jheh]bq_kdh]h h[_ki_q_gby� ba]hlh\e_gb_ fZd_lZ ijb[hjZ b _]h bkiu-lZgby�

< gZklhys_c jZ[hl_ ijb\h^blky j_r_gb_ aZ^Zqb h g_klZpbhgZjghf

we_dljhfZ]gblghf ihe_ lhdh\h]h dhglmjZ� jZkiheh`_ggh]h \ ijh\h^y-s_c fZ]gblghc pbebg^jbq_kdb�kehbklhc kj_^_�

F_lZeebq_kdb_ dhehggu� ba]hlh\e_ggu_ ba n_jjhfZ]gblguo fZl_-jbZeh\� h[eZ^Zxl� \ hlebqb_ hl kj_^ h[uqgh jZkkfZljb\Z_fuo \ ]_hnb-abd_� \ukhdhc we_dljbq_kdhc ijh\h^bfhklvx �σ ≈ 107 Kf�f� b \ukhdhc

fZ]gblghc ijhgbpZ_fhklvx �µ hl ���⋅µ0 ^h �� 000⋅µ0, µ0 = 4π⋅10–7=g�f ±

fZ]gblgZy ihklhyggZy�� Ihwlhfm \ h[eZklb l_hjbb ZdlmZevgZ aZ^ZqZ hih\_^_gbb g_klZpbhgZjgh]h we_dljhfZ]gblgh]h ihey \ pbebg^jbq_kdb�

kehbkluo ijh\h^ysbo fZ]gblguo kj_^Zo� WlZ aZ^ZqZ \Z`gZ dZd ^ey bam-q_gby nbabq_kdbo k\hckl\ kZfh]h \_s_kl\Z f_lZeebq_kdbo h[kZ^guo

dhehgg� lZd b ^ey ihke_^mxs_c ihklZgh\db b j_r_gby [he__ keh`guo

l_hj_lbq_kdbo \hijhkh\ \ebygby ^_n_dlh\ dhehgg gZ g_klZpbhgZjgh_

we_dljhfZ]gblgh_ ihe_� fhgblhjbg]Z bo khklhygby�DZd ba\_klgh >�@� ^bgZfbq_kdb_ oZjZdl_jbklbdb n_jjhfZ]g_lbdh\�

\ qZklghklb� kj_^gyy fZ]gblgZy ijhgbpZ_fhklv� k\yaZgu \ hkgh\ghf kh

k\hckl\Zfb kZfh]h \_s_kl\Z b k we_dljh^bgZfbq_kdbfb k\hckl\Zfb

gZfZ]gbqb\Z_fh]h l_eZ� I_j\h_ hij_^_ey_lky ijbjh^hc ghkbl_e_c fZ]-gblgh]h fhf_glZ� kha^Zxsbo gZfZ]gbq_gghklv� Z lZd`_ bo \aZbfh^_c-kl\b_f� kljmdlmjhc n_jjhfZ]g_lbdZ b baf_g_gb_f wlhc kljmdlmju ijbgZfZ]gbqb\Zgbb� <lhjh_ hij_^_ey_lky hkh[_gghklyfb jZkij_^_e_gby

i_j_f_ggh]h we_dljhfZ]gblgh]h ihey \ jZkkfZljb\Z_fuo l_eZo� \ qZkl-ghklb� g_iheguf ijhgbdgh\_gb_f i_j_f_ggh]h ihey \gmljv ijh\h^gbdZ�

J_r_gb_ aZ^Zqb ^ey g_klZpbhgZjgh]h ihey [m^_f ijh\h^blv \

ij_^iheh`_gbb� qlh k\hckl\Z kZfh]h \_s_kl\Z hklZxlky ihklhyggufb�l� _� hlkmlkl\m_l \j_f_ggZy ^bki_jkby� Z fZ]gblgZy bg^mdpby b gZijy-`_gghklv fZ]gblgh]h ihey k\yaZgu ebg_cgh�

IhklZgh\dZ aZ^Zqb

<_jlbdZevguc fZ]gblguc ^bihev k fhf_glhf Fz = J⋅S ihf_s_g \gZqZeh dhhj^bgZl �� Hkv z kh\iZ^Z_l k hkvx pbebg^jZ� Ihklhygguc lhd

k Zfieblm^hc J� l_dmsbc \ djm]h\hc jZfd_ fZeuo jZaf_jh\� \ fhf_gl

t = � \udexqZ_lky� <k_ ijhkljZgkl\h khklhbl ba lj_o h[eZkl_c� \gmlj_g-g_]h pbebg^jZ� pbebg^jbq_kdhc h[hehqdb b \g_rg_c kj_^u�

Mqblu\Zy kbff_ljbx aZ^Zqb� aZibr_f mjZ\g_gby FZdk\_eeZ ^eydhfihg_gl ihey� hlebqguo hl gmey� \ pbebg^jbq_kdhc kbkl_f_ dhhj-^bgZl

( )( )

∂∂

∂∂ σ

∂∂ µ

∂∂ µ

∂∂

∂∂

ϕ

ϕϕ

+

]

+

U(

(

]+

U UU ( +

+U U

U ++

]

U ]

U ]

U

]

− =

= ⋅ ⋅ = − ⋅

= ⋅ ⋅ + =

�

� � � �

GLY �

�

��

&

(1)

A^_kv b \ ^Zevg_cr_f lhqdZ gZ^ nmgdpb_c hagZqZ_l ijhba\h^gmx

ih \j_f_gb�Hl^_eyy dhhj^bgZlm z� ij_^klZ\bf dhfihg_glu we_dljhfZ]gblgh]h

ihey \ \b^_

(0

) U W ] G]

ϕ πλ λ λ=

∞

∫�

�

�

� � � �FRV� � , (2)

+0

5 U W ] GU

]=∞

∫� �

�πλ λ λ� � � �VLQ� � , (3)

+0

= U W ] G]

]=∞

∫�

�

�πλ λ λ� � � �FRV� � . (4)

A^_kv mql_gh� qlh dhfihg_glu ihey ?ϕ b Gz kbff_ljbqgu hlghkb-l_evgh z, a Ηr ± Zglbkbff_ljbqgZ�

Ih^klZ\eyy \ujZ`_gby ���±��� \ ���� ihemqbf \ dZ`^hc h[eZklb �� ��

� ke_^mxsb_ mjZ\g_gby ^ey Nmjv_ h[jZah\ F, R, Z:λ σ5 = )

U− ′ = , (5)

− =λ µ) 5� , (6)

�

U UU) =

∂∂

µ� � �= − , (7)

�

U UU5 =

∂∂

λ� � = . (8)

Hlf_lbf� qlh σ0 = σ2 = 0, µ2 = µ0 �\gmlj_ggbc pbebg^j b \g_rgyykj_^Z g_fZ]gblgu b y\eyxlky baheylhjZfb� .

IhklZ\bf aZ^Zqm ^ey R� ihkdhevdm wlh h[jZa ghjfZevghc ih hlgh-r_gbx d ih\_joghklyf jZa^_eZ kj_^ dhfihg_glu �fZ]gblgZy fh^Z��

Ba khhlghr_gbc ���±��� ihemqZ_f mjZ\g_gb_

′′ + ′ − + =5U5

U

U5 5

UU U

� � � �

�

λ µσ � (9)

b ]jZgbqgu_ mkeh\by gZ ih\_joghklyo jZa^_eZ�

[µR] = 0, [R + r⋅ ′5U] = 0, ijb r = r 1 = r 2. (10)

GZqZevgh_ mkeh\b_ aZdexqZ_lky \ lhf� qlh ihe_ \ gZqZevguc fh-f_gl \j_f_gb �ijb t �� jZ\gh ihex ihklhyggh]h fZ]gblgh]h ^bihey \

jZkkfZljb\Z_fhc kj_^_� Ijb r → ∞ b t → ∞ Zfieblm^Z ihey klj_fblky dgmex�

J_^mdpby d aZ^Zq_ RlmjfZ �Ebm\beey�

J_rZ_f aZ^Zqm ���±���� f_lh^hf� ij_^eh`_gguf :� G� Lbohgh\uf

^ey ]hjbahglZevgh�kehbkluo kj_^ >�@�

IheZ]Z_f 5 U W ; U HW� � � � �= −α\kx^m \ h[eZklyo �� �� � b \ ^Zevg_c-

r_f [m^_f bkdZlv R \ \b^_ kmi_jihabpbb j_r_gbc lZdh]h \b^Z�

<g_ pbebg^jZ

; U ; U, U

, U

� � � �� � � �� �

� �

� ��

�

� �

= λλ

r ⊆ r1, (11)

; U ; U. U

. U

� � � �� � � �� �

� �

� ��

�

� �

= λλ

r ⊇ r2. (12)

<hkihevam_fky mkeh\byfb i_j_oh^Z� dhlhju_ l_i_jv hlghkylky d

nmgdpbb X. < j_amevlZl_ ihemqbf�

ijb r = r1 ; U ; U, U

, U; UU

��� ��� ���� �

� �

� �� �+ ′ =�

�

��

� �

� ��

µµ

λ λλ

,

ijb r=r 2 ; U ; U. U

, U; UU

��� ��� ���� �

� �

� �� �+ ′ = −�

�

��

� �

� ��

µµ

λ λλ

.

LZdbf h[jZahf� \ h[eZklb r1 ⊆ r ⊆ r2 ihemqZ_f aZ^Zqm RlmjfZ�Ebm\beey ^ey X:

U ;U; U N ;UU U

� � ��� �′ + ′ − − =� � , r1 ⊆ r ⊆ r2,

; U ; D ;U+ ′ =� � , r = r1, (13)

; U ; D ;U+ ′ = −� � , r = r2,

]^_ D U, U

, UD U

. U

. UN�

�

��

� �

� ��

�

��

� �

� �

�� �

�= = = −µµ

λ λλ

µµ

λ λλ

αµ σ λ� �

� ��

� �

� �� �

H[s__ j_r_gb_ _klv X = A J1(kr)+B N1(kr�� A^_kv J0, J1, N0, N1 –nmgdpbb ;_kk_ey b G_cfZgZ gme_\h]h b i_j\h]h ihjy^dh\� Ijbq_f

; U; $- NU %1 NU NU ; NUU+ ′ = + =� � � � �� �� � �

Ba ���� bf__f kbkl_fm ^ey hij_^_e_gby g_ba\_klguo dhwnnbpb_g-lh\ : b B:

NU $- %1 D $- D %1

NU $- %1 D $- D %1

� �� �� � �� � ��

� �� �� � �� � ��

�

�

� �

� �

+ − − =

+ + + =

, (14)

]^_ - - NU - - NU 1 1 NU 1 1 NUL L L L L L L L� � � � � � � �= = = =� � � �� � �� � �� .

Hij_^_ebl_ev h^ghjh^ghc kbkl_fu ���� ^he`_g [ulv jZ\_g gmex�

qlh b ^Z_l mjZ\g_gb_ ^ey kh[kl\_gguo qbk_e k:

> � � � �@ > � � � �@

> � � � �@ > � � � �@ �

- N UD

N U- NU 1 NU

D

NU1 NU

- NUD

N U- N U 1 NU

D

NU1 N U

� ��

�� � � �

�

�� �

� ��

�� � � �

�

�� � �

− ⋅ + −

− + ⋅ − =

&h]eZkgh l_hjbb ^ey aZ^Zqb RlmjfZ�Ebm\beey� bf__lky ^bkdj_l-gh_ fgh`_kl\h kh[kl\_gguo agZq_gbc b kh[kl\_gguo nmgdpbc� LZdbf

h[jZahf� h[s__ j_r_gb_ ^ey nmgdpbb R _klv

5 U W & ; U HMM

M

WM� � � � �= ⋅=

∞ − ⋅∑�

α, (15)

]^_ Xj ± FR[kl\_ggu_ nmgdpbb aZ^Zqb RlmjfZ�Ebm\beey� αλ

µ σM

MN=+� �

� �

�

Dhwnnbpb_glu Kj ^he`gu [ulv hij_^_e_gu lZd� qlh[u m^h\e_l\h-jyehkv gZqZevgh_ mkeh\b_�

5 U W & ; U HMM

M

WM� � � � �==

∞−∑

�

α, (16)

]^_ 5 �klZpbhgZjgh_ j_r_gb_� m^h\e_l\hjy_l djZ_\hc aZ^Zq_

U 5U5 U 5UU UU

� � ��� �′′ + ′′ − + ′′ =� � �λ 0 ⊆ r ⊆ ∞,

> @ � > @µ5 5 U5U= + ′ =� � , r = r1 = r2, (17)

5 . U→ − ⋅��� �λ λ ijb r → 0,

5→ � ijb r → ∞.

>ey hij_^_e_gby Kj \hkihevam_fky hjlh]hgZevghklvx nmgd-

pbc U ; M � Mfgh`Zy \ujZ`_gb_ ���� gZ U ; N UQ⋅ � � b bgl_]jbjmy� ihem-

qbf

&

U5 U ; N U GU

U; N U GU

M

U

U

M

MU

U=∫

∫

�

�

�

�

�

� � � �

� �

λ. (18)

Bgl_]jZe \ agZf_gZl_e_ \ujZ`_gby ���� \uqbkey_lky ZgZeblbq_kdb

U; GUU

N U; N U N U D D

U

N U; N U N U D D

U

U

MM M

MM M

� �

�

�

�

�

�

�

�

�

�

�

�

�

�

�

�

�

�

�

�

�

�

�

�

�

�

�

�

��

��

∫ = + + −

− + −

� �� �

� �� ��

(19)

Bgl_]jZe \ qbkebl_e_ \uqbkebf� bkihevamy lZ[ebqgh_ jZ\_gkl\h

���� >�@ b khhlghr_gby f_`^m nmgdpbyfb ;_kk_ey J(x�� G_cfZgZ N(x� bfh^bnbpbjh\Zggufb I(x), K(x� >�@� < j_amevlZl_ ihemqbf�

U5 U ; N U GU

$$

U- N U , U N U- N U , U

N

U

U

M

M M M M

MU

U

�

�

�

�

� � � �

� �

∫ =

=−

++

� � � �

� � � � � � � �

λ

λ λ λ

λ

+−

+−%$

U1 N U , U N U1 N U , U

N

M M M

MU

U

� � � � � � � �λ λ λ

λ� � � �

� �

�

�

(20)

−++

−$%U- N U . U N U- N U . U

N

M M M

MU

U

� � � � � � � �λ λ λ

λ� � � �

� �

�

�

−+

+%%

U1 N U . U N U1 N U . U

N

M M M

MU

U

� � � � � � � �λ λ λ

λ� � � �

� �

�

�

,

]^_ ; $- N U %1 N U 5 $ , U % . UM M= + = +� � � �� � � �� � � � ��

λ λ Dhwnnbpb_glu :

b < k lhqghklvx ^h khfgh`bl_ey hij_^_ebf ba i_j\h]h mjZ\g_gby �����

hldm^Z

$ 1 N U D 1 N U % - N U D - N UM M M M= − = − +� � � � � � �� � � �� � � � ��

Ba mkeh\bc gZ ih\_joghklyo jZa^_eZ \ ���� gZc^_f dhwnnbpb_glu

:* b <

*

$ % I I = − −�

�

�λ ,

%I , U . U , U . U

. U . U I , U . U , U . U

> � � � � � � � �

> � � � �� � > � � � � � � � �@�= +

− − +λ λ λ µ λ λ

λ λ µ λ λ µ λ λ

�

� � � � � � � � �

� � � � � � � � � � � � ��

I, U . U , U . U

, U , U�

� � � � � � � �

� � � ��

= +−

� � � � � � � �

� � � � � ��

λ λ µ λ λµ λ λ

IU , U , U

�

� � � � �

�

�=

−� � � � � ��

µ λ λ λA^_kv µ ± hlghkbl_evgZy fZ]gblgZy ijhgbpZ_fhklv jZ\gZy µ1/µ0.

BlZd� \gmljb f_lZeebq_kdh]h pbebg^jZ j_r_gb_ gZc^_gh� hij_^_-e_gu \k_ nmgdpbb� \oh^ysb_ \ ����� < khhl\_lkl\bb k mkeh\b_f ���� gZ

ih\_joghklyo jZa^_eZ ijb r = r1 bf__f µ0 R0 = µ1 R1 � qlh iha\hey_l gZclb

j_r_gb_ aZ^Zqb \h \gmlj_gg_c ihehklb pbebg^jZ �r ⊆ r1):

5 &; N U

, U, U HM

M

M WM�

�

�

� ��=

=

∞ −∑µλ

λ α� �

� �� � .

Ba ��� b ��� gZoh^bf \ujZ`_gb_ ^ey w�^�k�� bg^mpbjm_fhc \ dhglmj_

jZ^bmkhf r� ihf_s_gghf \h \gmlj_ggxx h[eZklv ijh\h^ys_]h fZ]gbl-gh]h pbebg^jZ

εσ

λ λλ

µσ= +∞

=

∞− +

∫ ∑U

U*, U ]

, UP J & ; N U H GJM

MM M

P J W

U

M

��

�

� ��

�

���

� �

�

�

�

�

�� � FRV� �

� �� � � � �

� �

(21)

]^_ *0 J

U

]= �

�π� _keb bklhqgbd ± fZ]gblguc ^bihev� * -U , UN N= � �� �λ ,

_keb bklhqgbd ± dhglmj k lhdhf� & & J U= =λ λ�� �� , J ± Zfieblm^Z ih-

klhyggh]h we_dljbq_kdh]h lhdZ�

LZdbf h[jZahf� ihemq_gh bgl_]jZevgh_ \ujZ`_gb_ ����� hibku-\Zxs__ g_klZpbhgZjgh_ ihe_ ^bihevgh]h bklhqgbdZ b dhglmjZ k lhdhf

\ ijbkmlkl\bb ijh\h^ys_]h fZ]gblgh]h pbebg^jZ� <ujZ`_gb_ m^h[gh

^ey qbke_ggh]h ZgZebaZ� gh� dZd ke_^m_l ba ����� ijb fZeuo \j_f_gZo b[hevrbo fZ]gblguo ijhgbpZ_fhklyo koh^bfhklv jy^Z mom^rZ_lky�

H[km`^_gb_ j_amevlZlh\

fZl_fZlbq_kdh]h b nbabq_kdh]h fh^_ebjh\Zgby

Kj_^kl\Z ZiiZjZlmjgh]h b f_ljheh]bq_kdh]h h[_ki_q_gby wdki_jb-f_glh\ ih kdZgbjh\Zgbx f_lZeebq_kdbo dhehgg baeh`_gu \h \lhjhc klZ-lv_ wlh]h `_ Z\lhjkdh]h dhee_dlb\Z \ gZklhys_f k[hjgbd_� IjbgbfZy bo

\h \gbfZgb_� jZkkfhljbf kh]eZkh\Zgghklv j_amevlZlh\ wdki_jbf_glZ b

qbke_ggh]h fh^_ebjh\Zgby� Z lZd`_ g_dhlhju_ f_lh^bq_kdb_ \hijhku�

Ihemq_ggh_ ZgZeblbq_kdh_ j_r_gb_ j_Zebah\Zgh \ qbke_gguo jZk-q_lZo� L_klbjh\Zgb_ qbke_ggh]h fh^_ebjh\Zgby ijh\h^behkv g_kdhev-dbfb kihkh[Zfb� < qZklghklb� iml_f kjZ\g_gby k g_klZpbhgZjguf we_d-ljhfZ]gblguf ihe_f g_fZ]gblghc ijh\h^ys_c pbebg^jbq_kdhc ie_gdb

S� JZg__ >�@ ihemq_gh j_r_gb_ ^ey n pbebg^jbq_kdbo ijh\h^ysbo ie_-ghd� \eh`_gguo h^gZ \ ^jm]mx� >ey h^ghc pbebg^jbq_kdhc ijh\h^ys_cie_gdb w�^�k� \ dhglmj_� ihf_s_gghf \h \gmlj_ggxx h[eZklv� hibku\Z-_lky ke_^mxsbf bgl_]jZevguf \ujZ`_gb_f�

ε λ λ λλ µ λ λ

λ� � � � � �FRV� �

� � ��H[S�

� � � ��W

-U U

U, U , U

]

, U

W

6U , U . UG U

N

N= −∞

∫�

�

� � �

� �

�

� � � � � �

�,

a^_kv S ± bgl_]jZevgZy ijh\h^bfhklv ie_gdb� µ = µ0 = 4π⋅10–7 =g�f ±

fZ]gblgZy ijhgbpZ_fhklv \ha-^moZ� r1 ± jZ^bmk ie_gdb�

J_amevlZlu jZkq_lZ i_j_-oh^gh]h ijhp_kkZ \ ie_gd_ S b

g_fZ]gblghf pbebg^j_ dhg_q-ghc lhesbgu ijb\_^_gu gZjbk� �� JZaebqb_ f_`^m djb-\ufb kiZ^Z w�^�k� mf_gvrZ_lky

k m\_ebq_gb_f \j_f_gb klZ-gh\e_gby� qlh b ke_^h\Zeh

h`b^Zlv� GZ jZggbo \j_f_gZo

i_j_oh^gu_ ijhp_kku jZaeb-qZxlky� ihkdhevdm lhesbgZ

kdbg�kehy f_gvr_� q_f lhe-sbgZ kl_gdb pbebg^jZ� Ijh-\_^_gu jZkq_lu g_klZpbhgZj-gh]h ihey \ ijh\h^ysbo fZ]-gblguo pbebg^jZo dhg_qghc

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1. Sato M., Fuziwara J., Miyairi M. et al. // Directional Induction Loggingmethods // Proc. of 35th SPWLA Symp., 1994.

2. Cheryauka A. B., Sato M. Directional induction logging for evaluation oflayered magnetic formations // Geophysics, 1999, 23 p. (submitted).

3. Sato M., Cheryauka A. B. Magnetic formation evaluation by directionalinduction logging – A modeling study // Proc. of 5th SPWLAJ Meeting, Chiba,1999, 11 p. (in publ.).