UNIQUENESS OF SOLUTION OF THE PROBLEM OF ELECTRICAL PROSPECTING

• Document Type: CREST
• Collection: General CIA Records
• Document Number (FOIA) /ESDN (CREST): CIA-RDP82-00039R000200030030-4
• Release Decision: RIPPUB
• Original Classification: R
• Document Page Count: 9
• Document Creation Date: December 22, 2016
• Document Release Date: April 11, 2012
• Sequence Number: 30
• Publication Date: May 23, 1952
• Content Type: REPORT

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~~'~~' ~,~~rl~~~~.~ry
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STAT
 Declassified in Part -Sanitized Copy Approved for Release 2012/04/11 :CIA-RDP82-000398000200030030-4
r UT:CCN OF ~H~ P Ra~3LEM QF ~!~FCTR~CAL f'RCSP~C~'J~G
UNI~UL'~ES~ OF SIL
'1'ikhono~', Corr.Mern Acad Sc~. USSR,
~. N
~m~.~. Nauk SSSR, ~'ol b'9, Nn ~~
~ag~s ~~~-soon
~ L~ra~.ngrad 4 2~. December 14~~.
Mos co /
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STAT
 Declassified in Part -Sanitized Copy Approved for Release 2012/04/11 :CIA-RDP82-000398000200030030-4
nUni ueness of Solution of tb~e Problem of ~I.ectrica~. Pros ect~.ng~~
A~ N . ~'ikhono~v
Corresponding Member of
Academy of Sciences i~SSRa
t
~ `' ~ '' Declassified in Part -Sanitized Copy Approved for Release 2012J04/11 :CIA-RDP82-000398000200030030-4
~;ote; The follo~.ng report appeared in the regular ~Geophy~sics section of the
Ll~
' e-month/ Doklady Akademii Nauk SSSR, Volume 69, No 6 (21 Dec 1949 ), pages
thr~.c Y
797x800 ~
' :f an. electric current generated by a point source, located
The 9tat~onary f~.eld o
ounda of a conducting ha~.f-space x ~ 09 is determ~.ried by
at a paint Ma of the b ~' J
fiction (pntentxal) u(x'y,~) sat~.sfy~ang the equation
ax ~~ ~ ~
si ma ~ conductivity of the medium) and the condition ~ u./~ ~ = 0 at
(where g cT'
0 M ~ M and possessing at the point M a singularity of the type
~ , o' o
Here ~'"p
o"(M )s r is the distance of the point M(x,y,~) from Mo(xo'ya~~o
0 0
u is a function bounded at M~'and regular at infinity.
e~*istics of ~, medium is often studied by measurement of the
.The electr~.c chara~ct
'ts d~rvat-ivies determined by apparent resist~aees)
field _ of a po~.nt source (or o~ r ~
he ose of this work. is to sho~? that for laminated media
on the surface ~ a? '~ p
~ the value of the superficial potential cannot correspond tt~ varia~ss
' ~'or ~ laminated rr~cdium the equation for ~ has the f~sr~
electric crass secta.ans.
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~t of the prab~.em it ~,s evident that
In ~~.,rtue of the cyl~.ndxica~l s3~ ~'
et us discuss. the auxiliary function
~~}~ ~'
~a ~
~~
~~ ~ ~ ~
f'
(~( ~
where J
p
uation
all show that this function sat~.sf~.es the eq
We sh
~a~ ~~~
C
~- ~~ ..
p, The integral determu~u~~ ~~14 ~,
d additinnalconditions ~-~ W ,
e
n
rom the asymptotic b,~havior of
converges ~nifor~.y. 'phis follows f
function.
~ (~ ~) and fmom
a
? antinuaus function of z(o { ~ { ~xa) ~ It is easy
~~ence it follows that Z~~~~) xs a c
that ~(aa ,~) r p~ The integral
to be convinced ~'
-~ ~ ~~ ~ ~ ~
~~~
a ~ ~ ) has a con
? Tt foll?ws therefore that ~( ,/~
converges absolutely and un~.formly.
ch (2) is .continuous. At points
ug derivative along x far ~.1 x for wh ~"'
t~.nu~
roduct (~)d~/dz i~ continuous. It fo1lc~~s
where ~"(L) is discontinuous, tha p c~?'
a~rd ghat' d(~ ,~ )d~ ~ a.
also from this formula what d~/d~ ~~~ ~.
sha11 sho~r that the integral
We
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p ~~.~ ~,.
~~
~~~~~` ~
. coon of zero order and 1-st kind.
} ~.s a Bessel fun
 Declassified in Part -Sanitized Copy Approved for Release 2012/04/11 :CIA-RDP82-000398000200030030-4
r ~ ~~Y '~ ~
t~a~l b part
by Integra Y ~~, ~~ . ~, ~ ~~ ~ ~ ~ ~ ,
...,, ~ ---~ (,l a
-~ ~ ~~ ~
~ ~ ~ ~ ~ P~ ~
~' J as we].1 as the
totic orders of u, ,~ u~~ , JO and 1
in into account the asyrnp
Tai ~
d that the
? determi.n~.ng Z, we arc convince
uraif orm convergence of the a.nt~gral j ~, ~~ ~.
erivative ?` ~~~
and that the d ~ ~ ~'
integral I converges uniifarmly
p and 2 - ~ , we find that
..
ex~.s tS ?
~'assing to the lxmxt, ~. ~ A ,.~ ~
~ ~ ~~ ~ ~ ,
~~ ~-? a
~'"' , ~ CJ
'atin the equation far ~, we ?btain
pif f erenta. g
~ ~ ~ ~~~
~~ ~
~'~
~
su
where Zl = - ~ d~/ o~? ~?,
...~ ~ W~
J
~ ~ ~ ~'. ~ a I ~ ~ ~
,~,- ..."" ...,, Cb
~, ~,~ ~ ~~a ~I
~ ~' ~ ~ wo different funcp
the ex~.stenee of t
nda.tioras uniquely deterrfd.ne ~l~ Indeed
These co not ident~.cally
d mean that their difference Zl,
tuns satisfying these cond~.tions wool _ ~~.
~.1 ~ ~ Z1
` the same equation and the conditions ~ ~ ~
equal to zeta, satxsf~.es
annot have posita.~'e maximums and negative
he function ~ , by airtua of the equation, c
T 1
.. .~ ~ It follows at once from these
~ ident~.cally' ~ ~ ~
minimums. Hence it follows ghat ~1 ~ ~ function.
is a not.. ancreas~.ng
at Z , . p for all ~, and that ~~~ ~ ~ ~
cans~.derat~.ons th lug
ns~.der the nonhomogenou5 equation
het us co
ill
tuting the new variable, we obtain ~
bst~
-- ~
.~-- ~
same. consid~'ratians that ~ ~~, 0 if f
T~t follows from th
We shill siaow that if
~~' 4 ~~
~. ~ ~'
A..
,C),.
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__ _ __ _
~ ~~ ? ~. ~.
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~~i~z~ it fol~.ows from 4" 1 ~ ~ that u -~ ~~ ? Tn~a~d, we have
~ _
.~
c~ ~~ ~~
~,
where Z ~ ~ - ~ ; .therefore ~ ~ ~ or ~ ~ ?
l 2 / 1 ~' 2
Tf the right term axpresses the local function for whicb~
~p~~ ~
~~~
~ ~ ~-
then we m,~r perform the passage to the l~,mit at ~ -~ 0~ and within limits of ~
we shall obtain the function K (`~ ~ ~) s of the point source satisfying in ,~ ~~~
the homogenous equation, while at the. point ~ ~ we have:
xt is ~sn.dent that, fox equations .with various ~"' ~ (i ~ ~., ~) ~ ~ ~~
we have for the source function the inequa~.ity K (~a ~~) ~ K2~ a~~)'
1 ,~
3. Basic theorem. xf the functia~n ~(~, ~,) is defined as the solution of the
~ '~
e nation ~~, ,~,., ~ -~? ~~ ~,
then
C
R~~ ~,) pannot correspond tea different
Let us assume that to some funcions d"'l~z) and ~'"~(~) correspond identical
~ ~~ ~
are piecewise analyta.cal functions, ,~' ~~~ ,`~"~/ ~'"~ ~ ~
is a~niquely determined by the values of
C~a ~. C
Tn other woxds, identical values of
functions ~ ~ (~) and ff'2 (~) ?
values of R (~)
1.
zee us normalize the functions ~~(~,~) by setting
~ (4 ~) ~ 1 (~- ~ ~,~)~ The funetiora. Z(
equation
f
Declassified in Part -Sanitized Copy Approved for Release 2012J04/11 :CIA-RDP82-000398000200030030-4
~ (~~~) ~ ~~(~,~ )..satisfies the
1
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and the conditions ~(4' ~) '" ~ ~"
' the f ?rrn
The function Z may be expressed 1n f
yr?n. M~~ ~ ~r M ~ \ / ~ M ~ Cl 1 ~ ~ 1
~ C ~- rr J Fw_ 1
~~~~~~
~ , ~ ~~~, Jiff era from
? nera~,ity we may consider q(~) "' ~-2 ~J"~
Without 1im~.tatxon of ~e d } ~ da then
f ~,?~ were not ~o an q{
fox values of ~ as small as des~.red. T
sera
.. - ~ ~ d~(~, ~ )~d~ ~ d and the ari~in
~ ~ it i5 evident that ~(~> )
for 4 .~~ .. ~~
of should start from ~? M
of reckoning , an.t~ ~
is as Burned nnl~~ sn ordex to ~~'' .;~
~ The iecewise analyticity of ~' { ~ }
ate. ~
he class of adm3.ssible functions
near 0. T
the sign-constancy of q(~) ~ ~
ry
could be transfarm~d in such a
hat the sign constancy of q(~ }can be-prop
way t
utilised. Doubtl~e~sly the neceSSitY
of. proof
this assum~ti~n is connected with the method
of
cations with constsnt coefficients we have:
For eg _ ~'~,~
~ ~ cr
~ ~
~ ~~ ~ r