er i ni
blrlestlrerek,
v v.F -vF F +kzF+v .e = o
e91Ll igi
bulunur.
fonksiyorr
kosulunu
Burada
oldugundan
kullanmak
lYard ISAZ).
skaler
dolayr
oldukga
Bu durumda,
p
( 13)
A
uygun
fonksiyonu
nin
keyfi
seclmirrde
olacakLrr.
(Vanyan
blr
LorenLz
1967 ve
ul' =o
vtt -k
f'
'l r-:l ft dal ga. {Hel
grdtier.:
i 15-r
rroltz)
!i :.ser (11)
denklemine
dOnti3;ur.
}-hrrye+-i k
rlenlcl"emlnden,
?:1,-i t;u
i.i= *.k
1t . 1
ci;-rr:rl.
orir-unr-':'iVarryan
Fur aeiak i
4'
denr-tr e.in:,nr saglarligrnl
I \'.
M ; - ^ x r r ' e Il
.alarr
denl:l'eminde
ci6]
1967,
Fat.ra
ve
MaIl. 1ck
f onk sl yonunun
sk a I er
96r'eblI
yerine
mek
igin,
1g8f,-'.
da
(11)
da:. ga
d*nl'l.r'tni tii
yazr lrrsa,
v'rktl'*V"{r)=O
f ,p "u"\z'=c,
we cira (.L4>
egllJ-iginda:':
. a ^k=4=r-,
c) dugt:
gr:l- rrJ r-bj i i :- "
de
yararlanarak,
{3.7}
Penelrasyon
derinl
(<5):
idi
".3.1.
,iC-ir"r tJ;rlr.;;r derrklemirrder
I:
k
seklinde
yer'
(r.ialE;l
lt
"rJ,lr:
sayr
sr)
r= -J
t^rp{ cr+l <rr)
lanrmranrr.
Yer
---:idugu 191rr ihmal
degigLirme
edlIebilece{inden"
akrmlarr
dalga
qok
saytsr
koqtrk
Ck)
k =( _i urpr'>t/z
olur.
F- lqin
dalga
cJenkleml
a$ad.rda.ki
rl.rbi
bir
cc)zrlme
:;ahj-pt-l r.
-kz
F=F
Burada,
()E
E
k=a-i/-l
tiicelikleri
rwt
birbirine
/u
s =fs=l -
191n
eSit
( lvard
daj-ga denklemlnl
o
E
e ''-nl n
dc)nogor.
orup"
a
ve
f]:-e*rr
olarak,
E
.-Lf>
Bu
dur unrla.
Ir
CclzOmC,
i e^rt
*i<oL
B gerqel
ktlqormesl
Hohma.nn l gaz) .
ve
n esas
-(a+L ft)z
=F e az
()
sekline
sayr
|
Lanr ml anr r
F=F
bir
l, o\r/z
\A)
sekl i nde
karmagrk
seklinde
bir
tlalgan:,n
sayl
oldugundarr
z
s6noml-r*nms,sLni
brlyudrJkge
gOslerlr.
Herhangr
brr
fakLdruyle
genlrginrn
dalganrn
EM
azalm;rst
o.l-ar ak Lanr mI anl r' .
penel-rasyon
Pont+tr'asyon
Penetrasyon
f
I
2.
Sekif
da
azaldrkqa
anl a5r I acaQr
orLamlara
hassasdr
dO5trk
r.
fnekanslardan
alcak
gereklr.
yant
inmesinde
et.kln
bir
nrn
EM
frekanslara
rol
deQeri
degisimj
lncelendtginde
gdrtllrJr.
artt.rQr
f rekansl
dal gan:, n
o
olarak
a.1.
dalgalar
o] arak
sIra
baQIr
$ekiI
6
f rekansl-r
r gi n
Bunun
o, ya
yt1kse'k
Uzere
Sc'nr-rq
yaprlabilmesi
ve
riepLh)
skin
)
g6sterllmekLedir.
wa o de{erlerj.
da
f
-
dorlru
6,
der i nl I {i
L
derinliginin
(6
derinliQi
(L]"'
i*1"'=5os
o=[
--f <t
L. p o
)
cleljnl.ere
clal gaI ar
r
ise
dr*ri n
derl
Buradan
nI i k
sr O
orLamlara
rondaj
l. nr r)
f rekans:
nl n
dogru
deQiglirrime-sl
de
dalganrn
yuksek
derinIer-e
ovnar.
a)
Seklf
a- l.
Penetrasyon
derinllelnin
olarak
deElSiml.
C6)
f
ve
ya
baSlr
l/r:
-
3.
HOI'{O.IEN VE
]'ABAKALT
t,,l:ZFjRI
NI)E
OR]'AM
Bt [..
l)tJSEY
MANYET1K DI POI-
3 . I . l - | , : ' r 1 , : ' . j e nO r t a r n
!
Eksenl
(Ie'*")
ele
La5ryan
alalrm.
sillndrrik
z-y6nunde
ktiq0k
Bu
koordinat
nokt,a.sr
vardrr
dlpoJ-den
oIsun.
bunlar,
bir
Burada
akrm
f
ctr poJ
manyetik
gdsterrildlgi
gl bl
yrJksekliQinde
bir
olasr
we P,
bl rincil
kaynaklanan
da
h
Pr,P.
aI ternat.l
9
:-j.1'de
Sekll,
sisfemlnde
yerl"ergLirllmlS
ki,
ya
halka
dipol
noktaya
t':r-
blr
drlzenlenml
tig
noklalarrd:
manyellk
adet
r .
t>l gum
Btlyl
e
:rIan,
-kR
+
F =C-
| 18)
A.Z
I
Pr(r,Q,'.-in)
I
T\
Sekil
3. i. Yeryuzunden
yerl
P ,P
L23
h
eptr ri 1 mi 9
ve
P
ytksekliQrnde
bi r
manyeti
dl cum noklaL arr .
b:. rk
nok Laya
d i pol
ve
i,r
(Keller
verrJir
Seklinde
v(-.
F'rjschLne{ht
C=I d;r.,'A.trCI d:l rnanye.Lrk mr:nrerrt-.-)ve R=.(-r'
Maxwe,i l
clenklern)
denkl
verkL6rel
koor-ciirratlal'daki
ve
yc)nUnde
r
o}acakL:.n.
durumda
\{ar d
r
gd)z0mu
ol-arak
yal.n:.2
ayrrmak
al rnmrgtrr.
r
nl n
suretLyle
peki
l de
s6z
bi I ergeni
denklemi,
O
gibi
selc-r-iip, gerekli
,F
dR
=l
ar
her
iki
Laraf
o"
R
ye
mevc ut.
aCrsl,na
gdre
(PaLr.a
veril"ir
ve
slnrr
saelayan
$arllarrnr
bul unabl
Li r .
Bundan
ol_acagrndan
numaralr
nln
bl 9lmde
olan
dolayr
1kl
azF
, -=2.
orz
b()lunerek
arz
azF
=
R--.
,
ozz
F"=F
fcnkslyona
alrnarak
duR
F
denkrem
C6ZOIebl-Iir:
-
tam
son! a.
dj.fe'ranslyel
fonkslyonu
Lurewleri
1 ar\
z
F=RCr) ZCz)
oi.arak
ve
1 987) :
konusu
c1s)
lzleyen
Q
olmayacaQrndan
oz"
uygun
z
s;ilir:'lir-1k
/
ve
yalnrz
ve
rrr.rm;rr:*I r
82F
Er
z*bl-Iegenl
C( 1 5)
denkle,mj.nde
O"
sadece
a$aErdaki
we, Hohmann
qc5z0mri aFagr daki
trl n::adece
degrgim
1
aF
z+__z+t=kzF
arz
Bu denklemln
bin
dalga
(dzaq=O)
dolayl.
EzF
skaler,:lalga
iiurad;i
,ir , ^
clerrkla;rrnlnin
o"lan
rcl acaSr ndan
sr f r r
1 g8O,
Mal I i ck
d:ricr.r
kar5rlr$r
Bu
simeLriden
EM
''-r'".'j"''
rrf <l*; t-:rli l t;rr
rrdt'rl
potansiyelde
t Ur ewl- er 1 de
bj r
eml eri
19n8l.
62z
azz
C19) denklemincJe ye,ri n*
yazt lrrsa,
1d"RIdRtrJzZ
.+----.---+PaCeo)
R
denklemi
elde
de$lSkenlerine
Larafrnda
sadece
Sekllnde
1945).
z
rR
drz
<ir
dlr.
Buna
1
dzR
R
drz
de$l gken
r,
sadece
sag
^Arfken
(1Sas)
1
dR
7
cJzz
rR
dr
z
dzz
gore
yazLlabilir.
gc)re;
prensibine
ayr-rlmasr
denklemlerln
Dlferansiyel
edilir.
ba$rrrslz
,.,1.",2
Z
Burada,
\
aylrrm
denklemin
sol
t-araf rnda
lse
derrr CaO)
sabltidir
denkleml
,
(Arfken
Buradan,
t
dzz
z
dzz
( a1)
=x2+kz
ldzRldR
+ R
drz
+71z=O
rR
dr
veya
r=}.r
de$isken
ddnuSumtini
crzR
1
denk ] emJ er i nr
(et>
e
+R:jO
xr-
{eA)
d(Xr)
eI de
erder i z.
denkl
e r m in i n
Cd)zumti
r +z( k2+^.2)r'/2
clnslnden,
Cee)
clnslndendlr
karma.5r k
,
dR
+ d(\r)2
R r I e qarparak
yap.rp,
CMcLachlan
q()ztlmu
genel
Ct)zumu jse
denklemi.nin
Bu durumda
1934).
ve
CPaLra
fonksiyonlarl
Bessel
l'la.]I i ck
C1g)
de'nkleminin
198O) :
,= f*foa\)e-zc\2+kz) "t *rrx)*z(\2*kz)
J L
^"f
., cxr)d\
JO
o
c e3)
Sekllnde
veriIlr.
havadakl
CF )
o'l
agaElCakl
gibi
$ek1I
ve
3-1!cieki
yerdekl
yezrlabllir
dipole
vekL6r
CF )
(PaLra
ve
Mallick
aiL
havadakl
potanslyelleri
1980):
-kn@
eo?
F =(oPJo
+ | acx>*-^o"J
c).r)dL
<?4>
o
cn
f
^(\r)d\
F =l BC\)e"r-J
o
'J
o
('e5)
Burada,
potarr:;i
{e4)
yel I ,
yansrtan
denklemincleki
k.,
ilk
havianr n,
k,
orup,
(e4)
sabiLIer
d*:
ter"im
yeri
birlncil
rr manyerl-i k
ve. (2s)
wekl.cJr
frzel .l i kl er i n j
denk]ermlel-.rncle
srr-as-l
iIe
n
2>r/z
=(L'+k
r)orl
gek I r ncierdi r ;
AC ^)
ve
g a r L l a r : -r r d a . n
hesapl
anacak
denk I em.l er i nden
Lan,;:r.nsl yel
ol duQurrdan
k
ve
ma.nyetik
derecede
ma.nyeti
ffi
AF
tz
ilz
saSr an1 r
permeabiriteslyle
f arkl
oI an
fonksi
sirnlr
yonl ardr r .
sr nl r
k
1 se
Maxwel l
Sar LI ar r rra
a.I anl ar
sr nr r da
96r- €',
sr.ir ek L i
z =O cla ,
<1olay:.
$ar tl ar r
yonL arr
fonksi
edi I en
F_F
ot
sr nr r
BC \)
el de
el ekLri
z)t,z
n =(),.2+k
ve
c KeL l. er
yerinki
oI madr gr ndan
r
1 s66)
.
Bur ada
qok
blrbirlnden
dol ayr
ha wanr n
6nemll
po=l:^=p
ol arak
alrnmrglrr.
llr nr r
i"ntegral
edile.cek
e
kn
R
formu
garLl arl nl
uygul amadan t)nce
Sommerfeld
formolo
bi rt nci I
yardrmryle
al anr n
ifade
olursa,
k( r2 *='>tt"
cr" *='>"'
ft - e
r
-z(rrz*kz)''"
J cr." +kz>1/2
J (Xr)d\
o
( a6)
3. 1'den
SekrI
ll=(r'2+
f=*n l')"t
i ser P
noklasr
z
manyet,ik
formul
gdr0lecegl
de
d!..
Burada
i. qi ndi r .
momenLine
uzere
Cz-h)
ve
P,
P,
Bdyl erce yerrderi
sahip
birincll
P"r:z.rklrklari
nc.:kLagr
h
iq:in,
y{rksekl
kayn;rk
rctrl
lCin
Ch--z-)
i Si nde
l cla
Sommetrf'eld
u,
KR
e
I
(-(
R
c)"2*kz)
lc()
geklinde
vektd)r
verilir.
Bundan
potansiyeli
agafr
@.
A
rr
o
Daha
(5nce
uygul
anr p
e
llJLn
oo
edllen
Jo
bi rbi
\
n
g'
ri nl
-^oh
-n
denk -l-emi ndek i
yazL I abr- l i r .
o l=-t1 l+Rrlr6.'-r-ro^i
]r"axr)d\
lqin
nl n
ve'rilen
katsayr
gclLCrreceSinden)
+ACL) =BC).)
o
cLe-noh
(?.4)
anarak
gi bi
daki
z=O
f s (xr)d).'
ler
Q-
-n
ll
I fade
o
o
c.-no=t
=L
yararl
A(\)=n
ot
BC\)
:
s:.nrr
I ar r
caa)
5artlarr
eSi t.l enl rse:
clenklemleri
B( X)
edrlir.
elde
ve
ACA)
denklemlerden
Bu
q(5zrJlLlr se,
An-n
A( A) - C-
c'
I
nn+n
oc,
e-n()h
I
e\-hh
-C-
BC\)
oIarak
n
bulunur.
A(X)
yerine
denklemLerlnde
wekL6r
-r-n
e
o
()L
ve
(ZA)
sabiLIerini
tlCX)
hava ve
yazLlarak
yer
ve
C25)
manyeLlk
lCin
poLansl yelleri,
s
F
-o =( t **-^ol=-nl*
J
L"o
^
^o,^,
.o
^o*^i
.-^o(z*h)1.;
J
o
(xr)dx
( 2S)
F =C
1lo
sekllnde
flax
I
elde
edl11r.
ESer dlpoJ- yer).Oz0ne
oI arak
c 30)
C\r)dX
.tr"-toh.J
Jn+n
ooa
a} r nr rsa.,
denleml)
agagrdaki
1940,
Slnha
vekLd)r
harradaki
gekllde
ve ColIeL
indirllecek
ifade
1973):
edillr
olursa
potansl
CPaira
yani
h=O
yel 1
CCeS)
ve
MaIllck
1e,
It+"
7.01
o+-
F_C
o
^
n-n
*-""ot]
nn-tn
oor
. J,rcxr ) dA
( 31)
3. 1. 1. Manyeti k
Hesaplanmasr
Homojen
olacagrndan
{.H ve
?
Bi I egenl er i ni n
Al an
H)
r
!
ortam
we do-I aYr s: Y] e
t-ant ml anan
clenk I eml yI e
C31)
ko=o
clurumunda
n=\
o
wek L6r
poLanslyeli,
['-+]"
@
F=f,1
r
'"
-)ry
@
2X
r
=cJ
-o
,\+n
Sekllne
d6n0gecekLir.
CSeklI
3. e)
C16)
glbl
Hohma.nn
1"918'7):
- ^ z- . J
e,
-
o
asaEldakl
.J
o
CrJ
(\r)d\
C3a)
CXr)d\
a
Manyetik
denkfemlnden
cKoefoed
hesaplanlr
AF
H=V9'F=VC
o)
alan
ya.r ar I anar ak
wd
1se
bllepenleri
LS7e,
( k =k
o
Ward
c 33)
&z
g=J
t
azF
ozF
H=J
&='
t
=O)
c34),C35)
&zdr
ve
11
(34)
ve (35)
manyeLik
yararlanarak
denklemlerinden
alan
H =2C
z
clii5ey
ve
r adyal
bile5enleri,
J
-rrz
eo
J
o
(3?,)
(Xr)d).
@
H =2C
r
peklinde
elde
mcnyetrk
J
edilir
kuvvet
*-to=.J
( lYard
(xr)d\
c 3e)
1
1967)
.
gizgtleri
l t
\
.: J.
/.f
.,/ Hr
/H
ttetksn
Dugsy
Sekil
3. e.
D(rgey Manyetlk
Cizgileri
ve
monyeti"k
Dipole
yorL-€on9uz
orLcrn
di.PoI
CD|'{D) ai t
6lqCtm noktastndaki
manyeLik
kuwef
alan
bile5enleri
i8
P
I
ot'lt
az'l't
'a
Seklf
3.3.1kl
Labakalr
nokLalarr
2.2.tki
Tabakalr
Sekit
Burada
olarak)
B6yle
3.1'deki
blllnIr
orLamdaki
fonksiyonlarr
ve
dipol
P ).
iki
pL
b6lgelerdeki
1 ' r od e { e r i n e
dlpol
e5if
iqln
olsun
deQerleri
kabul
olarak
blrincll
Ozerinde
orLam
Labakalr
yerleSLlrilmig
nokfaya
bOLOn
manyetlk
F=c
oLarak
bir
bo5lugun
blr
CPi, P ,P
234
6l qum
DMD
OrLam:
h ytiksekllginde
3,3).
{rzer indeki
ortam
vekt()r
Cpo=u.
edilmistir.
potanslyell:
c 3a)
e-kn/R
CKeIIer
OC bOlge
Ft
CSekif
ve
Chava,l
Frischknecht
.
Cl =O,1 ,?) ,
ve
2.
C15)
1S66).
Labaka)
denklemlyle
lqln
tk1
Labakalr
potansiyei
verllen
c 3s)
I .:!
dal ga
denk I emi ni
1966).
Btr d*:nkl emin
Csilindirik
oldugu
saQI amal r dr r
sr I ir:cij ri k
koordinaLlardaki
icin
ve
Pafra
utt,
1
orz
9ek I 1 ndedl
r .
+
dr
+CC\2+k
dolayr
' =t1r'.
&z-z
denk -Iemi n
Bu
n:rt-larclakr
genel
I
poLansiyeli
2)z)
( 4L)
C41)
denklemindeki
dak 1 gi bi
zlO
3.3'deki
yonlarr
Kel-Ier
orLama
vekL()r
potanslyelinin
YazL 1 abl ] i r :
L91n,
tiC
birincil
vekLc)r
(1955),
CBhattacharya
a$agr
Sekf l
fonksi
Oz'dt$::77
q6ztlmu ,
io
veril.rr.
I r {r
(4o)
I
F=erJLz\fJ
geklinde
kar5r
1980):
utr,
dF,,
r
koordj
Fr j sr-hk n'.:r.:l-it.
ve
si melriden
MaIIick
+-
( Kel I er
ve
aiL
vekLdr
potansiyeliyle
toplamr
Frischknecht
Seklinde
1966)
2A
-Ch+d)(z(
-h
igin,
co
f
(
f
ez ( \ z + k z )Ir ' / z
t"=tLJ{o"cx>
'2-'1'/2\
"2
d(X)e-z(A{-Ktr
o
z.( -'( h+d)
}'rr.x.>axJ
i qi n
ru
l' (
f
I
r '. = c f I { o c x > ez ( ^ z + k t )z" t
LJ L'=
\
1
l - s oc \ r ) d \ l
)
J
o
c 4e)
Burada,
eLde
C=Ida/An
edllecek
olan
Ca6)
yer
olarak
ve
\
ya
baElmlr
notasyonlarrnr
ya.z:-lbillr
verllen
deplgLlrme
c\2+kr2)
glbi
Qt,Q.,Q",Q.
de'nklemiyIe
yararlanrp,
alrp
u.
we
05
CKeller
FarLlarrndan
fonkslyonlardrr.
Sommerfeld
akrmlarrnl
ihmal
t/2 =vr.,
cr.2+k"2>r/2 =v",
da kul.Lanarak
srnrr
C4e>
lnLegra.linden
ederek
k
o
=O
*,t=t oLpo.
denklemlerl
ve Frlschknechtlg66):
a5agrdakl
.:.
cn
,-
' rn z
r
I
_Iz'l
F -c I
1
J
l.
L
F =c |
J
"
^tl . (
., xr ) crA
[*^= +@2( \).L
J o
t"=.
J [+".
t. =.
f [*r.
re (\)e-^'l.
I
J
l c\r )ct\
.'
^, .u r=*e.(r.>*-'.=] ' Jo( \r ) c!\
^, .u ,=f. Jo(xr ) dx
i'r3)
q5,
|
6- 2,
O,
{1366)
3
da
6
1
w e 6' 5
veriLen
a.l-anlarrn
Labaka
dayanarak
elde
fonksiyonlarl
Tanjansiyel
sonuglandrrrlabilir.
C1366)
edilen
bu
surek1i
s,lnr r
s:-nrr
de
sarllarr
KeIler-
atr
1
= -
AF
z=-(-h+d)
gartlarrnda.n
1?tzoz
F=F
, 2s&zoz
da
prensibint+
AF
aF
z=-h
1rk
Manyet"i.k
da,
AF
z=OdaF=F,i=-
irriscirknechtve-
ElekLrrk
agagrdaki
56zO
ve:
KeIler
gegerken
srnlrlarrnr
edllen
we Frischknecht
,
AF
F =F
e'azoz.
( 44)
veri
5ekI i nde
kullanrlarak
( 43)
I mi 5t..i r .
elde
edjlern
rle'nkl erml er j
( 44)
ve
.ait-
Qi fonksjyonlartl)a
a5:edrdakr
denk I em si stc.m.l .
!
Cramer
C45)
denklem
denklern
yerine
aI arak
g()re
kuralrna
gclzulUp
7.
sisteminde,
slstemlndeki
yazr.l-rrsa,
2.
d.
denklemclen
(45)
denklem
bul unabilir.
fonksiyorrlarr
denkJeme
gdre
Q.
sist-emi
AC
Qr=Q.
Ceki lip
daha
3.
basiL
dir.
Bu
denklemde
bir
hal
,
h*ra
r ' , + ( v , - \ 1e t ' h @ . .A ) = - 2 \ e - ' ^ ' h
"-u,
(h+d ) -e.(
(h+d ) =0
(h*d )*y'oc
\) evr
x> e-vz
er( tJ
"-t'
(h+d )
(h+d ) :v205(\)
vl (h+d ) -vlr4(A)
,rLr(;\.) e
6v1
e-vz
- (^.+vl)
c46)
gekline
sisLerninl
verllmlstlr.
d6n0;tir.
Cramer
Bu
kuralrna
lineer
q6re
homoJen
qdzCtmu ise
olmayan
izleyen
denklem
bigimde,
23
-(\-v
t
-v
e
I
I
)e
-r."
e
I-'
o
r"
1
(h+d)
1'
e
Vr(h+d)
v e
-
rz
( r. -\)
v (h+d)
JC
v (h+d)
-v
-V
2
I
k
1
2'
e
I
( h+d )
-v
(h+d)
.,X
I
-AJe
Lv
t
_e-Vz ( h+d ) .-v,
1'
-Ve1
_.,
-L/\-T-V
I'
V e
Fvl
(h+d)
( h+d )
_v evr ( h+d )
I
t
(
-vh
^*', _) e
I
[t
I
-v
v(h+d)
€ r
-AJe
(v
-v
./h
e
1
I
f-,
LlI
(h+d)"-v,
2'
V e
2
(h+d) (\+v
"t,
(h+d)
(h+d)
2'
v e
1J
-v
1
( h + d )' ll '
)*-'rh-
. JJ
. . r e - u . ( h + d ) e - v , ( h + d ) ( w , - \ ) e - r^,l)
-l
-'a^._g
)-,
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rr
l-'
L
l
o
e
o
-€,\.e
o
1'
-ve
-v
(h+d)
t2
D =Z\e-)'h"t.
72
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vz ( h+d )
e
-\)e'r"
(v
rr
F.
I
v (h+d)
-vel v (h+d)
- X h- ' -
+ v evl ( h+d | .-u,
evl
( h+d )
-v, evr ( h+d )
( h+d )a\e-\h
( 4a)
lrr'r,
D
2I
D
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-l
i
( v
t
o' 3
+v
-)
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.iv
tztt
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hr
o
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e
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2'
2
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e
2'
e
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s
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_
|.
it h
-t-rl
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-v
D
l-
1
I
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(hrd)
1'
e
\
/
=
(v
D
Hawadaki
( h + d )' P - v
2'
r, *^) --?u
nd a.,
aX**(
'
(\)=
+w )CX+v
12tl
wekL(1r
potansi
yonl ar1 nr n
)-Cv
-v
yel I eri
, ( h+d )ar..-).h
( 5f))
-v)
1Z
-/\Je
J(v
--2v
1
(51)
d
gerekli
igin
olan
Qr.,Q.
eOzUmu,
-(v
--\)(w
+v
tt
CrC X) =CzC\)
Cv
)+(v
+\)(v
)-(v
-v
21t
+\)(w
+v
21
tI
)(v
*-v )e
z
-2v
d
1
--2).ir
-}.)e
2A
( 5a)
ol arak
\
-d4..:
I
1
e
fonksi
-v
-v
2-
aa
)e
-rr
o
_]Ih
f) =cA_a:
-rr
-(\+v
o
Vr(h+d)
I
2l
-z\c
v e
( 1l!,1)
cl
,\J€'
el de
edi I i r
CK e l l " e r
ve
Fri schknecht
1966).
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orl.am iein
Labakalr
3.e. 1.lki
m a n y t ' : tl k
aI.ln
t-'iI
o<orrl
rr
l nr
rl
tresabr :
i qi rr F
Keller
aI r p
o1 ar ak
r=F r=F ro
r; Oz r-ttnundeln
f'c>nksi Yonl arl nr r)
Qrvu 6"
havadak
C1966)
we Frischknecht
polansi
vek t.r1r
i
t.) "Yl
:lonra
yel
i r:i
dan,
()J
F
-Clf-r^
:o
f
|
+
LR
o
J
R(\)e-'r"(z+2h)J
t ^ . ., O o
]
i ki
Labakal
F r . r r . r , J a I;:CX)
SekIinde
yazrlrr.
qeki:Cek
f'onksi yonu
(v
)+(v
-v
)-(w
2t
-v
+v
2t
1t
R(\)=
+X)(v
+v
11
RC^,d,o,f)
lki
bilegenleri
tabakalr
CH
z
ve H )
r
2l
+\)e-2vr
)Cw
-\)e-zvr
fonksi
parametrelerin
ilgiIi
seklinde
A
)Cw
Cek i r dek
Lanr ml anmr S L.l"r .
ol ar ak
I qr n
orLant
r
oIup
-\)(v
1w
s3)
h=o
orLamda
yonu
RC\)
fonksiyonudur.
igin
olursa,
hesaplanacak
d
manyetik
C34)
rre
alan
(35)
denkl emleri nden yararlanarak'
fl ='
az
(
a*="
JL c lL -R
a2
H--
" a-.a.
4@
tL
.@
( f r
lLc l L- R
f
- x= J
+ lRcx)e
o
oJ
-\=
+ rl R c x ) e ,
oJ
J
o
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c 54)
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( 55)
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|
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()
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r
g = Cl - '.LR3
3.3.
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vd
Koefoed
CSekil
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vekfdr
or t.amrja
trer' frangi
( lla)
o(\r)d\]
r q(l)
(\r)dX
I
o.l.ur
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n*TabakaIr
(|j7)
) d^]
bj, 1e5enJ" er i ,
RC\)J
*, \ ) J
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J
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i ki
ak
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J
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J
sek j i nde
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1
al an
k
manrreli
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H=C
Rf ),)e,
_\ R( \) e "'',J ( I.r
z=O
br r- nok Ladak i
I
,z
^'
( Koef oed
Vekf 0r
wd
1 97e)
PoLansi
yel i
ve
ManYert.i k
:
(1972)'de
potansiyell
n-Labakalr
bl r
orLam
I cr n
"
0
f
,r
F = = C L - O--
f
J
R(\,d,? .,f)
.-\=Jocxr)d\
( 60)
$eki
I
3.4.
n
Labakalr
di pol
.
ortam
Qzerindeki
d(tgey
manyeLik
;::s
j f arle.:i yl e
vrl
ve:rr.l mi;rLi r'.
ilc'l,ker:.l jk
sayls-rrlr r-)t.abakalr
Rurada
(:-=J,r3,3,.
de$errler.idir'
q*:kirdek
f J t r ra c i a k i
orL;rm
bagr nt.r sr ndan
iqln
hesapl
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o,n
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t
a
n,n
1,5'7?) .
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L
t
'
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K
rrh
L
.Je
r
v
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C\) =O
t
k z = i Z n L 'tO o ' "f
L
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(\Je-td.t,
+R
L-
I --
u =(\z*k?)t'z
Laba.kaIr
n
gC5sLermek Lerdi r .
sayL s:. nr
i-t,n
R
t.abaka
R(\,,J.,s,f)
yerytiztrndekr
indis
v
v
kl arl
CX)
birinci
i ncji :-; cle t.abak a
vd
n
verilen
biqjmde
CKoef oed
anr r
()l{.tp
,r)
1,,:rrksiycnrr
121 eyen
kal rnlr
v€i L1r Lahaka
d.
ve
orLam
!
C35)
iqin
v.
i,k
=( v -v.).2(v
r
hesaplanacak
alan
+w.)
i"
k
yararlanarak
denklemlerinden
manyetik
k
bilegenlerini
Cz=O
of ursa,
@
t
Hz =
clLReJ."oJ
*
[ ^t *ax,d, o,r)
J cxr)d\l
@
?f.-t
I{ ==cl | \2
r
LJ
R(\,d,o,f)
r
L
J (\r)d\l
t
(62)
J
n
(61)
.:,cr
el d€t edr I : l- ( tloc'I t:r-Ll .''f1 :l !l'7e)
g:ekirr:Jek
Si nha
ve
R(X),
fonksiyc:rru
yansr
modunclak i
k aLsayr
rna
yukarr
n
dakr
l,.alrlriral,r
gl br
-frarrsvers
rndan
laraf
C1!)73)
Col leL
'cle.
(1972)
l(r:r+fc:ed vd
Sek I r ncle
sr
r(. lI)
ortaln
t . , ]n t a k L a
t;er'aLle*r
(1'F-")
FlIekt.r'rk
gl bi
aSaQr dak i
Lanr ml;rnmr 5t.r r:
N-Y
K
o1
LAJ=-
( O..J ,,
TEN+y
ot
Burada,
Y
+N
M+l
M
Lanh(u
d
M
M
)
( 64)
Y_N
MMN{-YLanhcud)
MI|{+1MM
x-i),1t:,
I
--l\
",.
ri.rr*.1 )
,
NN
N =Ll /ir"r,u,
Itll
-.2.,
(l=L,\+K)
l|L{
K
,
=L L@UO
Ml|ll
2-1./2
_ 1,/?.
)
tf=o,t,2,...,!)
( 65)
Y
,
( trli-)
1 r . :L I t
t-al:aka
Ortt-{r t,"'italt.t.;"kr
llk
rJr"rrl l etnl nr.1{Jr)
N
( 64)
r q-i rr
e,:f mek
r*l rle.
Lrtr cie(lerr
Ar ,:lr rr,l;n ti
hers;r pI a n r r' ,
y 6 j r I n{i
detnk 1 elni nde
baQ:rrt. I
ikj
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anr.r.
oI ar a l1 lit:.:;apf
yi l-rr:l emel- i
cic'rJr+ri rti
t,
crl --rr' .e k
solrllq
l: orr;lt';;
L- ,
er5i i.
st.rt lug I aI
iIc:
ver.iIeri
ur et.i r .
3.4"
(}nceki
R(\)
p:arameLr'€].1 eri
[-r-r
ayrr
Karmagl k
hesaplanlr.
par'amelreL
eri
degisik
ol"arak
ayr'-i
ol arak
t'abraka
RC^) ,
o]an
baQl i
frekansa
ve
sayl sl
Labaka
Cd , o.) ,
yop
f onksj
brr
f ' c . : r r k sj y r . - t r r t j
Cek i r ,lek
.
{
vL]
o.
,'
X t n t r r f u r n k s j -y c r n u
] Ci n
de{err
f rekans
bi r
tl
X,
fonksiyonlr
f orrksi yonudur
nj n
ba{rnt.r:sr
ylneleme
b15lumcle
gekrr<leP-
EM
:
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EM Cekirderk
almakt-adrr.
deQerler
\zBc x> =*-t"RC y)
fonksiyonunun
arLtrkga
fonksiyon
buyumekt-e
ve
kaymakLadrr
yapr 1 mr gLr r .
yat-ay
dugey
eksen
(Sekil
vr:nurrrun t.abaka]
boyunca
orf am
k r sr mI ar r-nl n elavr arrr pI ar r
gdlst.er
i. .I.rnr.:k i.e-.,-lt t- .
ve
da
j ncelenmesi
gd)re
vt:l
r.
Fr ek ans
boyunca
eksen
3.5.
r
o
da
3.6.
uzerr
ndeki
$ek r |
3. 7.
ydnde
y()ne
).
reel
3,b
I er i
de{er
neglaLif
negatif
$ekiI
geki, rdek
olan
fonksiyonu
paramelrelere
ilgili
bi qi mde
i zJ- eyen
lonksi
girig
suzgece
$eklinde
doQru
Cek i rdek
ve
satral
ve
c
de
J}
-'it.'lr'lr; {:.rlaf .1}.
j n
f5qrlsLn
rJt:''Qr'-;} Ildt'k j
ek:,c:t) lroyunt-a
yat.ay
s i k I I qe
de{i
t"7
ned€n
ol makt.;icir r
l'a{.I }
t1) ge"k r.Je
t:lu5lry
kayrn;r:',rlra '
(Kos*f',rerd
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I
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f=1@
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3-5 - o =O. 01
fslooo
Hz
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o =O. OO1
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Cb )
tl
sahi.p
degerine
reel
f onksi yonunun
homojen
k:, sr ml arl
.)rf amlar
iiz-er j nder Qekirdek
gore
de$i si mi
rrr n f rekansa
0.0
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I
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r ir-r-:-:-r---Tl--T--r-"-T-T-l
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ro
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l-.-
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rlll
Y =I n ( 7 / x )
Seki I
3.6.
o =O. 01
mho/'m
homo3 en
--+
L:j r
or Larn
uzelr i nde
1
( -.--)
r t:e.i
f onk si yonrtnun
Qek i r dek
fl'-'1";ttr.;;r
k:S:nilartnltl
C- -----_)
sanaf
( f = . 1O ' ; e
1 OO Hz.) C a) , ( i' -1 ()(;O ..,i, j r.)',-){)i)lI.:.)
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n
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al
hl,o/
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il
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o
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3. 7.
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4
u
rr
d
O
l
ve
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f=1O
orLam
uzerinde
Tabakalr
qekirdek
reel(--)
fonksiyonunun
iqin
sanal C -----)
k r s r m l a r r n t t ) d a v r a l ' r l .5 . 1a r 1 .
Hz
ve
4 " l'lANY[:T1Fi: AL-AN B I L.-l::LNt..-t:ti:1Ni
N SAy I SAt- Hi:5AE{i
qr-:nt:l
bin
denk,l
.rnrl .+ri y.l e
f'ra.iiy1 ..t
ver j -ier)
lresal--.r n r
yapma.k
dc)rrr-r5umti
yap:. I r p:,
mar:yet,i
i <;i rr
L-:u
B=R,rd
(i-1)
k
al an
al r nar ak ,
ver
l{r
. . . 1 2, ' r r r z
de;er 5l. rlrr
}>i I e5enl
f
eri
,
]
j
t
=c[L
\2
ro'i,,F',
| *R(g)
-
Ij"
ir)t.rit;lnl
L:i I t:Serrl er j nt rr say.r s;il
de.rtk I elml +t.d.)
-
H,=c[
('i,.)
vrr"
*
I o t" R ( q ) -"r ( q Bo) a o l
53 J
Bt
(66)
J
.c(t
F L a
t"=.[
Seklinde
elde
,. o J
s'' R(E) J.(oB) on
edilir
CKeIIer
ve
(67>
]
Frischkne'chL
1966).
denk I eml- er de,
@
TocA,B,D
,k )=
T-cA,B,D,k
I
L
'
)=
| *an,
gt
| *an,
gt J.cge>
J--o
J
J^cgB)
i-
dg
dg
( aP)
c 6s)
Bu
tg
.:J
(.t )
R
z+17
A=
-
L).---t
F=--r
6
o
-,
_ 1/2
d>:-L -)
.f
o
i=1
2
?
<Dpq
k u I l a.rrr I ar ak
nolasyonlarr
Sekilde
agagrdaki
1966
Si nha
ve
H
r1
)l
iA,B,D,k
rJ
!
' 0,.']
daki
H /H
zr
oranl,
(:.72:)
T r , . A r B r D . r l::, .)
LL
Yukarr daki
edi .1"
i- r .
H /H
uzere
zlo
kuramr
hesabr
Anderson
yardr
h t - s a p r l a l r n r ; j i " , - : )C ; l l r g : , I a c a k L r r .
T
T
ve
T
de
vt:
t.
anmasl
n:-
inLegralleri
OL
(1979)'dan
mr yl e
eml- erden
hesapl
oI ar ak
Ru qa,l r grnada
C79..7?) ve
denkl
oranrnrn
say:- saJ.
I j nee+r' suzgeq
Lrr r
FrischknechL
ve
T o C A , E , D i , l , :. . ) - 1 / f J 3
z=O
gerek Li rmek Ledi r.
vd
o
z=h=O
inLegrallerinin
Koefoed
I
T (A,B,D
t
sonra
el de
anlagrlaca$r
f
":t
r"=
t
L n IF=o
oI ar ak
( Kel I er
L
basi
daha.
1 973):
+-
H =C l_
L6.
"
r
err I
1',1
|L K O ^ 3
Bu iglemlerden
b.i I esenl
edi 1j r
el- de
we> C o I l e L
r'
-Cl-z
gi bi
aI an
Ckonwol
yar:rrlanrlarak
usyon
gekt i ncle)
36
4.1.lkr
Kaynak
( 1oo;:)
ol arak
halka
brr-inin
ile
alrcr
yarrgaprnrn
ikisi
1 954) .
de
Bu
bi ri nci I
1t<i
halkalr
halkada
c 1 lg r i l r r r ' .
1ki
j.krncil
c > r a n t "n d a n
zi yade
CmuLual
(Keller
halkaLr
induklenen
volLaj,
I
empedans
empedans
veya
Z=Y/1. olarak
Burada
1966).
verici
c5rrUne
k r : l ,l a n r l . r r -
kavramLarr
ise
gbz
al an
LopJ.amJ,ar.rn:.n,
I r kl r
kargrlrklr
(Wait
redi I mi s
sj.sLemler
kar5r
we FrischknechL
haLkalarrn
davranrr
alan.larrn
coupling)
igin
bu
nor mal j z.e
eger
halkalardan
mesafe,
gibi
dipol
ol arak
sisLemLer
fanrmLanrr
olarak
ve-
kuplaj
met.odl ar r nda
daha. briyukse
ni ce,l j.k
birincrl
al ana
kargrl:"kIr
kaLrndan
maLemaLik
5icJclerLi Lerimlerj.
sorrcla.j
lral kanl n
bi r
k uquk
LraJka ar-as-rnciaki
beS
mef odda
a-l.rnclrQrnda
i nrtrjk tl f
()ranlarl:
Krrpla3
t.agr yarr
akr m
l . :r r l I a n r l c l r g r
verici
her
vc. F.argrlrktr
Soncla.lI
Halk:l
halkaya
V
al:.cr
uygulanan
akrnrdrr.
Alrcr
duzenl
1954).
duzenl
enmesi
nde
rre
en
EM deri-nl-ik
emeler
ver-rci
cok
halkalarrn
kul l-anr l an
sondajLarrnda
gunl ardrr
1. Sistcrfn:
YaLay
a. Sisfem:
Dik
3. Srsfem:
Dugey
4.SisLem:
Dugey
(Sekil_
4
birbirine
si sLem
gogunlukla
wardr
06re
r
kullanrlan
( l{ai t
bu
4. 1):
eS-dtizlemli
CHorizonLal
cop-l anar)
(Perpendicular),
eg-duzlemLi
e5
eksenli
Cwertj
(verlical
cal
coplanar),
coaxial).
37
ttzayda
.'.i:r l:r:st.
k r . r t r :al . j
r 9 j r'r
Z
gekl:-nde
z-,p
z
('
(Ke-l-ler
yaz.rlabilir
karg:. 1r klr
' de
a.9agr dak i
gr bi
dik
kuplaj
birincil
alanr
ise
iqin
srfrr
verici
olduQundan
(74>
1z,p
C73)
dogrulLusundaki
ve
toplam
H
r,a
we {74)
)
alanlar,
Loplamr
i{, ve
denklemlerirrcle
Lopl am al anl a.rr
dogrul Lul ardaki
z,e
halkalar
=H zH
)
yazLlabilir.
Cl{
r .
oran:' ,
OfI
alanlarrn
Frischknechl
c73)
yd)ndeki
(Z/Z
r
v e r i I m - rg t i
ve
=H /H
birbirine
yaLay
kargrl":-klr
Keller
icirr
halkalar
Olzz,p
e. SisLemdeki
bobinin
g6sLerir'.
e5--dUzlein,l-i
Z2o
Burarcla
1S66).
FrischknechL
aLanr
yat.ay
oranl
)
ve
birincif
kuplaj
CZ/Z
i I gi I i
al:rrr
nc"i I
biri
halkada.ki
z.,p
dogrulLusundaki
Seklinde
hal k ar ar
duzl eml ,i
=*Cr,R3
=H
1. Sistemde
( 1 966)
( e$
Zo
al r cr
karStlrkll
arasrndaki
ol ar- ak ,
Z
t{
-i I i r .
c.l ,Jr:Qr-rrrrlan)
{?.-,9Oo
5 t c l d t : . t .j
gdster
uyl e
,sembol
hal k aI a r
SekIinde,
gc)sLermekt.edi r .
birinci
I
ve
H.
z ve
ikincil"
^n
t . sisrrv
vATAv
r:q-r.li_rzr-nvr.i
(Ht)RIZt
NTl\I-,
CoPT-ANAR)
ln
@' h -Y - - - . - - - - - 1 : ) 2 .
i.----+--R--;
oix
ffi
-;
Li/-
(\/ERTI(]AL
nn
$ekil
.r. sisrEv
DU$Ev es..rxseNr-i
|( +.
(\/ERTI(:AL
COAXIAL)
i i , ,i I
I - J- i i
(7ri)
.ril
I
DMD
F r - r ra d a
L : ' r t t : n t l ; r t - t - !r
halka
kullarrrlan
sonclajlarrnda
halka
lki
duzenl emel er i
4.1.
r ' - r la. t ' ; , ' l '
CC)PLANAR)
rrrm
v*--i ;'v
ffi
-----r-
<prnpr:NDTcLTLAR)
:. sisrEv
p\isev rq-ocizr-evr-i
^ \l/
f---l--x*;
lrlmrrfn
SISTEM
r ,p
i - - ;r i t r r : . i I
rqr-n
derrJe'rl r:'r'r ,
ql
II
--- t - 2
t,
_1\.,__ 11-)
-
= C . ' F - -l -
z,F,
-j(_-l L::
-ll
[- l \
tt )
,)
^2
r.5
( T!-.)
21=,f1::Q
ol drri!unr-lan
,
( 79)
=-C,,R3
II
;-el., I j titlt:
ol up:,
H
-O
orl at'r
Lkiirr-i-L
I E{,)')
r.l)
;
tlerlr,:.t- I r:t
r
:rY.
c
H
.,JT
z,a
()
,k
T (A,ii.[r,1..
ii
r.t,
1.rl,i
(.A,f,,ll
6n
lill rirr
tlL
r':l
t1
i'
|
.1 i 1: ,.
i
' - - r lr i
i- f:i
)
F J l r c J e r r r kl . ; t n ] r . l
72".2--
()
kar'5r
l.: kl.r
cl<+rrk
I eml erri ndern
l:,;rr'l: nr.l.r l .
krr;:l a.j
or anl
- L^l - K _ ^ 3
)
()t!
cC,/63)T
C.A,B,D
=-BUT(.A.B.D.1,.
L
[-rt-r1 i-rriurr
reel
ve
FrischknechL
sistemleri
igin
ayrrntrlr
Ke]ler
birgi
ve
ntn,
we sanal
4.2
ve
CoI I eL
ortam
i 9i n
arryle
Ker] I ,ar-
ve
Ig73
I .
we rT.
de
g(5st-eritmigLi,r
ve
lyait
1s5s).
kargrlrklr
Sinha
kuptaj
ve
coll
c1966)
denkl
eml eri
eL
Laraf
i ncel
rnclan
n
Rt ytr
(Keller
Dider
oranlarr
c1gz3),
vu=
s i s L e ; n r le r e
a m p . 1i l i i d l e r r n i
we 4.3
1966
(74)
B;
L
krsrmf
Frischknechf
( 73)
orant
ve
Homojen
Sekir
)
T,
i.Si nha
1966).
gc.:re clegigimi
)
,K
-C,'R3
.rjr,
rn
{g.ll
''tL
t: --
z2o!
..','4.)
vr.,
)
,k
Olr
=1 *R3T cA,R,r),k
aiL
i"i"\.t
i,, j ri
har.ck r:t l e,
LzJ,=
Fr j.scl:knecfrL
'-.r:.i..,;tr1.,r
l I
" r r . . tn r
-.C,,R3+CC,,63)T (A,R,D
.z-
,:,I ,.ii-..ri.'
vel
bobi n
hakkrnda
wai t.
veri
encll gi ncie
c1 s5s),
lmi5t.i
lf
z/Hr
r.
.'io
N
N
I
E
l f
:[
(!
Y
Y
.^
tJ
J
a
a
o
-o,f-,- \ ,,
\-1--.t
I
.i-,.'V,--,=-,---'-l
olz).56
I9rOIr2
;
SekiI
4.A. Homojen ort.am tizel-ii-iclt- T.
v€.
Ti.
:;i=t.em
igin
kargrlrkl
r
kuplaj
\/e]
orarrj,rlr-l nt n
rr.',:l
sana.l
(KelIer
k:.srm.l-arr
ve
Fr-i,schl:nechL
1966'clan
alrnmr 5Lr r) .
t.
o
N
N
o.5
Seki I
4.3.
Homoj en
yer yUzu
uzer i ncle
igin
kargrlrkli
krrprlaj
Cl{ai t 1955 clen al r rrrn: slt r ) .
rzr?
T.
or ii rll
T I . si sleml- er
amp} j. t-tici.l er I
r{ )
( 7..,,'. .)
(t
!t
--ii--
( 7.,'7. )
t
i.r
r ' . 1r j r r Q t . r k o J a y i - t k l ; l g r : r t - r i i e l ) . i l - 1I
C8 4 )
clenkI eml er i
I'l /ll
zr
or anl
yuk:rr': rJaki
1--BeT
n
- 1^1 3
i
I
l r . : l ' : 1 " :,
[ J r . J r ' : r r j . .t ,
i I '.r ( il3)
gi bi
i ' r r . , : . . ; Ir ; i
\rrj
k < - . n r _ ar lr a k
traGr trt,.t r:i;r yr:,t l I r.:a
ciak r
aga{r
H
I
;rnabi,l i r' :
'-|
1
t{
( "r.
I* r H
T ( A. B. D
)
lh=o
()IL
T CA,[f,D
z=A
kargrlr
kIr
A'-l
ciaki
Yukarl
kuplaj
,k
'F':l
( a:i)
,K )
hesapl
oI ar ak
sayr sal
uzere.:
<la J,,
hes.abr
sayr sal
oranlarlnln
96r itJ eceQi
cle
emi er-deD
denkl
i nLegral 1eri ni n
)'1
ve
T,
anmasi, n:
gerektirmekledir.
(7e)
clenklemlerin
sonug
kuplaj
k uI I anl I a.r ak
Korke'alaakso
gC5rulebilir.
DMD frekans
sorrclajr
oranlarr
(Z/'Zo)
el cle
ve
ol cJtrQu
egiL
Saksa
ndan
edi I ebi I ecedi
1S86).
cli r-
t>u
I ncc:1 endt Qr nde ,
denk 1 eml er i
Ca5)
birbirine
qtkarrlacak
kar6ilrkl:"
ve
r"
mo<lel
rrr
Buradan
e$rilerinrn
(7./7-i'),rt Y*
(, lii rrh::
:l-t?7S
oranl
ve
-aF
I
vr'I
4.4.T
itrtrt;l
.rji,'t
-i I .r
fiayl
:i.'rl
Oi ar ;rl.
Y ' r t - r l i n tI
llr.tr ;r lnt
i rrj l'r l,i trr''.t',i' l:.tt;'. it
a)I
l J e r : ; a 1 r) ; r . r r r r ; , rr: .:
4 . a ' : . 1 . S l r ; r g : + g K r r r r r . in r a s . r
Bu
bdl timrle,
k aLsayr I ar r nr n
d$nti:;
nas;r l
gaJ r 5r I acakLr r.
C e + kj r c l e k
Hanke.l
n
i:,i r
fc_'rnksr y(rnun
uml er- i yl e:
hesapl
arrcir g:
i l me-'yr,:
lldst.er
r:-l ma}.- uzel'e
Lamsayr
<\uzge'(
i i nee-r
k ( )'
der- c:,.-eden
tt.
II:;liker-l dtlt-rr,i5umtt,
(l]
=
KCb)
o.larak
LanrmIanl
Besse.l
f onksi
birlikLe
Bu
degigkenin
deQigkerr
( b\) \d).,
BrrraclaCok
k ar gr n
k C X)
karma5tk
J", a.
sayLcia
b >o
( B(i)
derreceden
araSLl
(Koefoed
1A7?,
dtSnugum degrgkeni
b)O
dol ayi si -v1.e
fonksiyonlarr
cldnugunil er i yl e
^
rmac:t
vd
ve
srf r r r rrct
Cekj rdek
Seklinde
sonucunda,
i rrLegrasyr-ln
yonuyl
f onksi
Anclerson
gerqel
K( b)
ci ns
e;
1S79).
b.i i' sayr
ger gei
bi r
olabilir.
sayr sal
sr-izgeg
k urmak
i gi n
(86)'de
x=J,';Cb) ,
y=J r,(7 /X)
ddnU5umrj
kul larrr.l arak
doQr u-l Lusuncla
l'rer i ki
r.
I.k{ \)
Hankel
denklem
-.
k ( _A ) . r
Lanrmlam.rglard:r
ddnugurn
o1 masr na
II
yonr,rdur.
j-ni
parameLres
I
I
yarrl
s j met-r' j
**
el de
ve
Cher
erLmek et.mek
I g j r-r)
i 1r: r;arpr I ar'ak ,
i kr
Lrr.t
apsi
S
rlt+nk -l emi n
+"j
CD
x\
x,,-
g
c:|
K(
J
-s
i nLerqral
bu
formtrna
i nlegr
sahip
) L
lilv
(
l ) e r r r k . le m C 8 7 )
edi I i r .
i ncel
oI.duQu
gClruIebilir
: Girj.9
fonksiyonu,
/
/
endi Oj rrcJe,
konvolUsyon
lineer
tl
inLegrali
(Papaoulis
Br-r
1S62).
af de ,
*K(
(>
.*)
t
**-YJ
Konwolusyon
bilinen
:
Qr k:. 9
c.*-v:.|,
Leoreminden
bul unabj
fonksiyonlarrn
aza.l"an
girig-qrkrp
c797e)
osyon
genel
nden
fonksiyonrarrnln
sonuqlar
suzgeq
urelen
da mumktn
Anderson
oldugunu
c1g7g)
clgzg)
ve
bl linen
suzgeql
suzgeqler
yanl
yl e
s1ra,
en i
erlcre
hlzl
r
cek j r ciek
suzgeqboyunurr
belirt-mjs.lercljr.
agagrdak:,
ve
oLarak
ke_vf i
edilmesi
bx_r
hrzrr
edi I en
Bunun
i 9i n
Anclerson
yanrLlarryIe
konvorve
dolayr
suzgeq
oIu5an
er de
gc)stermiSlerdir.
suzgeg
kul-Lanrrarak
gaIrgmalarrnda
I eri
uyumlu
1g6a).
bi r
fonksiyonlarrnclan
s6nen
krsa]Lrlamsrnrn
qifLi
cPapaouris
6nemlidir.
yapLrkrarr
ordukga
sebeplerden
oldukca
i nLegral
eciile,biie,ce'Qini
fonksiyonudur
yararlanrlarak
Hedef I enen
seqimi
vd
yonu,
fonksiyonlar:"
1i r.
Koerfoed
kul"lanarak
f onksi
Suzgeg
^J
girip-9rkrp
yanr Lr
oIarak
e'l de
x-\'
(- (. l
L^l
denkleminjn
k Ce-Y)
konvol
J
clenkJ" emi
inl.egral
r1
I x-y ,
.,
le
f
I K(e
iki
Bu
Hankel
'14
dAnr
( (ir- n rl1\lthl
r< iirnr r
kul I anar
ai
1|
g j Lr
\,/r'l
{:ryl'r
F : y z fr : |
s t tT r;r'r; l r'1' 1
k r.tr'tnrt:'t.ltr
r -b
.^'',
l '^.'J'
7
f-^'*
J
LD/\.,)ON
yat.ay
_2
ve'rlik
bobinler
lCi.n
benzer
fonksj.yonlara
(I97A>
wd
Koefoed
yapfrklanr
kurarkern
suzgeq
olarak
Chrzlr
tisLel
segmiqlerdir.
adr mlar
aSaQr daki
gdre
C197?)'e
Koefoed
kurmadaki
suzgeq
sonra
seqimi.nden
Fonksiyonlarln
diQer
/zlo,
( 8{r)
lda)
fonksiyonlar
azalan)
7
JD
<lrr'.
l:)O
galrgmalarrrrda
yukarrdaki
./1c.,
( tJtJ J
t
J
a)O,
.
(:a).
(-o/\, z )
Burada
i!j7q)
Arrt-ler-snr-:
t b A )d tr=
( -L,
o
19n5:
sekildedir:
7 .
en
k0Cuk
sabiL
hale
bir
Kuramsal
deQerinden
absis
Burada
(197Sj)
en
fonksi
Ax
ile
deSerine
drneklenerek
kullanrlacak
6rnekleme
yonl arl nl n
absis
buyuk
aralrgr
Ornekleme
getirilir.
Anderson
gi ri g-qr kr 5
stizgecin
herbi
ri
kadar
sayrsal
kurulmas;tnda
aralrdrnr
Ax=J i:( 1O) ,/7\ - 5j,3=O. 2
seqmisLir.
her bi r'i
elde
Bunun
i ci n
sebebi
ise
i nl-er pol asyon
edi lebilercegi
ni
girig
hat.asr
g6sLerebilmek
ve
rrr n
qrkrg
fonksiyonlarlnrn
1 O-d cian
iqindir.
k uqtik
ol ar ak
.1 s
"+J
Ksr-:f'Oeqiv,lclgTil)Y.r1-'itl.l.rlrl"'rr'"r':l'-.ririrrlr-l,ir
rryg(trr ()lar'al.,
t 5 r I ' r r : k i r * m < .a: r a l r g r n r
en
. , . l r j i r .rr. ; , 1 e r - c l i .l .
eme
Olnekf
r:lur !.Imunrla el r,le e,li I en
()1:rcaE-r nl
(
,.;ebep
1O) /6
Ax=ln(
l'Jer iki
f azl a
aLrnlr,
apclrnda.n
spekLrumu,
dOnu5umu
allnrnlF
hat.ay:r
karmasrk
sr f rrlal'a
c\rnekleme
ip]emine
uygun
grriS
bag,l anmasr yl €) krrrt.trlLtnabiI
r;tkt1-;
edj I i r '
e1 de
spektrumu
1' adrntda
iSlemint-len
se'q:ilert=k
de$e'ri
ba5langrq
(Koef oecl vd
inir
blJ]utrr'1r
spekLrumuna
b6>lul)rne
btr
i:r"rttr.t'+t
tJ$nrrlE;tJlnit a] lnnll 5
suzg6?q yant t, f onk:;i yonunun
Bu adrmda
'eyrrk
fonksiyr;p11r1
$rn'=klerren
ddnu$umu
BOyt ece
l-rr-r,yuk
bel i r f mi 91 er di r -
oJ acaQr nr-)
e.
daha
'l'
irt-t'"rli'
t.;l rir Ii:!.r
at' .l Ii
liaLal
1 OO-2OO kat-
iken
"l'it
L : , r . r l e : t j r " lr l + t i ' i ' r h ; r
ara.llgt-nrl)
se,;.i I me:;i
Ax'ln(-io';.1il
ve
197e
Ra5oktrr
1 EA4) .
3.
suzgeq
I
Sinc-ya.n:.
orneklenmi$
spekLrr-rmr: ile,
siec
si nc( y) =si n( ny/Ly:-
qarpll
spekLrumu
4.
e;lmek
'inc
j qj n,3.
yat-alr
ve
sjnc*yanrL
dr k
ve
4 . 5' de' Or- nek
bulunabiLi.r
=yanl
L
al r n1 r .
bobi
f az
ar ak
c'r.L
nl_ er
yani
{Bagokur
suzgeq
yonunrln,
1Sa4)
Koef oed
i qr r.r
vrl
(797e)
ha.zr r I a.rran
gara.f ikleri
i I mr gi er cli r' .
e l de+
k af say: I ar r nr
spekLrumunun
siizgeg
spektrrrmu
g()st.er
f onksj
2. adr nrdak.r siizr;eg
/(ny/LY)
arak
acitmdakr
Four i err dClnugunru
spekt.rtlmr-: '
L6jrs
aYrrk
! den
aL L nml $
suzgec
Sek j i
I '.-r-j n
4- 4
vt3
'{ L-r
Kl':efcrecl
ayrI
nt,r ] I
Ct karnlr
ve
Brrtra
n€iqat-if
yakl.rSLrQrnr
kaymasr
rregaLi
rJ(l1*:,
absjs
f
bir
bi r
Bi5yI ece
t>rneklenmig
Sek i I de
scJnuml
yc>l olduQu
sLnc
(Bagokur
Ax=i
. : ,J l i c - y a . n t
enmesi
srJzgercit-,
),
I-i
LlZeri
Lru
ne
at'al
yonttrrrlr)
yatda
t {r ,lr r' .
b,j r
, - : I r * l l ' -I, e m e r
m P t . o i i l.
sr f r r
i l-yrJrlrr
rme'trj tr
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6 , l.
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a
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tli nt.r ir;r,'1) I
rncrcir,ll
ayrrrnl
\/f-'1"l.'t
etJj 1 nrei. t.eil r r
ol
.a kolri.t
:,^
.t,
.
"'
.r.lt"
kc,nlrr|t,r,
ilo1
1-I ei<arr-l
iJt'rt.i.jrrl.jk
tlti
! ) c , . t t ( + 1 . 1 ' 4 i : i y ( r l r 1 . li - i l - F
yprln
i$.i rl-)
t.':.*
.\r-.'Vr
Tt
tl I 11.,Lr ll-^L**
8. Dl'1D Fii:Ei:.r\l'15
l 'l I
72
ti. 1
l.i-'t:i 1
l.al:i;rka
t-'r: r ki
,ira:.;.lrldaki
uzakllk
qc>s:;t.errrrrek{...+rli
r'-
i sr:: :ryf i rnl I I I k
1a
kri(,iik
F,f)
r>r-:,r,r
. : ij l , - l
, : .I 1 1 1 1 1 r
1 . 11 . i r .
t.t
i . : , r - r ; ' r tnl r a l I a d t
2tli..l-r
k't.;,tt}.
:.1 t^\,
! i . r i r - ; r r 1-
vFaI't'^l
" , ' r J rt l c ' t
t l r r r lrr.- ' l
ir'jrrr',
i,irJr;t'Jri
e t ' 1i l r r r i i i l l
|rrtyrrl
q , l lI n t r : : . . 1( l u r i . . t l T t ( I J t r j . i , , , : ; L i ; - r
<:rar)1.n1
Ru'I-l
f)::Ililttt
'', - li-:
,,,
v'.. h:r.t'.lrl.:,rr.rt)
qrubLr
ayrlmIrl.lqln
6?r.larr
t-: i.il
, - l e - : f l . i : .i ,t i l ' . : t . , : l
CRJ
ijr.r e+rJli
rnc*rlendi,]rnd€i
clr.rrurncia vt:
qrttlrrl
l r r . r . - l l€ f e $ r i
fJ. 1:).
CSekjl
ir,
, : i r - , - 1 , ' \ ' ' 1 . t '1l i . ' L
rr'l ry1;.,,;1
litJrtrlr.l
( ) l , 1 1 - .{1J l r l l
ill.lt-rltrttlli'ia
t-.
z
-a
, \
f a ' ,
l l _ . .
r0
-
?
r'^' J
t'J
T
1
l_:*,1".
r
'l
IJ, I
(-/
. Cl i-t1 nrltrt.
t?1
r rl .;1; ;;.p
r-f,,-.
,
llt ,
t - t ^ . -
-\-;
,|
/'(r
a I j )l.t i-,
';(),
f li-';.
eri t I ,.;t-,
s'l ,:li-.
()().
i,li,r
t
.
,
(
j
,
-1
R
i
r . - 1 r..' r i : ; i r r . t r
€ : ' r = lri ,' ' 1 r . i t l - : r i .
ii()t tnr'r
1 '. lr i "
73
t_i. i
i ,]Je li,r]ri Jr Or i.arn.larcl:'r
[)rr5r,rk I i et.teni
Met.r:rcJtl|r Penet.r ;1:'vr')r)(t:
i".ahakanr r)
Llst
rJegeri
ner
bir
ve
c'l dr-lQrr
sahrfJ
i-abakalr
t
claha
r)nce
duSuk
t.ai-rak anr
n
A. 2)
o.ro
e+cli1 en
A- V;
model
r=1
egr
ir:;t.
i rrc--.
sahi;r
ol citrk$a
gug
yapt.r
kl ar t
nokLaL
arcla
Schl umber ger
DMD
enebi
i r r .
FS
hem
de
sonda j r ncla
i^)
inc:e
s c ' n d a ' i1 a r r n d : r
FS
,-1r-r:;;irk
saplad:
r.tml:erger
Labakayr
saplayamaml
i l et. k r:rrl
vltl* -;€ti<
aYn.l
9 i . r r ' ' - l ' rr .
i rle''
nl t-l
a 1 3ij7)
sahadar
rir rSr:!
Ca] r s.mal ar
k.i ar t
ori..atrl;it',:l"r
fji ulrltr
r'e
araS't-r rrlr kl.rrr
Schl
erl rle'
q d , ? - . r l n i lc l n m e < l
i I tl
Munclr',2
Br-t
trs{-
qt- tlpl al. I l)l r)
s:r,hi 1,
qc.>k
ve
t-rMir
ar,r ri
yapmr sl ar dr r .
iteLkenl-ik.i
DMD f r eakans
sd;rl
rlj
.;al I 5malarrnda
arazi
sonciaS I ar r
CduSiik
t,abaka.l
o] acalJl
hem
v€-'reC€,qi
'is1
*ai'r.i 1r
gr)r'u1 rrrii5t iir" .
iletkenliEe
dUSrtk
c;(lr i1,:l' jrr
clr:Q:' 51- l" r r- i er ek
j g i
benzed
D,
nt tr:'
Y^Jrll,Jr'\;trtl.r-'
t - r - j t -j
Rtt
or anr
R,'D
Er.r
dz,liretr,-e)
r .
si-l
gr ubrrna
i
ar
:lq;nuCl
c:I dugu
OO
5o6r.1g olarak
i yi
annlanll
r;f sl.I
.1l.k i 5;.i rir:
i:).
yr-rr'llnrrt
Cyuksek
iIetkenliQe
( $ek i I
vr}
E..f.a
-:l
i - a l : : ; r k ; 1r r r r r
i i : . .1
.ikir
gt"k-rJt-1.- rgra.f.ik
agrk.ian,l.i{,r
r:ldrlkCa
meLc';ciun
($el.'i j
jmr:;{.rr
ara5t.trl
all
ol aral
tn
ilelkr,'rrl
clugrlk
kalrnlrQr,CeSigfir-ifr-:rek
et.kisi
f).=1 f)
uzerinde,
orLam
r:l rlrtCrrr
OOOO, i OO, I
o*/t|^=I
ayrrl
el ,Ekt- r- t k
sc)rlrJ{^ t)nr);)
ozrii r errf I i
r r r - , i ' 'ai i ; r r c i a k j
74
.-.-'-...-....-'--.-;
i
f
I
-J
I
I
-1
I
I
r'*'lo
I
lJ
-l
I
I
I
-l
|...1
-i-
I
I
-t
-
I
I
.'1
l
l
..,]
i
-.1
I
I
I
-.1
I
!
I
l
I
IC
-.)
-Tf
-T
-aJ
Sekf l
8. a. ^-
or=O. OOO! mho/m,
R-3OO
b-
n
, D, d e E l g k e n
a.=O' OOO1 mho/m,
R=1@
am/or-IOOOO,1OO,1
m , D
2
m,
C-37. 5,50, fr,7.O0,15O,aOO,
a^/o^-LAQQO,
dcgf gken
; Dr=1 O
1OO,t;
D.=5 m,
C -33, 50, 75, 1OO m).
3OO
m).
75
9. DT''DFS
MODEL EGRILERINTN
SC}ILT'MBERGER SO}.IDAJI
lrr]nFl
EGRiLERi tLE KARSILASIRILMASI
Bu bt5lOmde rlq Labakalr
sondaJr
model
kargr I agLrrr
efrllerlnin
modellere
DMD
alt
model
FS
Schlumberger
egrileri
iIe
yaprlmr gLrr.
I nasr
Model A
q = O. OOO1
,,
D=1()m
t
gpr=
mho/m
1OOOO ftn)
o.=O. o7. Cp"=1OO)
4
Ir,lode}
p=
2
-
37.5
50
-
78
-
o"= 1 6p"=
l'lodel A
sondaJr
modell
e.erllerl
yansrtan
rrerllmlsLi.
Sektr
100 - 150-aoo-300
1)
esas
arrna.ra.k
g. 1
sekir
DldD FS
g.1
rastranmayan
yoksek
R./Dr=3o olan)
1.
Egrl
de
hazrrranan
g6stertrmcktedlr.
egrlleri
de
lncerendlglnde
6zdlrenqrt
Labakanrn
etkrsl
r*$r i I er I nde agr k qa 96r ttl mekLedi r .
schlumberge.r
$ekll
DMD Fs
c1o
m
Ayn:.
g.2. a
da
egrltertnde
karrnlrgrndakl
schrum.berger
sondaJr
No
76
10'
10!
itl
d
1oI
a
arl
o
Sekf I
9. 1.
l'4odeJ A tsas
alrnarak
SondaJr
nodel
e$rllerl
CC--
-)lkitabaka,
snAB/2
o
hazr rlanan
Schlumborger
C3 tabakal"r
ortanr)
egrlsl).
?
()
T7
f'lodrl
a * O. 01
r.
1OO m
1OO Cltt)
Cp.=
1
urho/n
B
5432Ir
50 rn
-
O. 1 - O.OO5 -
-
o=
7. -
-p2=
L
e
5t
10
eOO
6
5
1
3
?
2
O.5
O.2
Model
O.OOI
1OOO
--->
I
O. OlCp =
Schlunbar6tcr
sondaJr
g6sterltnektedlr.
DMD
FS
3.
tabakanrn
3.
Bunun
ve
etklsl
yanr
sr'ra
Scilumberger
bellrglndlr.
o"/or)1
sondaJr
vG
9- 2. a
Fektl
or/or(1
lncolend-ldllnda
egrllerlnde
nodel
e.grllarl
c,grf lcrl
9. e.
Dl'lD
hazrrlanan
alrnarak
rndeL
So&II
g6r0lrpnrektedlr.
rpdel
1OO)
Gtsas
B
I'bdel
oldu$u
rrodal
FS
ve
b
dc
olan
a{.klsl
tabakanrn
No
Sch.S-
l.todal
o"=
Egrl
DMD
e$rllerinde
F:S
Egrrl
No
e
e
{
10
-'
Fr rk.nr
10.
b
,
6
a.
lor
'
-g:r'A8/2t
Sekf f
9.a.
iiodel
(a)((
e{rllerl
C-
-)
Labaka
-)horrcJen
tabe.ka nodel
lkl
rre Schlunberger
lkt
hazrlanan
alrnarak
B esas
sondrJr
model
!
t-A8/2
e$rllerl)
DMp
npdcl
l''s
nodel
er$ri '
egrl}orl.)
nodel
e$rllerl
(b)
CC-*--)
79
Hodel
D=
I
c r = O . O 1 mho/m
1OO m
C
1Pr=
I
1OO fl'n)
o =O. OO1,
=O. OOO5, =O. OOO25
p2 =1OOO,
=2OOO,
2
=4OOO
Model
D_
z
Egri
No
10 - a5 - 50 - 100 - o.o
ar= O.O1 gp"=
liodel
parametreler
C dekl
DltD FS ve Schlumberger
g6sferllmektedlr.
kalrnlrgr
sahlp
ragmen
56z
konusu
gd)r0lme'mekLedir.
$ekil
9.4. a daki
numaralr
9.4. c
egrl
dekl
ve
1
dururndadrr C$ekll
egrller
IIe
Cakrgnrg
gakrgma
$ekfl
3 numaralr
egri
lle
9.4. b dekl
9.4. d).
ve
egrilor
9.4
e.
olan
FS
aynr
farklr
durumdadrr
Sekll
3 numaralr
model
Labaka
kontnast,r
S.4.
de
tabakanrn
DMD
Schlumberger
srra
numaralr
bu grup
lleLkenlik
Bunun yanr
Sekll
ve
=O. O25
hazrrlanan
9.3
Sekll
edllen
=O. O5,
blrblrl
alrnarak
kontrastr
elde
or/o^=O.L,
esas
efrileri
tlet-kenllk
kalrnlrklarrna
9.3).
model
degfstlrilerek
egrllerlnde
olmasrna
1OO)
Cseklf
eSrllerlnde
lncslendlglnde
9.4, b
efrl
birbiriyle
dekl
lte
e
$ekil
Cakrsmr$
s
Lodal
D -
111
o = O. Ot
1OO m
n*loln
C*1
Cp = 1OO Ctn)
o =O.OO1,
2
=1000,
p
'2
L23
Model
Egrt
No
D=10-50-1OO
2
a"=
O. O1 qp"=
1OO)
t{odel
D = 1OO n
o = O. Ol
I
I
mho/n
C-2
= 1OO Ctt)
Cp
'i
a =O. OOs
2
=aOOO,
P
,2
I
2
-----)
3
Sch.
Model
D=10-50-1OO
2
ar=
O, Ol
7p"=
1OO)
Sond.
Egri
No
81
10
z0
L
o
tlt
10
0
!
f
r
I
o
c
$ekll
9.3.
l'lode1
C
nedel
eSrllerl
c----)lkI
esas
alrnarak
CC-.-.-.-)
tabaka
spdel
hazrrlanan
Dil{D FS
honroJen model
efrlsl).
egri
f32,
Model
C-3
cr,- O. 01 mho/m Cp = 1OO ftn)
D-= 1OO m
ltt
o -O. OO25
2
'p2
=400(),
L23
D_= 10 - 50 - 1OO
2
Model Eerl
o"= O. 01 6pr=
1OO)
Model
D = 1@
itt
a = O. 01
m
o =t
22
No
D
mho./rn Cp = 1OO Cln)
Ce =L)
I234
D_= 5 - 10 - aS - 50 m
?
Model Eerl
No
Model
No
I234
o= O.Ol
a9
Cp=
lletkenllk
kalrnlrgrnl.n
hazrrlanan
9- 5.
DMD FS ve
g6steri
de
altrnda
yer
R,zD_=6O,
2
egrllerlnde
1OO)
kontrastlnln
deglgken
oIduSu
l mektedl r .
ytksek
30,1a,6
olan
oldukqa
sablt
Model
Schlumberger
alan
EQrl
RrDr=3
D
model
oldugu
esas
1".
lIeLke.nllkIl
CdUSok
tabakalarln
eLkllerl
bellrglndlr.
alrnarak
egrllerl
ol duSu
D"
gekil
tabakanr
n
5zdLrenqLl)
DiltD
FS
sekll
9.4.
llodel
c-1
hazrrlanan
ca),
c-e
cb),
Schf urnbrger
c-3
cc)
sondaJl
Qsas
alrnarak
rrodel egrllerl(d).
r-A.8/2
Sekf l
9.5.
l-lodel
eErllerl
D esas
(a)
Schlumberger
alrnarak
CC- sondaJr
hazrrlanan
-
-)
homoJen
nodel
DMD
rrodel
egrl]erl
FS
efrl)
(b).
model
ve
85
Sonug olarak,
sahlp
ince
eQrllerlnde
usf
rasLra.nmrgtrr.
e,frl I erl nde
konLrastr
a.yrr ml r 1r k
deglldir.
sondaJr
or/or)t
sondaJr
olup
metod
her
lkl
Dl'{D
schlumberger
FS
da lyl
vrs
durum
Dt'{D Fs
t I etkenl
AB4=5OOO
ortamrarda
nazaran
m) .
o./or)1
ordukga
konusu
DhtD
oldukga
FS
ve
y(rksek
Bunun
alrcr-verici
agrrrm
i k
oldulu
s6z
vermigtir.
FS
sondaJr
durumrarda
ayrr.nlrlrk
kurranrlan
Lahml n edl I nrekf edl r .
Dl,{D
ortamlarda
sonuglar
sondaJrna
etkisine
CO"/Cr)L)
bu
kullanrlan
tabakalr
Ozdirence)
azaL makta
tabakalr
CR=3OOn
DMD Fs schtumberger
oldugu
or/orlL
egrilerinde
nda
sahlp
-
(y0ksek
schrumberger
e{rllerLnde
sondaJrnda
k0gOkt0r
ileLkenllge
rnrecegl
o"/orlL
ol dukga
oldugu
Schlumberger
<a5)
ra0men
saptanamamaktadrr
Schlumberger
ol dukga
(R/D
tabakalartn
rasLrannamasrna
e.prllerrnde
srra
dOSOk iletkenliSe
yanr
uzaklr{r
uzaklrfrndan
A)rr ca
oldugu
lyt
driSrlk
zaruln
sonugrar
86
1O.
SOf.ftCLAR
\ e
HorrcJen
bl I cSenl
erl
tabakalr
hesaplannaslnln
blllnrnektedtr-
DldD FS
potansl)€Ilnden
vektor
lntegra.l
Ilneer
denkleml
suzgeg
yez:-lmrg
olan
g()re
oranlarrna
tkl
tabakalr
hazrnlanan
nrodel
yaklagr.rala
)mumunun
sondaJr
e$rllerl
sondaJr
Schlumberger
lletkenltse
(y0ksek
6zdirence)
2-6
t
sonuqlar
wermlgtlr-
e$rllerlndekl
oldukga
arasrnda
oldulu
OLe yandan,
ayrrmlrlrk
dOSOk
saptanam.amaktadr
olmakta
r.
g6rolnogtor-
arazl
eQrllerlnln,
graflk
oranr
anlagrlmrgtrr.
, R/D
oranr
ve
sondaJrna
or/or(.1
Schlumberger
orror)1
durumunda
eErllerl.ne
llet-kenllk
d0g0k
nazaran,
oldukga
durumlarda
ye
sonucunda,
ortamlarda
sahlp
R
Schlumberger
kar$rlagLrrrlmasr
Dl,{D frekans
R.zu-, oranr-
olacagr
oldufu
kolay
R/D
ve
or/o,
DMD FS eQrllerlnln
lle
g6re
yararlanrlarak
dr., o"/o^
g6rolmo$tor.
oldugu
FS
egrllerlnden
DltfD FS ayrrmlrlrgrnrn
bagrmlr
uygun
Dl,'lD
oldukga
dlllnde
paranetrelere
e$rtlerln
olup,
ait
FORTRAN
hangl
hazrrlanmasrnrn
ortama
hesabr
sayrsal
yaprLmrgtrr.
lle
eQrilerlnln
aragtrrrlnrg
Blrer
hesaplanmlgtrr.
yararla.nrlarak
program:-
de
blle5enlerl
bllegenlerinln
alan
kuramr.ndan
DMD FS nodel
ya-ylnl ardan
alan
yararlanrlarak
bllglsayar
hazrrlanaca$r
gerekll
191n
kullanarak
olduQu
uygun
oldukga
yel I
potansl
rrckt6r
EM
nl n
alan
manyretlk
lqin
ortam
rae
hassas
DlrD
FS
nazaran
konLrastr
a7
KAYNAI.II.AF:
ANDERSONT tl,l-.
1975.
Irnproved
for
drqttal
frlte,rs
eval uat i ng
Four i er
and
Hankel
transform
rnterlrals.
U.S.6,S.
Rep. U.S.6.S-Gd-75-O12e
avai I
es
N TI S
rep,
PB-242-BCtCI/ LbtC,223 p.
A N D E R S O N , l . l .L .
t979.
N u m e ri c a l
i nt'eqrat i on
of
rel atecl
Hankel transforms
of orders (i and 1 by acJaptrve digital
f i I t e r i n g . G e o p h y s r c s 4 4 , L 2 8 7 - -t 3 C ) 5 .
ARFKEN,
G.
Ac aderni c
BASOKUR, A. T.
Ankar a
1985.
Pr ess,
f.lathematical
London.
f.lethods
1'144. D0gey El ekt r i tr
B H A T A C H A R Y A ,B - K . ,
two layer earth.
for
Sanda.jr .
Ptrysi,:ists.
yayrnr ,
TPAO
1955.
Electromagnetrc
induction
J. Geop. Research 60(3) , 275-ZBB-
KELLER, c. v- ve FRTSCHKNECHT, F.
c- ,
Methods in Geophysical prospecting.
York.
1966.
pergamon
rn
a
Electrical
press,
New
KOEFOED, o-, GHosH, D. p., poLMAN, c. J.,
rg72.
computatlon
of type curves for EM depth
sounding
r,ith
a
horizontal
transmitting
corl by
means of
digital
linear
frl..er
Geophysi cal prospect ing 1"4, 229-24I .
K O R K E A L A A K S O ,J .
ve
SAKSA, p., 1986.
Calcuiation
of
EM
field
components in frequency and trne domain using dipole
source
excitation
an
layered
earth
model.
Technical
Research Centre of Finland,
Research Notes 5g7.
MCLACHLAN, N. V. ,
1934 .
Bessel
Oxford Uni- press, London.
MUNDRY, E. ve BLOHM, E. K.,
using vertical
magnetic
35, 110-123.
PAPAOLILiS, A.,
A1_'pljcatjonPATI1A, H. P, ve
?. Elsevler
]987.
dipole.
Functrons
for
Frequency
geophysical
Engineers,
EM sound;.rrg
prospectlng
1962.
The
Fourjer
Integra)
M c G r a w H i 1 l L . r e s s , n - e v Y cr k .
MALLICii,
X.,
Press,
l-ondori .
19t0.
Geo:ri,uncirirg
arrd
I'rtrrclpl
aa
PI{ILLIPS, H - B,'. ,1964. Vektorel
A.U. Fen Fak. YayLnl, Ankara.
Anaiiz.
ERGUN,
Qerviri:A.N.
SINHA, A. K. , 1979.
Maxiprobe
wide-band
A new
EMR-16 :
tnultifrequency
ground EM system. Current Research Part
B,
Geological
Survey of Canada, Paper 79-IB, 23-26 SINHA,
A.
K-,
1980A
study
misorientation
effects
in
Geoexploration
IB, III-I33.
SINHA, A.
K.
oscillating
conducting
7 3-25 -
ve
S-,
1973.
COLLET, L.
placed
magnetic
dipoles
Geological
Survey
earth.
VANYAN, L. L.,
Consultant
1967Bureau,
Electromagnetic
New York.
EM
fileds
of
rnultilayer
over
a
of
Canada,
Paper
Soundinq.
Depth
WAIT, J. R. ,
Mutual
1954.
coupling of wire loops
homogeneous ground. Geophysics 19,290-296.
L/AfT, J.
over
and
sounding.
of
topographic
EM
roultifrquency
lying
R.,
I955.
Mutual
electromagnetic
coupling
a homogeneous ground. Geophysics 20,630-637.
WARD, S. H.,
In Mining
1967.
EM Theory
for Geophysical
Geophysics, V-2, S-8.G., Tulsa-
L I A R D , S . H . v e H O H M A N N ,c . W . ,
l_987. EM
Geophyslcs-Theory
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