Cagniard_1953_geophysics.pdf

BASIC OF l\IETHOD BASIC THEORY THEORY OF THE THE :\IAG:\ETO-TELLl'RIC NAGSETO-TELLURIC METHOD OF PROSPECTI:\G*tt OF GEOPHYSICAL GEOPHYSICAL PROSPECTISG*t$ LOl'IS LOUIS CAG~IARD§ C~GNIARD~

ABSTRACT ABSTRACT From Ampere's Ampere’s Law La\v (for aa homogeneous homogeneous earth) earth) and from :\Iaxwell's Maxwell’s equations using the concept of Hertz horizontal components Hertz vectors (for aa multilayered multilayered earth), earth), solutions are ohtained obtained for the horizontal of the electric and magnetic earth. The The ratio ratio of magnetic fields fields at the surface due to telluric currents currents in the earth. these diagnostic of the structure structure and these horizontal horizontal components, together with with their relative relative phases, phases, is is diagnostic true resistivities of subsurface electromagnetic elements subsurface strata. strata. The ratios of certain certain other pairs of electromagnetic are similarly similarly diagnostic. Xormalh', apparent resistivity resistivity and Sormallv, aa magneto-telluric magneto-telluric sounding is is represented hy curves of the apparent the phase telluric curphase difference at aa given station plotted plotted as as functions of the period of the various telluric rent components. Specific Specific formulae are derived for the resistivities, depths to interfaces, etc. in both the two- and three-layer three-layer prohlems. problems. For two sections differ sections which are geometrically geometrically sim!lar similar and whose whose corresponding resistivities differ apparent resistivities differ by only by aa linear factor, factor, the phase phase relationships are the same same and the apparent the same This "principle same proportionality' proportionalitv constant which relates the corresponding true resistivities. This “principle of similitude" such as as is given for use use similitude” greatly greatly simplifies the representation of aa master set set of curves, such in geologic geologic interpretation. interpretation. In addition currents (no need need for current current addition to the usual advantages advantages offered by the use use of telluric tclluric currents sources sources or long cables, cables, greater depths of investigation, investigation, etc.), etc.), the magneto-telluric magneto-telluric method of proslxcting resolves resolves the effects effects of individual individual beds beds hetter better than do do conventional conventional resistivity It pecting resistivity methods. It seems basins with with potential potential peseems to he he an ideal tool for the initial initial investigation investigation of large sedimentary sedimentary basins troleum reserves. reserves.

introduction INTRODUCTION

There is no doubt doubt that that the first first positive positive success in geophysical There geophysical prospecting flrospecting \vas obtained obtained by by electrical electrical methods. methods. These These have have always always appeared \\'as appeared promising promising both both and mineral mineral prospecting prospecting because because one can usually usually expect for oil and expect large large resistivity resistivity contrasts case of horizontal horizontal bedding, bedding, eleceleccontrasts in earth earth materials. materials. Moreover, Moreover, in the· the-case trical prospecting prospecting can give give information information at locations locations where where neither trical neither magnetic magnetic nor nor gravity anomalies anomalies can exist. exist. The The equipotential equipotential method, which gravity method, which involves involves the the mapmapping current is introduced introduced ping of the equipotential equipotential lines lines on the the earth's earth’s surface surface when when current into failed because because of difficulty difficulty into the the ground ground through through two two point point electrodes, electrodes, usually usually failed in analyzing Ohm's law, the the analyzing the the diagnostic diagnostic features. features. In In spite spite of the the simplicity simplicity of Ohm’ s law, theory One may may resort resort to experiexperitheory of current current flow flow in the the earth earth is very very complex. complex. One ments the shortcomings shortcomings of of the the ments on scale models, models, but but these these preserve preserve many many of the theoretical theoretical approach approach when when applied applied to a practical practical situation. situation. In general, general, petroleum petroleum and and mining mining geologists geologists were were not In not satisfied satisfied with with the the amamManuscript received by the Editor Editor September I, I, 1952 roja * Manuscript tt Translated Translated from the French by aa professional translator translator

for the :\Iagnolia Petroleum ComComMagnolia Petroleum

pany. pany. Translation edited by M. M. B. Dobrin, Dohrin, R. L. I,. Caldwell, Caldwell, and R. Van t$ Translation Van Nostrand, Nostrand, Field Research Laboratories, Laboratories, Magnolia Magnolia Petroleum Petroleum Company. Company. §QProfessor Geophysique and Professor at the Sorbonne, Paris, Past Director Director of the Societe SociCtCde Prospection geophysique Compagnie Generale G&&ale de de Geophysique. G6ophysique. of the Compagnie

60 5

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606

LO1’I.S CAGNIARD Cd GNIA RD LOl'IS

biguous interpretations interpretations which geophysicists geophysicists could could offer offer them them on the the basis of equiequibiguous which potential data. The use of alternating current is even less desirable in this respect potential data. The alternating current is even less desirable this respect Maxwell’s equations are considerably considerably less less manageable manageable than than is Ohm's Ohm’s because Maxwell's equations law. law. The introduction introduction resistivity methods methods was a step in the right right direction, direction, The of resistivity chiefly because the "apparent “apparent resistivity" resistivity” of a section section whose structure structure is not not too chiefly complicated can actually actually be calculated, calculated, or at least least estimated, estimated, without without too much much complicated risk ofof error. error. H-0\\~ever, Ilcwever, new met-hods-, methods, especia-Hy especia!!y \\!it-hwith respectrespect 1-0to depthdepth de~ derisk these ne,v termination, have not not proved proved to be as spectacular spectacular as they they first first appeared. appeared. Even Even termination, have the two-layer two-layer case, a large large amount amount of labor labor is involved involved in developing developing a master master for the curves and and one is seldom seldom able to match match his experimental experimental curve curve with with any any set of curves the curves curves in his catalogue, catalogue, extensive extensive as as it might might be. Moreover, Moreover, the useful useful of the depth of investigation investigation is limited limited to a few few hundred hundred meters meters in the the case of direct direct depth current and even even less less in the the case of alternating alternating current, current, especially especially at the the higher higher current frequencies. In In order order to investigate investigate to a reasonable reasonable depth, depth, it it is necessary necessary to use frequencies. direct current current with with such great great electrode electrode separations separations that that the method method no longer longer direct the advantage advantage of being being inexpensive. inexpensive. has the It is thus thus evident evident that that electrical electrical sounding, sounding, at least least in petroleum It petroleum exploration, exploration, originally promised promised much much more more than than it it has realized. realized. However, originally However, the the relative!y relatively recent discovery discovery of the the telluric telluric method, method, although although little recent little known known and and little little used used outside of France, France, offers offers more more favorable favorable prospects. prospects. Although outside Although the the principles principles ininvolved were were recognized recognized about about 30 years years ago by by Conrad Conrad Schlumberger,* Schlumberger,* no practipractivolved application was made made until until a few few years years before before World cal application World War War II. II. The The telluric telluric method has several several advantages advantages in that that it it does away away with method with a current current source source and and the the associated long long leads, leads, combines combines flexibility, flexibility, rapidity, and associated rapidity, and low low cost, cost, and and reaches reaches ordinary resistivity much greater greater depths depths of penetration penetration than than do ordinary much resistivity methods. methods. In In spite spite fundamental advantages, advantages, however, however, the the telluric telluric method of its fundamental method seems to represent represent only only a temporary temporary stage stage in in the the development development of more more advanced advanced methods. methods. The The magneto-telluric magneto-telluric method, method, which which is the the subject subject of this this paper, paper, answers answers the the evereverincreasing quantitative results. results. Actually, Actually, itit is not not a strictly strictly electrical electrical increasing need need for for quantitative method, method, but but rather rather a combination combination of telluric telluric and and magnetic magnetic methods, methods, a comcombination bination from from which which the the name name of the the technique technique has been been derived. derived. Essentially, the magneto-telluric magneto-telluric method method involves involves the the comparison, comparison, preferpreferEssentially, the ably ably at at one and and the the same same place, place, of of the the horizontal horizontal components components of of the the magnetic magnetic and and electric electric fields fields associated associated with with the the flow flow of of telluric telluric currents. currents. The The new new method method offers offers all all the the advantages advantages of the the telluric telluric method method and and even even improves improves on itit with with rerespect spect to tlexibility, flexibility, speed, speed, and and economy. economy. In In addition, addition, itit offers offers the the inestimable inestimable benefit benefit of of making making possible, possible, in most most cases where where the the bedding bedding is horizontal, horizontal, a truly truly quantitative quantitative interpretation. interpretation. Also, Also, the the method method can can be be applied applied without without particular particular difficulty difficulty to to submarine submarine prospecting. prospecting. ** E. G. Leonardon, Leonardon, “Some "Some Observations Observations Upon Telluric Telluric Currents Currents and Their Their Applications Applications to Electrical Terrestrial Magmtism Magnetism and and Ah. Atm. Elecb. Electr. 33 33 (1928), (1928), pp. pp. 91-94. 91-94. A A presentapresenta· Electrical Prospecting,” Prospecting," Terreshal lion 192 I under the direction direction of Conrad Conrad Schlumberger. Schlumberger. tion of a report report on work work dating dating back to 1921

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BASIC THEORr OF THE MAC.VETO-TELU'RIC Jfl,TlIOn SKIN SKII\ EFFECT EFFECT AiiU AI\D ITS ITS COSSEQUENCXS. COI\SEQUEI\CES, H.\RMOSIC HARMOI\IC SHEET SHEET OF OF TELLURIC TELLURIC CURCURRF,XTS S EARTII RE!\TS IS 1:'\ .iN .\:\' EI.F:CTHIC.~LLY ELECTRIC.-\LLY HOhlOGESEOI’ HOMO(;EXEOFS EARTH

By way way of of introduction introduction to to the the analysis analysis of of the the magneto-telluric magneto-telluric method method let let us us

By

consider consider a schematic schematic and and ideal ideal sheet sheet of of telluric telluric current current which which we we shall shall suppose suppose to to be be uniform, uniform, harmonic, harmonic, of of period period T, flowing flowing in in a soil soil electrically electrically homogeneous, homogeneous, of of conductivity conductivity 0. IJ. During During this this study, study, we shall shall only only use electro-magnetic electro-magnetic units, units, both both for for electric electric dimensions and and magnetic magnetic dimensions. dimensions. Let Let us choose choose a rectangular rectangular coordinate coordinate dimensions system system o, 0, s, x, y, z (Fig. (Fig. I)I) such that that the the origin origin is on the the surface surface of of the the ground ground and and oz is the the descending descending vertical. vertical. One One will will notice notice that that on the the ground ground the the angle angle ox, ox, oy oy is equal equal to -(r/2) - (7r /2) for for an observer observer who who normally normally stands stands with with his his feet feet on on the the ground ground and and his head head straight straight: up. up. It It is also useful useful to to remember remember that, that, ifif a current current circulates circulates in the the ground ground along along OX, ox, 01’ 0)' is at at the the left left of the the Amperian Amperian man man looking looking up up at at the the sky. sky.

o

y l

FIG.

I.

Coordinate systPIll. xy plane represents earth's surface. : is rositiw downward.

It is particularly s equations It particularly useful useful when when one employs employs Maxwell’ Maxwell's equations and and conconsiders siders a harmonic harmonic phenomenon, phenomenon, to bring bring in the the Hertz Hert z vector vector and and to make make use of of imaginary notation. notation. I shall shall use this this approach approach later, later, but imaginary but to handle handle this this first first parparticularly easy case, I prefer prefer to remain remain as elementary elementary as possible ticularly possible in order order to be understood by by those those who who are not not familiar familiar with with Maxwellian Maxwellian analysis understood analysis and and who who are are eager to understand understand the the principles principles of the the proposed proposed method. method. eager The term term "uniform" “uniform” when applied applied to the the telluric telluric sheet The when sheet we want want lo to consider consider rather inaccurate. inaccurate. As a matter matter of fact, fact, there there is uniformity uniformity only is rather only parallel parallel to the the surface of the the ground, ground, and and not along along a vertical vertical line. line. If If the surface the density density of the the current current represented on the the surface surface of the the ground, ground, for for z z=o, is represented = 0, by by = cos wi, wf, II, x =

Ia, = I, = 0,

(1)

the laws laws of physics physics show that that at depth depth z one must must have have the I,

= e-z’*““”

cos (wf -

-Z~ZPUW),

= Iz I; = = 0, 0, II, y =

(2) (2)

designating the the base of natural natural logarithms. logarithms. Formula (2) (2) holds for for what what is e designating Formula holds called the the skin skin effect. effect. When When z increases, increases, one notices notices an exponential exponential decrease decrease with with called respect to z at the the same time time that that the the phase phase retardation retardation progressively progressively increases. increases. respect

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608 608

I.Ol:IS CAGNIARD C4 GNIA RII I.OUIS

Under the the conventional conventional name name of of "depth “depth of of penetration" penetration” (understood (understood as as relatrelatUnder ing to to aa layer layer of of conductivity conductivity (J(Tand and to to aa telluric telluric sheet sheet of period lkotl 7‘) we shall shall define dehne ing 1') tiel)th p when \~ilenthe the term which which we we are are going going to to use use constantly. constantiy. It I~t designates designates the depth aa term amplitude is is reduced reduced to to the the fraction fraction r/e I/E of what what it it is is on on the surface. surface. amplitucl6 of

p =

.I

27r

.

“’

= _I

2/27md

u

(3)

fJ

is retarded retarded one additional additional radian radian each time time that that zc is is increased increased As for the phase, it is by p. by P. It is is obvious obvious that that for for z infillite, infinite, the amplitude amplitude of the magnetic magnetic field is is annulled; annulled; It otherwise the density density of the current current could could not be zero. At At the same time, time symsymotherwise '1 _ _ _ _ _ _...,0 B

f

o

c FIG. 2.

Path of integration

10

apply Faraday's La\\".

metry requires requires that that the the magnetic magnetic field field be horizontal horizontal everywhere, metry everywhere, parallel parallel to oy. Let us now now apply apply the the theorem theorem of Ampere Ampere to a rectangle rectangle .1 .4 IK‘ D with Let neD wit h sides AB AB CD parallel parallel to oy and and of unit unit length, length, with with side _;l H situated situated at depth CD AR depth z and and with with CD put put at infinite infinite depth. deljth. It It reads reads side CD

H, Hx = = o, 0, 7r

(4) 4

->

.

In o, In particular, particular, on the the surface surface of the the earth, earth, where where z= z=o, 11, H", == 0, 0, H, H

y==

4~ 471'

~$). Io0%; =2,:,cos(w~ 21:w (WI --:). S t oc J

.cdz

=

cos

(5) (5)

We We shall shall stress this this first first result, result, because because itit is the the key key to to the the proposed proposed method: method: On On the the surface surface of of the the ground, ground, the the magnetic magnetic field field oc :JC and and electric electric lield lleld E(E,= E(E.c = 1*/u) I xl (J) are are orthogonal. orthogonal. The The quotient quotient of of the the aml)litude amplitude of of the the electric electric field field by by that that of of the the magnetic magnetic field field has has the the value value

rl V2(J T.

I/~ZOT.

The The phase phase of of the the magnetic magnet ic field field is rere-

tarded tarded by by an an angle angle of of s/4 71'14 with wit h respect respect to to that that of of the the electric elect ric held. lield. It It is well well understood understood that that the the above above result result is valid valid for for aa telluric telluric sheet sheet flowing flowing

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R.tSlC THl".ORJ' OF THF. JfAG.Yl".TO-TELLCRIC JIETHO/)

In left hand hand in any any direction, direction, provided provided one always al\vays chooses the left measuring the magnetic fIeld. If, for instance, the component measuring the magnetic field. If, instance, component along along oy is of the form form I

side as positive posltJve in of the the !Clectric field electric field

cos wt,

(J

it will will be he necessary necessary to write write

11, = -

2,&os(_, - :).

with (5), since the the x axis indicates indicates with a change change of sign relative relative to the similar similar formula formula (s), the right yaxis, right hand hand side when when the the current current flows along along the the y axis. The represents the the toial total intenintenThe integral integral in the second member member of relation relation (5) (5) represents sity and unlimited, unlimited, going going sity of the the telluric telluric current current through through a rectangle, rectangle, vertical vertical and

o0

I;IG. 3. Section Section showing shoiving horizontal horizontal uniform uniform sheet sheet of current. FIG. current.

from width. The The magnetic magnetic field II from the the surface, surface, perpendicular perpendicular to ox, and and of unit unit width. IJ measures measures this this total total intensity intensity within within a facior factor of 471". 4”. This observation observation is of great great practical practical importance. importance. It 11 renzai~zs This remains sfriclly strictly valid for for any layered layered earth, and maintains mainfains approximate approximale 2'alidity zlalidily irr any ill many many cases cases ilateresli?lg interesting in in exploration. exploration. Remarks Remarks Assume aa horizontal, horizontal, uniform, uniform, extremely extremely thin sheet sheet of direct current Assume current of density I, I, flowing flowing at the horizontal planes planes with with sides sides z and z+dz z+dz (Fig. (Fig. 3). It It is well known and easy to depth zc between two horizontal show on the surface of the ground is is horishow that the magnetic magnetic field field produced by this horizontal horizontal sheet sheet on zontal, that it is 271:1dz. is directed to the left hand side side and that its value is mIdI. E‘or aa sheet sheet of direct current, current, flowing parallel parallel to ox, OK, from the surface of the ground down to tlel)th For depth whose density II I, would he he an5' anj function of z, z, one one would have Z, 5, and whose H, = ta

1 S

I,(a)dr.

(R) (8)

0

Because telluric telluric currents have an extraordinary extraordinary low frequency, frequency, since Because since the length of the wave is is relative to p, one one might be be tempted to apply apply to them relation enormous relative relation (8), (R), assuming their their behavior behavior to be be that of aa direct current, current, which would lead one one to write write

i +~ t-

Hy N, = 271: 2T

IAz)dz, I,(z)ds,

(9)

s 0

whereas the accurate formula factor PA. 271:. formula (5) includes the factor factor 471:, 4x, and not the factor

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610 610

I.Ol’I.7 CAG.VIARD CAG.VIA RII LOUIS

[Tnits and magnitudes

\i:e measure measure the magnetic magnetic field in 'Y, y, the electric electric field held in millivolts/km millivolts/km and the the We and period period in seconds. On the other other hand, hand, prospectors prospectors usually usually consider consider the re:3istivity resistivity rather than than conductivity conductivity (J. a. They ‘Ihey measure measure resistivities resistivities in ohm-meters. ohm-meters. pp rather JI

'y Y == 101065 em cgs cgs

1I

mv ‘km == 1I em cgs cgs mv/km

10) JO)

I km km = 10'> IO.+ em cgs cgs II

!Im = £1m

em cgs cgs

lOll IO"

ifiit!i the ne'oN new s-ys-t-em systemofof unit-s, units, one ()bt-a-in~: obta~ins: 'Vit-hI

_. __

,/'/ropT, p= --I VlopT,

P = 0.2T

p=

28 271'

E)2 (H

(II)

In order order to become become familiar familiar wit withh the order order of dimensions, dimensions, it is useful useful to conconIn sult gives the the values values of pp for difsult the the two two tables tables of numbers numbers which which follow. follow. Table Table II gives different values values of pp and T. 2’. Table Table 2z gives, gives, also as function function of p and ferent and of T, the the values values IZ corresponding corresponding to an electric electric field field of I mv mv,/km. of II /km. ?‘ABI.E TABLE 1I

DEPTHS OF PENETRATIOK GIVEK IN KM

P,\l’ . P. \1'. ----~

I

set sec

3 sec

0.2 0.2

0.22j 0.225 0., 0.5 0.503 I.) I .‘3 3 7.59 I. 59 3 3.5566 7.95 7·95 Tj.C) 15·9 35. 6 35.()

I

5 )0 10

SO 50 25 0 250 I,000 1,000

5,000 590°0 -~-----

10 sec

30 sec

I mm

2 min

5 min

10 min min

5 min

30 30 min min

------------- --

---~-----

0.7IZ 12 0.7 0.390 0 0.39 O.XfZ 1. 0.X72 I.59 59 6 I .95 .i 56 1. 95 3.5 2.766 0 2.7 5.03 5. " 6.16 11.3 6.16 II .3 1.3.x 2j.2 I.,. X 25. 2 27.6 27. 6 50..1 50 .3 hr.6 61.6 113 I ' .z

1.23 I 23 2.766 2.7 6.Ih 6.16 8.72 R·72 10 5 19·5 ;3:$ 43. 6 X7· 7 2 IO.5 195

I’ ..74 74 3.900 3.9

8.72 1',.]2 12.3 12·3 27.66 27. 61.6 61.6 12 I233 276 27 6

3 .900 3.9 x.72 2 1-:.7

2.47 2·47

5.5’ 5.5 1 12.3 12·3 '7.4·4 I7 39.0 39. 0 87.22 1',7. 174 390 39 0

'9.5 19·5 27.C 27·( 01.6 (JI .6 13 X 138 276 616

5.51 5.5 1 12 .. '3, 27· (J 27.0 39.0 39. 0 x7.2 87. 2 195 390 39 0 872 87 2

9.54 9· 54 21.4 21·4 47.7 47·7 67.5 67·5 Ijl 15 1

338 33 8 67, 67.5 IjIO 15 10

------

TABLE 2 AMPLITUDES OF TilE ;\1AGNETIC FIELD GIVEN IK 'Y 'WHEN I SW sec P. \T. Pi \T_+ - - - - - - - - - - - - - - - ---0.2

0.2

II

I

0.447 0·447

5

IO 10 50 2.50 250 1,000 1,000 5,000 5,000

0.2

0.141 0.14 1 0.0632 0. 06 3 2 0.0283 0. 028 3 O.OI4I 0.oI41 0.006,~2 0.006.F

3 set sec

3

10 xc 10 sec

-------------

I.73 1·73 0.775 0·775 0.346 0.346 0.245 0.245 0.110 0.110 0.049 0.049 0.024j 0. 02 45 O.OIIO 0.0110

30 WC sec

30

-

3.16 3. 16 I.41 1.41 0.632 0.6.>2 0.447 0·447 0.2 D.2 0.0x94 0.0~94 0.0447 0.0447 0.0200 0.0200

I

E

2 min mill

min min

2

--11 .o 11.0

--------

5.48 5.4 8 2.45 2·45 I.10 1.10 0.775 0·775 0'346 0<14 6 O.ljj 0.155 0.0775 0.0775 0.0346 0.0346

7.75 7·75

3.40 3.4 6 I.55 1. 55 I.10 1.10 0.49 0 0.490 0.219 0. 21 9 0.110 0.110 0.0490 0.0490

4.90 4.9 0 2.19 2.19 l..5j I . 55 0.693 0.693 0.3'0 0.3 10 O.Ijj 0.155 0.0693 0. 06 93

IS

I

Mv/KM

5 min min

10 min mm

5

-

'7.3 17·3 7.75 7·75 3.46 3.4 6 2.45 2·45 1.10 1.10 0.49 0 0.490 0.245 0.245 0.110 0.110

30 30 min min -- - - - - - - - -

10

---

24.j 24·5

11 .o 11,0 4.90 4.9 0 3.46 3.4 6 I.55 1·55 0.693 0.693 0.346 0.346 0.155 O.'Sj

42.4 42 ·4 X9.0 19·0 8.49 8.49 6 2.68 2.61', I.2 I.2 0.6 0.6 0.26X 0.26R

----------

From From now now on, on, one one will will notice notice the the extent extent to which which the the depths depths of of penetration penetration are are exactly exactly adapted adapted to the the needs needs of of petroleum petroleum prospecting. prospecting. One One will will also notice notice

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H.lSIC‘ TllEORI" i!‘JJI:ORI- OF HIP. THE .1f.1GSETO-T}:LLCRIC .11.1G.~I~T0-7‘Rt,L~‘RIC .lJRTJiOD H.L'i/C .I/P.TllOD

611 6rr

very ween which udes of the electric very large limits limits bet between which the ratio ratio of the amplit amplitudes electric and and magnetic magnetic fields may may vary, vary, which which is, is, of course, essential essential when when one wants wants to establish tablish a "precise" “precise” method method of prospecting prospecting in which which this this ratio ratio is is to be measured. measured. REL,\TION BETWEEhTIIE ELECTRIC ASD THE: :\L\.GXETIC M.~GSETIC FI1:I.D FOR .\ );0.'\X0XREL\.TIO:-.< BETWEEK TilE ELECTRIC AXD THE FIELD FOR A H.4RhlONIC TELLUKIC SHEET HARMOKIC TELLURIC SIlEET

If the components components of the telluric telluric current current no longer longer vary vary with with time time according according If to a sinusoidal sinusoidal law law but but instead instead vary vary in an absolutely absolutely arbitrary arbitrary way, way, as in natural natural telluric telluric sheets, sheets, the relations relations, obtained obtained above above are easily easily generalized generalized by by means means of operational operational calculus. calculus. I shall limit limit myself myself to give give the the result, result, which which does not not seem have any any great great practical practical interest interest in connection connection with with prospecting. prospecting. to have

E"(t)=-

If' f

... 27rYiJ

+x;

I

- -... 27rYiJ

II/(u)- du .. ~.-:

vi -

_00

H /(1 -

u)

0

u

(12)

dlt

-=.

yu

/. In this this expression, expression, 1Ix'(t) I{:(I) designates the the derivative derivative of H,(t) In designates [[x(t) with with respect respect to I. GEiVERALIZATION FOR AKY HORIZOKTALLY STRATIFIED SECTION GEKERALIZATION FOR AKY HORIZOKTALLY STRATIFIED SECTION

If the the earth earth is formed formed by by a number number of horizontal horizontal strata strata of arbitrary If arbitrary thicknesses thicknesses and resistivities, resistivities, shall start start from from the the equations equations of Maxwell and we shall Maxwell and and we shall shall preferably use imaginary imaginary notation, notation, stipulating stipulating tqat that all preferably all the the alternating alternating quantities quantities iwt • From From now now on, this depend on time time through through a factor factor ee-‘“l. depend this factor factor will will be underunderstood rather rather than than expressed expressed explicitly. explicitly. stood If the the harmonic harmonic sheet, sheet, assumed assumed uniform, uniform, flows flows along If along OX, ox, the the components components of the Hertz Hertz vector vector fI II along along oy and and oz are null. null. Furthermore, Furthermore, II* the fIz depends depends only only on z (and t). (and on t). The The equations equations of Maxwell Maxwell are are satisfied satisfied if

+

PII, V'2fI x + 4~ff~iII, 47riJwiII x = = 0.o.

(13)

e

The The electric electric field field i? and and magnetic magnetic field field X JC are are expressed expressed in a general general way way by by X JC = 4~~7 47riJ curl curl II, II, c = e

grad grad div div IIII -

(14)

VW, V'2IJ,

and, and, specifically, specifically, in the the actual actual problem problem by by

aI1 alI x I

Hy

H”=4==7 az

H,=H,=o Hx = Hz =

E, Ex = 4ZYTwiII,, 47riJwifI x,

E, Ey = E, Ez = o.

=

47riJ - - ,

0

(IS) (1.5)

Because, Because, in in this this case, E, Ex is proportional proportional to III,, fIx, we we can can choose choose E, Ex as Hertz Hertz vector. vector, so that that

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LOUIS CAGNIARD

012 612

(16) Il y

=

w

(JZ

Furthermore, we must IIy when when crossing crossing the the Furthermore, must assure the the continuity continuity of Ex E, and II, different different surfaces surfaces of separation. separation. In form of In order order to meet meet condition condition (16), (16), Ex E, must must be in the the form E,

=

,le”%“’

+

&“““,‘

( 17) (17)

defined as A and and B LI designating designating two two arbitrary arbitrary constants constants and and a being being delined a =

271" '

/~-

11

T

e-;,.-!4 =

__

2~

y'T

(1 -

i).

(I R) (18)

Let at the the surface surface Let us number number from from I to n II the the successive successive formations formations starting starting at will be necessary necessary of the the ground. ground. The The nth nth and and last last one is the the lowest lowest stratum. stratum. It It will in this A =0, becomes infinite infinite at at the the this layer layer to put put down down .A =o, because because the the first first term term becomes same time multiplied by by a constant constant time as z. e. Furthermore, Furthermore, any any solution solution can can always always be multiplied complex only definite deiinite as far far as complex arbitrary arbitrary factor. factor. In In other other words, words, the problem problem is only the concerned. For For this this the relative relative amplitudes amplitudes and and the the differences differences of phase phase are concerned. the z?z 211 constants constants A and and H. B. reason, reason, we can assign an arbitrary arbitrary value value to one of the We the bottom bottom stratum which We shall shall assume assume that that it it is constant constant 13 U corresponding corresponding to the stratum which is equal equal to unity. unity. In the same same number number of conconIn all all we have have 2(nz(fz- 1) I) arbitrary arbitrary constants constants to meet meet the ditions the two two fields fields at at each each ditions at the the limits. limits. These These conditions conditions are the the equality equality of the of the the nn- 1I surfaces surfaces of separation. separation. The what the the value value of 11, The method method of calculation calculation being being the the same no matter matter what 12, we shall shall only only consider consider the cases of n = 2z and and n = 3. It present any any other other complicompliIt is obvious obvious that that these calculations, calculations, which which do not not present cations the first first degree, degree, are are cations than than the the resolution resolution of simple simple algebraic algebraic equations equations of the done and division division and and do not not done exclusively exclusively by by means means of addition, addition, multiplication multiplication and necessitate necessitate resorting resorting to integrals integrals or series. SOURCE SOURCE OF OF CURRENTS CURRENTS

The the origin ongm of of the the currents currents The above above theory theory does not not concern concern itself itself with with the involved. internal to the the crust crust of the the involved. vVhether Whether the the source of these currents currents are internal earth sources are natural natural (actual (actual earth or whether whether they they are ionospheric, ionospheric, whether whether these these sources telluric currents), the the electroelectrotelluric currents) currents) or whether whether they they are artificial artificial (vagrant (vagrant currents), magnetic every case. magnetic phenomena phenomena inside inside the the earth earth are the the same in every In that the the telluric telluric In fact, fact, the reasoning reasoning depends depends only only on the the requirement requirement that current sheet be sufficiently uniform. But this uniformity is a matter of expericurrent sheet sufficiently uniform. But this uniformity matter experi-

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8.ISIC THHORr OF TIIH

.tr.IGNHTO~THUJ·RfC

.tIlUfIOD

ence. Telluric basins this uniformity uniformity Telluric prospecting prospecting proves proves that that in large large sedimentary sedimentary basins extends, way, expanse, often often some ten over a considerable considerable expanse, extends, in an approximate approximate way, over km should all the the more more if one only only km in width. width. Such uniformity uniformity should be expected expected all considers magneto-telluric comparison. comparison. considers the very very restricted restricted t1eld field that that enters enters into into a magneto-telluric Vagrant the sources which which produce produce \‘agrant currents, currents, because because of the relative relative proximity proximity of the them, the sheets sheets associated associated with with them, and and because because of the poor poor degree degree of uniformity uniformity of the such artit1cial On the the contrary, contrary, artificial currents, currents, are feared feared by by the the telluric telluric prospectors. prospectors. On they prospectors, because because they they they are looked lodked on as as a blessing blessing by by magneto-telluric magneto-telluric prospectors, offer the new method, method, and and they they offer sufficient sufficient uniformity uniformity to meet meet the requirements requirements of the usefully usefully enlarge enlarge the the spectrum spectrum of frequencies. frequencies. Readers mainly concerned concerned with with what what Readers of Geophysics, Geopl~ysics, as well well as this this writer, writer, are mainly is underneath tie interested what goes on on above above their their underneath their their feet feet and and are lit little interested in whal

FJ(;.4. Plane electromagnetic wave incident upon earth's surface.

heads. moments longer longer the the nature nature heads. However, However, it may may be useful useful to consider consider for a few few moments of the phenomena the atmosphere. atmosphere. the electro-magnetic electro-magnetic phenomena as a whole whole involving involving the In (r6) becomes becomes 3’ B2 E,/W=o. Ex/Bz2=0. In the the air, air, where where we put put down down 0-=0, u=o, equation equation (16) Ex E, appears appears as a linear linear function function of z, 2, IIy II, as a constant: constant:

+

Ex(z) E,(z) = Ex(o) E,(o) + iwzl/io); iwzH,(o);

ii:, = E, =

Hy(z) H,(z) = IIy(o); II,(o);

II,

= II,

=

0.

(4 0.

A solution One knows, knows, in particular, particular, solution of this this kind kind may may surprise surprise the the reader. reader. One that the earth earth undergoes undergoes quick quick that the the vertical vertical component component of the magnetic magnetic field field of the variations components of of the the same same variations whose correlation correlation with with those those of the the horizontal horizontal components field solution shows shows us that that II, lIz field or of the the telluric telluric [leld field is evident. evident. But But the the actual actual solution is null. null. Let the equations equations of hlaxwell, Maxwell, Let us not not forget forget that, that, in the expression expression (r3) (13) of the we have, speed V electro-magnetic have, from from the the start, start, considered considered as infinite infinite the speed I’ of electro-magnetic waves waves in empty empty space. For For waves in the the ground, ground, as well well as the the speed speed c of those those waves the ted from from the the start fictitious the real real phenomenon phenomenon of propagation propagation we have have substitu substituted start a fictitious stationary was quite sufficient for for the the calcucalcustationary phenomenon. phenomenon. The The approximation approximation quite sufficient lations picture of the the nature nature lations we had had in mind, mind, but but it it did did not permit permit an accurate accurate picture of the the physical physical phenomena phenomena involved. involved. Let wave spreading spreading in the the plane plane Let us suppose suppose that that in the the atmosphere, atmosphere, a plane plane wave oyz hits a (Fig. (Fig. 4). 4). In In order order hits the surface surface of the the ground ground at an angle angle of incidence incidence a: that the surface surface of the the ground, ground, that the the conditions conditions at at the limits limits might might be met met at at the

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LOl'lS CAGXIARD it is, first first of all, all, necessary necessary that that the expressions expressions for for the it the characteristic characteristic vectors vectors of the three three waves waves (incident, (incident, reflected reflected and and refracted) refracted) include, the include, respectively, respectivciy, the the folfollowing lowing factors: factors: Incident Incident Wave: Wave: e_iu(l_ . ( -tW t -

e

Q)

~~~ii~l~~~co~a~ Y - ~in -a. +~ co:-; C

Reflected Reflected Wave: Wave :

Refracted Refracted Wave: Wave :

The The constant constant K is chosen to satisfy satisfy the equation equation 2

V II

+ n ( 47rO"wi + .~~~)

o.

It It is thus thus necessary necessary that that (21) But, whereas whereas (C (c’2 /V) -sin* 2 a(Yis at its maximum maximum equal equal to unity, But, jV2) -sin unity, it it happens happens that that the the coefficient of i is enormous. enormous. For For instance, instance, for p= coefficient p = IO&Z Ioflm and and for for T = 30 set, sec, it it is equal to S.4XI0 5.4X1olO10 so that that in practice, practice, and and as an excellent equal excellent approximation, approximation, one may may write write

K = cd/aaT

er*14,

(22) (22)

bearing the imaginary imaginary part part of K must must bearing in mind mind the the fact fact that that the coefficient coefficient of i in the be positive. justify in the the form form itself itself of the the expresexprespositive. Accordingly, Accordingly, we justify the first first place place the sions (17) point. After After that that we (17) which which we have have adopted adopted initially initially as a starting starting point. notice that that an infinity infinity of possible possible waves waves in the the atmosphere atmosphere can correspond notice correspond to a given wave wave in the the ground. ground. Not Not only only is a(Y left left completely completely arbitrary given arbitrary since it it does not appear appear in (22), (zz), but but the the state state of polarization polarization of the not the incident incident wave wave remains remains also totally all kinds kinds of miscellaneous miscellaneous phephetotally arbitrary. arbitrary. One One is entitled entitled to imagine imagine all nomena in the the atmosphere, atmosphere, and and no particular particular condition condition is imposed nomena imposed that that the the vertivertical component negligible. component of the the magnetic magnetic field field must must be null null or negligible. SPECIFIC STUDY PROBLEM STUDY OF OF THE TWO TWO LAYER PRORLEBI

Let of the the upper upper formation, formation, and and u2 0"2 Let us suppose suppose 0"1 (~1 to be the conductivity conductivity that of the the lower lower formation, formation, h being being the the thickness thickness of the that the upper upper one (Fig. (Fig. 5). s). Following the the general general method method sketched sketched above, above, the Following the general general expression expression for for

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BASIC THEORr OF THE .lfAG1\·ETO-TELLURIC METHOD

6r5

o

2=0

. . .: .. : .. :. '.-,IS!, LAY~R:

a; '.,", '

~

--

FIG,S, Two-layer earth earth section. section, FIG. 5. Two-layer

will be as as follows: the fields will L In In the first formation formation I.

2, 2.

Ex

= Aeav·~

Hy

e i"/4

=

Z

+ Be-av;-l

Z

Y2 0"IT[ _Aeav~

Z

+ Be-av~ zJ.

In the second second formation formation In

The continuity continuity of E, Ex and H, Hy for z= Z= k It involves involves accordingly accordingly the two conditions conditions The

+ Be-a,'~ = e- av;;; Ay;:;- eav;-, + By;; e- av;-, Aeav~"

-

II

h

h

h

= Y 0"2 e-avu2 h

where A B

+ Y0"2 = -y;:;---..: -

--

eah(vUI-VU2).

2YO"l

The In this The result is is an expression expression for the fields fields on the surface of the ground. In expression penetration pr PI relaexpression we shall advantageously advantageously introduce introduce the depth of penetration tive tive to the first formation formation and we we shall be able to set aside aside a factor common to Ex E, and II II,,y , since since we we are only interested interested in the relation relation between those those fields. fields. One has has then E, = MeJ/f+”i Ex --

By .. /4-y,l,1 H, = = Y2;:;-T.Yei( 42alT _Vei(r/4-$1

(27)

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LOllIS CAGNIARD C.dGNIARD LOUIS

616

in which: which: in A4 cos cos¢r#l== M

(

hh + -

I

h

M sin sin ¢C#I == M

h)h

I

Ith

cos--: , cash-- + -I sinh sinh -cosh cos PI Pl P2 P2 Pl PI PI PI PI

-

h

h

PI

PI

sin -J - + + ~ - COSh~) cash i sinh ~ (~sinh sin~, ( Pl> Pl PI P2

PI

I

PI

P2

It + h

It) h

I

lr

h cos sinh -- + -I cosh cash-sinh cos - ,> > ( PI P, Pl Pl Pl Pl PI P2 PI PI h I1 h I It + -I sinh sin --lz .. cash-~.V sin sin if;$ == -LI cosh .Y sinh sin ( Pl P2 Pz Pl> .Pl Pl PI PI PI 'PI

s cos cos if;fi == .Y

(28) (28)

I

(

-L

h)

+-

(29) (29)

Whereupon : Whereupon: EZ

--

I

=

A4 M e--i(*/4++#).

-_

e-i(~/4+¢-f).

(30)

42alT .I y20'IT .Y

HV

The formulas formulas given given above above relating relating to to the the case case of of aa single single formation formation are are at at The once found found again again ifif one one starts starts from from those those more more general general expressions expressions and and puts puts once down: 0'1=0'2=0' u1= (TV= u and and Pl=P2=P pl = p,= p whereupon whereupon down:

M = .Y = -

I

P

eh / p

,

lz

¢=if;=-,

(31)

P Ex

I

_

= - - - e- i .. / 4 Hy y;'O'T '

conforming to to the the previous previous result. result. conforming

_. ;

.

.:

..

.

~

._

:

(,’ .‘.

..:

,‘_.

_’

.

.

,‘..

'. 2Nb LAYliR:

.‘..

:.

6; .. .:

FIG. 6. 6. Three-layer Three-layer earth earth section. section. FIG.

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BASIC JIETHOD B.dS’IC THEORY THEORY OF THE THE JIAG1'1ETO-TELLCRIC JI.4G~VETO-TELLL.RIC METHOD

61;

FORMULXS FOR THREE FORM.kTIOSS FORMULAS FOR THREE FORMATIOXS

In and Q, 0"3, when when the the second second In the the case of three three formations formations of conductivities conductivities 0"1,0"2, 61, ~2, and one starts 6), one uses the the following following starts at depth depth hl /z, and and the third third one at at depth depth h2 I?, (Fig. (Fig. 6), formulas. formulas. The The ratio ratio between between the the fields fields is always always in the the form form M

V 2 0"IT

e-' (,,/4+<1>-,,),

(32)

.Y

by by putting putting down down

(

he

I

121

>--=z4p

-+I

Pl

P2

PZ

(33)

i I (;1 - ;) + ~: = v, M cos ¢

=

(~ - ~) (~ cosh u P2

PI

-

P2

~ sinh U) h

+ (~ + ~) (~ cosh v+ ~ P2

PI

M sin ¢

=

P2

P2

P3

+ (~ + ~) (~ sinh v + ~ =

P2

pz

PI

_Y cos if;

sinh

h

P2

(~ - ~) (~sinh 1£ P2

PI

pz

P3

-

v) cos v, sin u (35)

cosh

~ cosh h

11 (34)

(~ - ~) (~ sinh 11 - ~ cosh U) PI

cos

u)

v) sin v, cos

11

+ (~ + ~) (~ sinh v + ~ cosh v) cos v, P2

PI

_Y sin if;

=

P2

P3

(~ - ~) (~ cosh 1t - -=- sinh U) P2

PI

P2

pa

+ (~ + ~) (~ cosh v+ ~ PI

P2

P2

(36)

P3

sin u (37)

sinh

v) sin v.

APPAREKT RESISTIVITY FOR THE CASE OF TWO TWO FORMATIOSS APPAREKT RESISTIVITY FOR THE CASE OF FORMATIOXS

E, and and Hy H, is made made on a ground ground which If the the comparison comparison of Ex If which is known known to be electrically homogeneous, homogeneous, the the relation relation between between those those two electrically two fields fields allows allows one to reciprocal, the know the the true true conductivity conductivity (or, if one prefers, prefers, its reciprocal, know (or, the resistivity), resistivity), the formation. formation. If If the the magneto-telluric magneto-telluric comparison takes takes place of the comparison place on any any formaformation, stratified stratified or not, not, whose whose structure structure is not not in general general known, tion, known, it it will will usually usually

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LOl'[S LO1 I‘S CAGII'IARD C.4 GiVlA RD

618

phase of H H,y with respect respect to Ex E, will not be a retardation retardation of 7r a/4. happen that the phase I 4. This will be the first indication indication that that it is is heterogeneous. heterogeneous. However, However, no matter matter This phase separation might might be, be, we can agree agree that that the modulus of the ratio ratio what this phase is equal to II 1/42(~,T, u, would be the conductivity conductivity of a homogeneous homogeneous is V 2IYaT, in which IYn formation which would give the modulus of the ratio between the fields fields whose whose formation value has has l;leen been experimentally experimentally observed. The quantity quantity IYo u, is, is, by definition, definition, the value apparent conductivity conductivity and its reciprocal Pu pOis is the apparent apparent resistivity. resistivity. apparent The apparent apparent resistivity resistivity is is usually usually a kind kind of average of the resistivities resistivities one The such that that density density of the current current is is not to be negmeets in a thickness of ground such with respect respect to its value value along the surface. However, However, it may may happen in lected with exceptional cases cases that that the apparent apparent resistivity resistivity might might be very very slightly slightly less less than than exceptional resistivities of the formations, formations, or on the other hand, very very slightslightthe smallest of the resistivities resistivities. Actually, Actually, one knows that that a similar similar ly greater than the highest of the resistivities. phenomenon occurs occurs for the apparent apparent resistivities resistivities that that are obtained obtained in the prosphenomenon use a quadripole quadripole of measurement measurement supplied by direct pecting techniques which use current. current. In the case case of two formations, formations, the apparent apparent resistivity resistivity is is easily calculated calculated by In In accordance with means of the formulas established above. In with this definition, definition, one has has I M I M

or

.Y ; (0M)" M”

Pa = = Pa

PI p1

s

;

and 2h

4 4 cos -~ Pa _ = PI PI

PI Pl ___

+--

I +

I

_-.___

m m

if one writes

9

2 11 I 21t cos --_. m + 2 cos m+~-

IP2!?+I d

(40)

PI Pl

~+ I

m= m= ___--

PI

I~-I

&v 1 e 2h // 1’ fll.

(41)

P2 --I PI PI

The The fundamental fundamental properties properties of the apparent apparent resistivity resIstivIty as defined defined in in the technique technique of electrical electrical sounding, sounding, with with respect to a certain certain length length of injection injection line line of current, current, appear appear again at at this point point in the apparent apparent resistivity resistivity defined now in in

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B:lSZC‘ THEORITHEORI- OF OF THE THE .1IAG.VETO-TELLCRIC 51.4G.VETO.TELLl.RIC METHOD B.ISIC ]IETHOD

6H) h1o

regard to to aa certain certain period period or or to to aa certain certain penetration penetration depth depth p. p. Indeed, Indeed, one one Imimregard mediately establishes establishes that: that: mediately I. 1.

pl = 0, 0, ifif PI

Pa =

2. 2.

pi = x, m, ifif PI

PC,== P2. P?. p"

PI,

EXPRESSION FOR THE RETARD.kTIOS OF PHASE PH,\SE EXPRESSION FOR THE RETARDATION OF IN THE THE C.kSE OF TWO TWO FORMATIOh3 IN CASE OF FORMATIONS

The other other parameter parameter to consider consider in order order to secure secure interpretation interpretation The is the retardation of the magnetic magnetic field with with respect respect to the electric electric field. field. In In the phase retardation case of two formations, formations, it is is expressed expressed by case 7r

+

8e=z+C#l-#, = cf> 44

if; ,

(42)

with with m-1 m I h $ == tan cf> ~ - - tan tan--,- , m-l-r I m PI $1

+ + tan h ~-.-- -.. tan-, $ = = tan if; m+I m

I

m-rm

I

-

(43) (43)

,

PI Pl

zh 2m 2h - - sin-, tan (cf> (4 - if;) +) == - 2113 sin - j tan rn?2 -m II PI Pl

-z,,-$5”

)

4>

4

having the the meaning meaning given given previously previously (equation (equation 41). 41). m having SPECIFIC SPECIFIC CASE CASE OF OF .I.-\. SECTION SECTIO)! WITH WITH TWO TWO LAYERS, LAYERS, ONE ONE BElSG BEING AK AN EXTREMELY EXTREMELY RESISTIVE RESISTIVE OR OR EXTREMELY EXTREMELY CONDUCTIVE CONDUCTIVE SUBSTRATUM SUBSTRATUM

In In these these specific specific cases, the the above above formulas formulas become: become: I. 1.

Extremely substratum : Extremely resistive resistive substratum:

I E,’ I---~ H,,

= -2 v’m1T

”-

e

0 =

zh,/pl

2 CDS

J

I+-

cash

.arc tan tan

nrc

4

( (

22

zIz,i’p,

-

cos

zh,‘fi, (44)

e211I,,, e2/'/Pl

)

--~_ . ----c4tl/Pl II C 4h /1'1 --

pl sin sin 2h,‘ 2h/pI

Ji

>

The The result result becomes becomes particularly particularly simple simple ifif h is very very much much smaller smaller than than p: p:

'~-= 1_ 1

H Y1

-

I I--

2vu 1T 2dalT

Pl PI It ’ It

0 8= = 0. 0.

(45)

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LOUIS I_OUIS CAGNIARD CAGNIARD

620 620

2.

Extremely Extremely conducting conducting substratum: substratum:

I Ex! /-----2Cos2h/ 1--1 =--=--=-1/ i Hy V l' ,cosh zh/ Pi + cos 2h/ Pi p~---

I.

1----------·----

2u l

I

w () = 4

+

(e~/M arc tan 2 ---- - - - - sin 2h/Pl e4h / P1 - I '

)

(46)

The much smaller smaller than than pl: PI: The result result becomes becomes particularly particularly simple simple if h is very very much

I I

-Ex -I= Hy

,

~-

--::I .--::--= h

~.'

() =

VUlT PI

2

LAW OF OF MAGNETO-TELLURIC SOUNDINGS LAW OF SIMILITUDE SIMILITUDE OF THE THE MAGNETO-TELLURIC SOUNDINGS

It of the electrical soundings soundings is It is known known that that the the interpretation interpretation the ordinary ordinary electrical made construction of theoretitheoretimade much much easier easier by by the the use of logarithmic logarithmic scales in the the construction cal templates the other. other. This This templates on the the one hand, hand, and and of experimental experimental diagrams diagrams on the use of logarithmic (geometric similitude, similitude, logarithmic scales is based based on the the laws laws of similitude similitude (geometric soundings. electric electric similitude) similitude) which which are applicable applicable to electrical electrical soundings. Laws soundings and and Laws of similitude similitude of the the same kind kind also govern govern magneto-telluric magneto-telluric soundings will play play an important important part in their their interpretation. interpretation. Before will part Before we explain explain how how to represent just obtained of master master curves, curves, it it is represent the the results results we have have just obtained in the the form form of necessary to establish establish these these laws laws of similitude. similitude. necessary Let stratified or not, not, being being Let us consider consider two two structures, structures, as complex complex as desired, desired, stratified geometrically similar, similar, the the ratio ratio of similitude similitude being being K Kr,. geometrically L. To To make make itit plainer, plainer, let let specify that that the the corresponding corresponding parameters parameters of the the two two structures structures will will be reprerepreus specify sented by by the the same letters, letters, respectively respectively primed primed and unprimed. sented unprimed. In In this this way, way, 1)’ L' and down designating corresponding corresponding lengths, lengths, we shall shall put put down and L designating L’ = KLL.

(48)

At two two similar similar points points of the the two two structures, structures, the the resistivities resistivities are At are p’ pi and and p and and postulate electrical electrical similitude similitude we postulate

p’ = K,p.

(49)

Finally, phenomena are T' and T, we Finally, if the the periods periods of the the electro-magnetic electro-magnetic phenomena are T’ and require require T’

=

K7.T.

(50) (50)

If y', solution of Maxwell’ Maxwell's If II'(x', II’(z’, y’ , z') z’) represents represents a Hertz Hertz vector, vector, which which is a solution s equations let us find find out out equations and and of the the boundary boundary conditions conditions for for the the primed primed structure, structure, let the the conditions conditions under under which which

lI(x, y', 11(x, y, z) 2) = II'(x', ]I’(%‘, y’ , Zl) 2’)

(5 I) (51)

solution for the the unprimed unprimed structure. structure. is also a solution

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R.I.';le TIIFORJ' OF THF. .lfAGNF.TO-TF.LfXRIC .lfF.THOTJ

It z) designates designates a function function It is necessary necessary to consider consider equation equation (51) (51) so so that that U(.r,)', 11(x, y, z) of x, II'(x', y', the coordinates coordinates .\‘, )', y, Z2 ohtained obtained when when one respectively respectively replaces replaces in II’ (s’, y’ , ::') z’) the x', the same same Iiertz Hertz vector vector which, in other other words, words, makes makes the s’, )",::' ?‘I, 2’ by I\I.X, K&Y, 1\1.)', A-L)‘, I\I.Z, KG, which, correspond correspond at two two similar similar points points of the the two two structures. structures. When \\‘hen one has w

(J'

W , =----,

Kp

K-r

(52)

the the general general equation equation

+

PII VII + 47r(JwiIl 47rawill -= 00

becomes becomes rWI

+

4ddi11

=

0,

if 1\2 Kl.? 1. = K,K7..

(53)

IV IVee shall impose impose this this condition. condifion. Besides, Resides, one has
t;,

t; = ----; K/.2

(54)

so met in one of the the structures so that that the the conditions conditions of continuity continuity supposed supposed to be met structures are also met met in the the other other one. The / H' of an electrical magnetic component component is equal equal The ratio ratio E' El/II’ electrical component component to a magnetic to the 'which real. The The the corresponding corresponding ratio ratio with with a factor factor of proportionality, proportionality, ull~iclr is real. phase ween those the same same in both bot h phase separation separation bet between those components components is, consequently, consequently, the structures. structures. On the Pa of the resistivities has the the value value the other other hand, hand, the the ratio ratio p,,',! pn’/pu the apparent apparent resistivities (55) (55) if we take hom one structake (53) (53) into into consideration. consideration. In In other other words, words, when when one goes from structure the same same ratio ratio as the the ture to the other, other, the the apparent apparent resistivities resistivities are modified modified in the real enough on the the basis basis of the the real resistivities, resistivities, which which moreover moreover might might seem obvious obvious enough principles principles we have have considered. considered. To apparent resistivity resistivity relative relative To sum up the the preceding, preceding, when when one knows knows the the apparent to a certain at once from from this this one certain structure structure and and a certain certain period period T, one one deduces deduces at apparent from the the first first one by by apparent resistivity resistivity relative relative to another another structure structure deduced deduced from geometrical similitude (ratio similitude (ratio (ratio I\p). geometrical similitude (ratio Kd KL) and and by by electrical electrical similitude K,). The multiplied by by the the ratio ratio The new new apparent apparent resistivity resistivity is equal equal to the former former one multiplied of electrical that electrical similitude similitude and and it it is relative relative to a period period such that

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622 622

I.Ol:IS GNI4 RD LOUIS C.4 CAGNIARD

KT

K2 KL2 L

(56)

KT= -i-- . Y

CONSTRUCTION AKD AKD DESCRIPTION DESCRIPTION OF OF MASTER MASTER CURVES CURVES FOR FOR TWO TWO FORMATIONS FORMATIONS CONSTKUCTION

A magneto-telluric magneto-telluric sounding sounding (in (in order order to to abbreviate abbreviate we we shall shall from from now now on on A MT sounding) sounding) will will be represented represented by by means of two two curves, namely namely those say MT indicating pU Pa and and 0() as functions functions of T. T. In In the preparation preparation of master master sets of of curves indicating case of two two formations, formations, itit is necessary to consider three three arbitrary arbitrary paramparamfor the case namely two resistivities resistivities and one thickness, thickness, each of which may may vary vary from from eters, namely zero to infinity. infinity. The value value of the law of similitude similitude lies in the fact that, that, in in order order to represent The MT-soundings, for two formations, formations, it it is sufficient sufficient to limit limit one’ one's the whole of the MT-soundings, s self to the specific specific case case of p1 PI = II and and h= h= I.1. In In this way way there only only remains remains one single arbitrary parameter, parameter, namely namely the resistivity resistivity pz P2 of the substratum, substratum, so so that that the totoarbitrary tality of MT-soundings MT-soundings is represented by by means of two systems of curves. tality Indeed, when, in a more general way, way, the resistivities resistivities of the two present present forforIndeed, mations will will be pl’ PI' P2' and when the thickness of the first formation formation will will mations #l ~ I and pz’ p,‘ ( T’ ) it will be sufficient to multiply be h’ # I, in order to obtain = h'~I, obtain the curve pa’ Pa'=Pa'(T') it will sufficient multiply I. by 1. by ~1’ PI' the ordinates ordinates of that that one of the curves p.=pO(T) Pa = PaCT) characterized characterized by by the ratio ratio p2’ P2'/pl’ / PI' equal in magnitude magnitude to the value value of the parameter parameter p2. P2. 2. by h’ h'2/ abscissas of this same curve. 2. */plPI ' the abscissas Furthermore, it will will be sufficient sufficient to Furthermore, in order to obtain obtain the curve (}'=(}'(T'), O’=O’(T’), it abscissas of of that multiply /i'2j PI;li the aoscissas t na t of of the tne curves B=K(iT) fJ = fJeT) multiply by the same same factor P/p There will will be be no reason characterized by the value value p/ p2’//p1’ . There characterized PI'. reason to modify modify the ordinates. Rather obviously much easier Rather than than to carry out these these multiplications, multiplications, it is obviously to choose choose for each each of the systems systems pa pa and ()0 the logarithmic logarithmic abscissas abscissas representing representing the logarithm logarithm of vI'. J-F. Furthermore, Furthermore, for the system Pa, pa, the ordinates ordinates will represent logarithm of p". pa. The The two sets sets of curves reproduced here were constructed constructed in the logarithm scales as as indicated indicated in Figures 7 and 8. this way, with scales With the help of these these logarithmic logarithmic master curves, the expansions of the With abscissas abscissas and of the ordinates described described at the beginning beginning of this section will will amount from now on on to aa simple translation. translation. A translation translation will be carried out amount as well as as for curve (), 0, and md this parallel parallel to the axis axis of the abscissas abscissas for curve Pa, pa, as translation tramlation will will be be oj of the the same amplitude amplitude in both both cases. cases. Furthermore, Furthermore, in the case case of curve p" p,, aa second second translation translation will be be carried out parallel parallel to the axis of the ordinates. nates. of whole of of the curves curves of of system system Pa, parcorresponding to the changing values of The whole P2, ~2, have an infinity infinity of of points in common, defined by 2

Pa Pa = = I, I,

cos -

PI PI

=

0. O.

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/ PO:toonr Po'IOOJlm

200

~

[)

CL

2 (!)

o

...J

Po

-IO.Jlm 10--

5------~---------------

__-------2---------------_----------1.5----------Po

-I

Jlm-+Iii§'""'~~~;::::~~~

- , - --1--- .,---.bQ..G_IOf'f: ,---,-.:::::::::-...!..:-""'----~T-.:'!.!10~0~0'__"_0.75 T ' 10000" T' 100 000·

-----0,5--------0.25--_ _ _ _ __

0,1_._ _ _ _ _ _ __

Po-O.lJlm

po-O.oIJlm

o

\ ;. Master Master curves of apparent alq~~~l resistivity resistivity for magneto-telluric magneto-telluric soundings over a two-lay3 FIG. 7. two-layer earth. Apparent Apparent resistivity resistivity plotted as as aa function function of period of the telluric telluric component for various reearth. sistivity contrasts. Numbers Numbers on on the curves show show the resistivity resistivity of the lower medium in ohm-meters. ohm-meters. sistivity resistivity of the upper layer layer is is always always II ohm-meter. ohm-meter. Resistivity

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LOliIS CAGNIARD cnc*vr:lKLI LOUIS

624

I'IG. R. 8. Master Master curves curves of phase I)hase differences differences versus versus the period of the telluric FIG. telluric component for various sholl" the resistivity resistivity of the lower resistivity resistivity contrasts in aa two-layer two-layer earth. l\'umbers Kumlxrs on on the curves shou medium in ohm-meters. I~esistivity ohm-meter. Resistivity of the upper up,xr layer layer is is always II ohm-meter.

Whereupon Whereupon 2

+

7r

t/,=__.” -ylT=

-:, = (21t (211 + I)~I) “. ;

8

+

212 + 21l

2

. II

nn being teger. being an in integer. Of their their common common points, points, the the one which which is situated situated the Of the most most to the the right, right, and and which by which is marked marked /1 A on the the chart, eliart, is consequently consequently defined deiined by Pa = Pa =

-\/T=S.= 8. ylT

I, I,

The points ID common The curves curves of system system e 0 also have have an infinite infinite number number of points in common which which are defined defined by: by: 7r

e -, 4

4 --,

n

nN being being an integer. integer.

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BASIC THEORV .'vIF-TElOD BASIC TfiEOR E’ OF Of; THE THE !"IAGNETO-TELLURIC MAGNETO-TELLURIC .2ZETZZOfl

625

The farthest to the the right, right, are are The coordinates coordinates of the the point point which which is situated situated the the farthest consequently consequently 7r

% = 2., 0=--, 4

yT z/T=4.=

4.

In have marked marked on the the set for for In order order to make make use of the the master master curves curves easier, easier, we have the the coordinates coordinates

o% the the point point A, 11, having having

71" r,

,g == - , 0 4

,..11'= d/r=

8,

which point A of the the curves curves for for po. p". which means means the the point point having having the the game same abscissa abscissa as point An resistivity, equal equal to An examination examination of system system Pa pn shows that that the the apparent apparent resistivity, unity The general general configuration configuration unity for T = = 0o approaches approaches P2 pa when when T becomes becomes infinite. infinite. The of system abacus for for two two formations formations system Pa pa is, consequently, consequently, the the same as that that of the the abacus in classical from now now on as E-soundE-soundclassical electrical electrical soundings, soundings, which which we shall shall designate designate from ings in order approaches zero, zero, the the apaporder to abbreviate. abbreviate. Let Let us notice notice that that when when T approaches parent oscillation on both both sides parent resistivity resistivity only only approaches approaches unity unity by by indefinite indefinite oscillation of its limit. apparent resistivities resistivities limit. In In this this way, way, it it is sometimes sometimes possible possible to observe observe apparent the formations formations which which are very slightly slightly greater than than the the greatest greatest real real resistivities resistivities of the present, smaller than than the the smallest smallest present, or which which are, on the the other other hand, hand, very very slightly slightly smaller of those resistivities. This phenomenon, a little paradoxical, is also observed, those resistivities. This phenomenon, little paradoxical, observed, as formations. one knows, knows, in E-soundings, E-soundings, but but only only starting starting with with three three formations. The 71"/4, as well well for for T=o The examination examination of system system 0 % shows that that {j% is equal equal to r/4, T=o as for T infinite. This set of curves, which has no equivalent in E-soundings, infinite. This curves, which equivalent E-soundings, is evidently going to provide one of the most useful means of control MTevidently going provide the most useful means control in MTsoundings. soundings. PRACTICAL USE OF CURVES FOR TWO FORMATIONS FOR THE THE INTERINTERPRACTICAL USE OF MASTER MASTER CURVES FOR TWO FORMATIONS FOR PRETATION OF PKETXTION OF MT-SOUKDI:>rGS MT-SOUNDINGS

All this memorandum memorandum All the the calculations calculations and and theoretical formulas formulas developed developed in this imply any sort providing they they imply the the use of electro-magnetic electro-magnetic units, units, which which may may be of any sort providing are consistent: which electro-magnetic electro-magnetic consistent: cgs for instance. instance. We We have have said previously previously which units results (Formula (Formula 10). units we should should use in the the expression expression of the the experimental experimental results IO). Those classical nor nor self-consistent. self-consistent. Those units units are very very practical, practical, but but they they are neither neither classical Therefore, longer want want to consider consider Therefore, it it is necessary necessary to specify specify now now that that we no longer our two formations formations with with our theoretical theoretical master master curves curves as relating relating to the the cases of two resistivities Inm and and p&m. P2nm. The The depth depth resistivities II and and h. r2. The The resistivities resistivities in question question are 1!22nt of the A is not not 8 but but (8/d10) (8/,..110) the stratum stratum is not I but but I Km. Km. The The abscissa of point point n (sec) (set)"'.112. This the experimerttal experimental results results This being being established, established, when when we represent represent graphically graphically the of a real l\IT -sounding we shall the logarithms logarithms of the the square MT-sounding shall plot plot as our our abscissas the square root of the the curve curve will will be logalogaroot of the the period period expressed exnressed in seconds. The The ordinates ordinates of

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LOUIS WUIS CAGNIARD CAGNIARD

626

rithms of the numerical numerical value value of the apparent apparent resistivities resistivities expressed expressed in in Sn. flm. rithms In In addition addition to this, this, we shall adopt adopt the same scales scales as for the theoretical theoretical curves. It is convenient convenient to draw draw the experimental experimental curves on commercial commercial tracing tracing paper paper It which cross-section lines are printed. printed. The The master master curves, on the contrary, contrary, are on which drawn drawn on plain plain Bristol Bristol board. board. In order to know know ifif the two experimental experimental curves pa Pa and and 0(J are characteristic characteristic In involving two formations, formations, and and in order order to know know the thickness of of a subsurface involving other words to carry carry out out an interpretation, interpretation, one must must try, try, by by the first one, or in other suitable translations, translations, to bring bring the two experimental experimental curves into into coincidence, coincidence, on suitable hand with with curve curve pa, Pa, on the other other hand hand with with curve curve 8, (J, of the theoretical theoretical the one hand set of curves. If If we are to be entitled entitled to consider the result result as as satisfactory, satisfactory, itit is necessary that the two theoretical theoretical curves with with which we compare the respective respective to insure that experimental curves correspond to the same value value of the parameter parameter pz_ P2. FurtherFurtherexperimental translations which which are to be executed parallel parallel to the axis of the more, the two translations abscissas must must be identical. identical. From From then then on, we shall be able to calculate calculate the resisabscissas tivities pi’ PI' and pz’ P2' of the two formations formations at at the same time time as as the depth depth h’ h' of the tivities second second one. Point seen through Point A of family family pa, Pa, as as seen through the transparent transparent tracing tracing paper paper on which we plot plot the experimental experimental data, data, has itself itself an ordinate ordinate whose whose numerical numerical value value is the logarithm %z. Likewise, logarithm of pr’ PI'flm. Likewise, the asymptote asymptote of the theoretical theoretical curve pa, Pa, considered as as sufficient, has OIZ on Gze Ihe tracing paper an ordinate ordinate whose whose numerical numerical value value is the logarithm Qm. In logarithm of p2’ P2'Qm. In other other words, the value value of pl’ p/ and and p2’ P2' can be read at at once on the tracing tracing paper paper if if one does does not not care for a precision of expression which, in this case, case, has the inconvenience inconvenience of making making things things which are very very plain plain look extremely extremely complicated. complicated. The The depth depth h’ h' remains remains to be determined. determined. Point Point A of the one or the other other abaabacus, seen seen through through transparent transparent tracing tracing paper, has an abscissa abscissa whose whose numerical numerical cus, value is is X(sec)I/2. X(sec)r’*. Conformably Conformably to the laws of similitude, similitude, one finds, consequently, value consequently,

Whereupon Whereupon KL2

=

IO ---x2

.‘

64

” ’

x __ --,

x

h’ = = s h'

8

V’IOP~ km. VIOp/ km.

INTERPRETATION IN THE THE CASE CASE OF OF ANY AKl’ STRATIFIED STRATIFIED EARTH. INTERPRETATION IN EARTH. RESOLVING POWER OF OF MT-SOUNDINGS MT-SOUNDINGS RESOL VING POWER

Let us us now supose supose that that one one has has to deal with with three formations, formations, of resistivities resistivities Let ~1, pz, and ~3. The depth of the second one is hr and that of the third formation PI, P2, P3. The second one is hi that third formation substratum is is h h,. If the ratio ratio hdhl h2/hl is is sufficiently sufficiently great, the influence influence of the subor substratum 2 • If

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B.4SIC THEORY THEORY OF OF THE THE .lfAGNETO-TELU!RIC .MAGNETO~TELI,l:RIC ,}fETHon METHOD BASIC

e 8

627

(DEGREES) (DEGREES 1

60~-------+--------~--------;-------~~----~--~------~ 60

45r---~~~----~~----~~==~~r-------r-----~

30~-------+--------~--------1_--------+_--------~------~

10~--------~--------~----~__~------~+---------+-------~

5

2r-----~~~~------r---------+---------+_----~--1_------~

II I

I

I

IO 10

IO'

IO’

IO'

IO'

10.

Computed curves for hypothetical hypothetical magneto-telluric magneto-telluric sounding over three layers in which FIG. 9. Computed which thickness of second second layer layer is is 900 goo times that that of first and in which PI, pl, P2, pr, and ,T~ P3 are in the ratio ratio of 2:2: IO: JO: I.I. thickness

stratum starts to be appreciable stratum appreciable only only for such large periods that that the apparent apparent resistivity resistivity is already already practically practically equal to p2, Pz, while while 80 has already already regained, regained, to a close close approximation, approximation, its initial initial value, value, 7r/4. 7T"/4. In In other other words, the influence influence of the third third formation formation only only starts to make itself felt felt for such periods that that the influence influence of the first formation formation may may be neglected. In In order to determine determine the termination termination of a graph for three formations simply led to construct formations of this kind, kind, one is is simply construct two graphs (Pa (paor 8) 0) for two formations, formations, one after after the other. In In the second second of the graphs for two formations, formations, the formation formation which is from from now on to be known known as the first one has the resistivity resistivity p2 P2 and the thickness hz, hz, while while the formation formation from from now on to be known known as as the second second one possesses possesses the resistivity resistivity p3. P3. An An example example of this kind kind is furnished furnished by by Figure Figure 9, in in which which the ratio ratio hz/hl h2/hl is supposed supposed to have the value value of 900, 900, while while the resistivities resistivities pI, Pl, ~2, P2, and and p3 P3 are proporproportional tional to the numbers numbers 2, 2, IO, 10, and and I.1. This This highly highly favorable favorable circumstance circumstance in which which the master master curves for two formaformations at at once allow allow the interpretation interpretation of a sounding sounding carried carried out out over over a section section

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WClS C.tCSI.tRD

involving more more than than two two formations formations does not not occur occur if strata involving if one is dealing dealing with with strata insufficient thickness, thickness, either either for for E-soundings E-soundings or for of insufficient for MT-soundings. MT-soundings. Let us imagine, imagine, for instance, instance, a subsoil subsoil of three three formations, Let formations, such that that hz//rl hdhl equal to 10, IO, while while the the resistivities resistivities PI, P2, ~1, and and Po pa are is equal are proportional proportional to 9, I,I, and and rex;.. Figure 10 IO represents represents the corresponding corresponding E-sounding, E-sounding, while Figure while Figure Figure III I represents represents the the two two curves curves for the MT-sounding. MI-sounding. h, I00 100

50 50

--- - - - - - - - - - - - - -

---~----.___-------

-

20 20-

/ 9 _____I~_I---------t-----------'If------5

2 _ENGTH OF LENGTH OF LINE 2/j, 24 2

5

IO 10

20

50

100

FIG. conventional type over three FIG. 10. IO. Computed Computed curve for hypothetical hypothetical resistivity resistivity survey of conventional layers in which thickness \\'hich p,, PI, p,, P" and p3 p, are in the thickness of second second layer is is 10 IO times that of first and in which ratio of 9: g:r:I: and 00. cu. ratio

each of those those diagrams, diagrams, the the apparent apparent resistivity, resistivity, equal short On each equal to 9 for for the the short lengths or the (l\IT-soundings), decreases decreases at at lengths of line line (E-sounding) (E-sounding) the small small periods periods (MT-soundings), first the period, period, reaches reilches a minimum, minimum, first when when one increases increases the the length length of the the line line or the and increases increases indefinitely indefinitely afterwards. afterwards. This This minimum minimum is not and not equal equal to I,I, either either on E-sounding or on the the l\IT-sounding. MT-sounding. One will will notice the E-sounding One notice however, however, that that while while itit practically equal equal to I in the case of the the MT-sounding,* NT-sounding,* it 1.25 is practically it is only only equal equal to 1.25 the case of the the E-sounding. E-sounding. In In order order to obtain, obtain, in the in the the case of the the E-sounding, E-sounding, with the the same resistivities, resistivities, a minimum minimum practically practically equal with equal to I,I, itit would would be necesnecessary that that the the ratio ratio be at at least least 25. 25. sary We shall shall conclude conclude from from this, this, at at first, first, that that the NT-sounding separates the We the :'IT-sounding separates the individual better than than the the E-sounding, E-sounding, individual effects effects of the the different different strata strata of the the subsoil subsoil better and times higher. higher. Also Also bearing bearing in and that that its resolving resolving power power is almost almost two two and and a half half times mind phase curves, curves, itit is consequently consequently mind the the additional additional information information furnished furnished by by the the phase

* And even slightly paradoxical phenomenon pointed slightly less less than unity, unity, because because of the somewhat paradoxical out when we described described the master curve for two formations. formations.

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R.ASIC THEORY THEORY OF OF THE THE }fAGNETO-TELLURIC MAGNETO-TELLURIC METHOD METHOD BASIC

ee

629

(DEGREES) (DEGREES)

60

45

30 30

15 15

o0

-

/r

~

'"""

~

20~---------------+----------------1---------------~ 20

IOr-------~----~-+----------------1_9,_"! - - -

5

2

Frc. III.I. Computed Computed curves for hypothetical hypothetical magneto-telluric magneto-telluric sounding FIG. layer configuration configuration of Figure Figure IO. IO. over three layer

already very very obvious that that the MT-sounding MT-sounding allows one to arrive already arrive at more precise precise than the E-sounding, E-sounding, even if one is is satisfied with semi-qualitative conclusions than with semi-qualitativr information. information. But But still, still, in thecase~of the-ca5eof MT-soundings, MT-soundings, -when when the problem problem calls for it, it, there~is there is nothing to keep us submitting the semi-qualitative semi-qualitative hypothesis we are refer!lDthing us from from submitting referring to here to the test of exact calculation. calculation. When When one has suspected suspected the existence of a certain certain number number of strata, strata, when one has been able to estimate estimate approximately approximately the order of magnitude magnitude of their their thicknesses thicknesses and of their their resistivities, resistivities, one can perform the complete calculation calculation of the results that that one would would obtain obtain if the subsoil presented exactly If there is disagreement exactly the supposed supposed structure. structure. If disagreement between between calcucalculation lation and experience, one will will alter alter the values formerly formerly assumed for the resistiviresistivities and and the thicknesses thicknesses so so as as to obtain obtain an entirely entirely satisfactory satisfactory result result by by a method method of successive successive approximations. approximations. In In other other words, the MT-sounding MT-sounding can be analyzed analyzed by by the same method method of interpretation interpretation one can apply apply in in gravimetry gravimetry and and in in magnetism magnetism which which is so so satissatisfactory factory for the prospector, but but without without fear fear of the disastrous consequences consequences of the fundamental fundamental ambiguity ambiguity which characterizes characterizes those last last two two methods. Indeed, Indeed, the calculation calculation in in question question does does not not involve involve integrals integrals nor nor series, series, as

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LOUIS CACNIARD CAGNIARD LOUIS

630

we have have seen. seen. ItIt can can be be readily readily carried carried out out when when the the general general formulas formulas for for four, four, we five, or more more formations formations have have been been established established in in advance advance in in algebraic algebraic form, form, as as five, have demonstrated demonstrated here in in the the case case of of two two and and three three formations. formations. we have However, itit is possible possible to do much much better better and and to save save much much time time by by use use of of an an However, almost exclusively exclusively graphical graphical method method which which is based based on on the the results results which which will will be be almost obtained in in the the following following paragraph. paragraph. obtained APPARENT RESISTIVITIES RESISTIVITIES AND AND PHASES PHASES AT AT THE THE DIPPERENT DIFFERENT LEVELS LEVELS APPARENT INSIDE AA HOMOGENEOUS HOMOGENEOUS FORMATION FORMATION INSIDE

Within a stratified stratified section the complex complex relation relation E,/H, E;rj Hv has a specific value value Within at each level level of depth depth z. We We are going to obtain obtain a formula formula particularly particularly important important at . .

.

.

. ,

. '.1' .

pa*_: :,;

.

.~

.‘

.

.

&i, . . .,.. ,: -* . . n -,.~~~~: : y_ '.' I. :: &, . ‘ * ..:;:R;...:_R:..::..:....I_ _~~....IIL...' Z =?o, = z,tS 11. t P, z-z,= /id&. ..

.: . ,• - .

.(

.

•

'. . I" . •

• 1

-

.

1.

.

..

.

.-

.'

. .

%=C, ~ = -t., .*.

.

_ _._· ..

-.

. . .

- . .

,_

.

.

I2. Geometry Geometry for computing computing relationships relationships between two two levels in same medium. medium. FIG. 12.

in practice same practice by by considering two levels Z1 z1 and Z2 22 at a distance h, inside the Same formation We 12). We shall put put down formation of conductivity conductivity (Ju (Figure (Figure 12).

iay-;' Ex ia-\/; E’, . R= = - R = __ - ~-= - i2 tan r.

(57)

Il 11,y

w w

The R and r are functions functions of z which represent respectively respectively the The complex numbers Rand values R1 = Z1 and R2 = Z2. RI and r1 rl for zz=zl Rzand r2 r2 for zz=z?. A and B B designating two constants, we have learned that that the expressions expressions for the fields fields (formulas (formulas 17, 18) 18) are of the form

Ex = Aeav;;-z

I

+ Be-av;;-z,

: ay-;'(Ae aV;;- z _ Be- av;;- z)

Hy = -

.

(58)

One deduces deduces from these these that: that: R =

I +

XemzaqTz

I

Xe-2a4””

_

B

+p, = A ;

A

= Cr

'

(59)

or or in in another another form form

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BASIC THEORY THEORY OF THE THE .lfAGNETO·TELLURIC MAGNETO-TELLURIC METHOD BASIC METHOD

R-I ___

e2a+Tr =

x =

631

cr.

( 60) (60)

1

K+I

Consequently, Rj : Consequently, one calculates at once once R2 Rz as a function function of RI:

+

+ +

2a v'.-h) + (I (I __ e?ad”h) e2avu h) + eeZaQ/,h) -------=-------=-2a v'u h) + (I Rl(I (I + e-‘ Rj(1 -_ ee2d-h) e2ao ,'u h) h) ’

R = Rj(1 RI(I

R22

=

(61)

+

and, afterwards, afterwards, r2 r2 as as a function function of rj: rr: tan rj rl - tan (iay-;; (iud/a h)

tan '2 r2 =

- = tan (rr -

+

h) 1 r + tan rj 11 tan (iayO" (iad;

itid/a h),

(62)

from which finally finally r2

=

rl

-

iad;

h =

rl

d2

+

(h/p)e-3iT14.

(63)

The since we are only only interinterThe indeterminacy indeterminacy of the argument argument r does does not concern us, since ested ested in the value value of R. R. It is is easy now to go go back to the apparent apparent resistivities resistivities p. It Pa and to the phases phases 80 defined by (64)

Consequently one has has Consequently

d

Pa -

e--i(o+*/4)

tan r, tan r,

(65)

tzn r2 2 __ ( tan rl > *

(06) (66)

=

P

finally and finally (Pa) 2 ol

2L(82~01) = e-

calculations of the prospectors prospectors are not usually usually carried carried out to 20 20 Since the calculations decimal places, places, a simple chart chart of the complex values of the tangents decimal tangents of a complex argument allows one to calculate calculate an MT MT-sounding argument -sounding very very quickly quickly for %+I n I formaformastarting from a sounding for n formations formations when the (n+ (n+r)st tions starting I)st formation formation is situated on top of the nth nth one one.. situated ..

+

REMARKS REMARKS

In the calculation calculation of a theoretical theoretical MT-sounding MT-sounding by an operation operation of successive In successive approximations, approximations, the geophysicist, by constructing stacking of strata strata laid down one one constructing his theoretical theoretical section through through the stacking strata of the ground on on top of the other, proceeds proceeds exactly exactly in the same same way way as as nature nature did when the real strata successiveprocesses processesof sedimentation. sedimentation. were laid down by successive a. At At two stations over a sedimentary sedimentary basin, the section only only differs, in principle 2. principle and as as a first approximation, through certain number number of superficial approximation, through through the addition~r addition-r through the subtraction~f subtraction--of a certain I. 1.

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632

LOUIS LOUIS CAGNIARD CAGNIARD

strata. calculate the complex strata. Consequently, Consequently, it will will often be convenient as as a first working working hypothesis to calculate ratio whose whose ratio ratio of the complex quotients quotients Ex/H., E,/H,, obtained obtained experimentally experimentally at the two stations, a ratio interpretation different for the two stations. interpretation involvesonly involvesonly the thickness of the superficial layer, layer, which is different 3. When, re~istivities of the ground to a When, for one one reason reason or another, another, one one knows with with certainty certainty the resistivities known part part of the certain certain depth, it it may may be be easy to omit, omit, through through calculation, calculation, the influence of this known portion. ground and limit limit the interpretation interpretation only to the unknown unknown subjacent subjacent portion. 4. This MT-sounding over a body of This circumstance occurs occurs in particular particular when one one performs a MT-sounding water calculation allows one one in such such a water for which the depth and conductivity conductivity are known. known. The The former former calculation one to obtain, obtain, case case to correct the MT-sounding MT-sounding for the influence of the sea; sea; in other words, it allows one that one one could have deterthrough a very very accurate calculation, calculation, the diagrams for the MT-sounding MT-sounding that mined experimentally experimentally if the water water were to have been been drained away. away. VARIA TIONS APPLICABLE TO PERFORMED UNDER THE THE SEA SEA VARIATIONS APPLICABLE TO MT-SOUNDINGS MT-SOUNDINGS PERFORMED UNDER

The any specific The measurement measurement of the electrical electrical field at sea sea does does not present any specific technical technical difficulty. difficulty. The The line of measurement measurement is is maintained maintained on the surface of the water as fishermen do with with their their water through the use use of floaters, in the same same way way as there is no difficulty whatsoever in carrying out a correct galnets. Moreover, Moreover, is difficulty whatsoever carrying vanometric recording on board of a ship tossed about by the waves. One need vanometric tossed about only only be suspicious suspicious if one observes observes phenomena phenomena which have a period the same either electrical either as as that that of the marine marine currents currents or as as that that of the swell, since the electrical currents magnetic field of the currents induced by motion motion of the conducting conducting water water in the magnetic earth forth. earth do not meet the requirements requirements of the theory theory we have set set forth. The serious technical technical diffidiffiThe measurement measurement of the magnetic magnetic field offers more serious culties if one is not willing magnetometer on a willing or able to install install a self-recording self-recording magnetometer series bottom of the series of piles forming forming a foundation foundation or in an immersed box on the bottom sea. sea. One way magnetic field on way of avoiding avoiding the difficulty difficulty consists consists in registering registering the magnetic prospectors who the the ground ground and the electric field in in the the sea. sea. The The daily daily experience of prospectors use beds, the line use the telluric telluric method, has shown, indeed, that that in sedimentary sedimentary beds, of the telluric expanses as as large telluric current current keeps keeps an almost constant constant direction direction over expanses uniform in a precisely as as 20-70 20-70 km. Besides, Besides, this direction direction would be strictly strictly uniform stratified stratified earth. earth. N ow the telluric current, is, approximately approximately Now telluric current, current, even if it is a variable variable current, conservative already pointed pointed out, conservative because because of its very very low frequency. frequency. As II have already the magnetic stations not not too dismagnetic field is very very approximately approximately the same same at two stations tant it represents, except for tant from each each other on the same same straight straight line ox, since it a factor 471", through a stratum stratum of unit unit 47r, the total total intensity intensity of the telluric telluric current current through width width starting starting from the surface normal normal to ox. The situated on the The argument argument essentially essentially implies that that the two stations are situated same component one is considersame straight straight line ox, perpendicular perpe.ndicular to the magnetic magnetic component ing. Consequently, condition if possible. possible. Consequently, it is is very very advisable advisable to adhere to this condition However, is not not always strictstrictHowever, experience shows shows that, that, in practice, practice, this requirement requirement is ly ly binding. binding. These observations observatory geophysicists. observations will will not come come as as a surprise to observatory Through that the meaning meaning of Through experience, they they are well convinced convinced of the fact that their electrical resistivity resistivity their magnetic magnetic data data does does not depend particularly particularly on the electrical

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BASIC THEORY THEORY OF OF THE THE MAGNETO-TELLURIC MAGNETO-TELLURIC METHOD BASIC METHOD

633

of the the geological geological strata strata in in the the vicinity vicinity of of their their observatory. observatory. But, But, for for over over aa hunhunof dred years, since the first observation of telluric current, it has been found that in dred years, since the first observation of telluric current, it has been found that in telluric registrations, on the contrary, one has had to be greatly concerned with telluric registrations, on the contrary, one has had to be greatly concerned with the local local geologic geologicstructure. structure. the Another way to avoid the the difficulty difficulty consists consists in in observing observing that that in in the the homohomoAnother way to avoid geneous medium formed by the sea water, where a telluric sheet is flowing parallel geneous medium formed by the sea water, where a telluric sheet is flowing parallel to ox, ox, the the relation relation between between any any two two electro-magnetic electro-magnetic dimensions, dimensions, depending depending to linearly on the Hertz vector, is expressed as a function of parameter only linearly on the Hertz vector, is expressed as a function of parameter AX only (Equation 59). The ratio E,/H, is also expressed as a function of X. In other (Equation 59). The ratio Ex/ Hy is also expressed as a function of A. In other

c

0

--,------------------+~------------l=Z2

--, I

---+ I

-:~ I I I II

-~t

--_

A

A

l

----

-

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J

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# -B

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t=i!, 2-r

-

I

FIG. 13. 13. Configuration Configuration ofofelectrodes electrodeson onwater water bottom bottom for for submarine submarine MT MT measurements. measurements FIG.

words, the the study study of of the the relation relation of of any any two two electro-magnetic electro-magnetic quantities quantities isis absoabsowords, lutely equivalent to that of the ratio E,/H,. lutely equivalent to that of the ratio E,j Hy. One can, can, for for instance, instance, substitute substitute for for the the measurement measurement of of Hy, Hy, the the measurement measurement One of the electromotive force induced in a large vertical ring parallel to ox, ox, this this of the electromotive force induced in a large vertical ring parallel to ring being constructed much more easily on the sea than on the ground. Yet ring being constructed much more easily on the sea than on the ground. Yet one knows that, if the vertical height of this ring is small so that the magnetic one knows that, if the vertical height of this ring is small so that the magnetic component HH,y inside inside itit isisalmost almost uniform, uniform, the the measurement measurement of of the the induced induced eleceleccomponent tromotive force is a classical way of measuring H,. tromotive force is a classical way of measuring H y. It may may be be easier easier to to substitute substitute for for the the measurement measurement of of the the magnetic magnetic field field H H,11 It that of a second electric field. Let us go back to Figure II, supposing this time that of a second electric field. Let us go back to Figure I I, supposing this time that level level ZIz1represents representsthe the horizontal horizontal sea sea bottom, bottom, level level Z2 z2the the surface surface of of the the water water that (or, in a more general way, any level between the bottom and the surface of the (or, in a more general way, any level between the bottom and the surface of the sea). It is easy to measure the field E, on the bottom of the sea by means of two sea). It is easy to measure the field Ex on the bottom of the sea by means of two immersed electrodes electrodes AA and and EB connected connected with with recording recording equipment equipment on on the the boat boat immersed by the two lines AC and BD (Fig. 13). by the two lines AC and ED (Fig. 13). It can can be be shown shown that that It

(67)

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634

LOUIS LOUIS CAGNIARD CAGNIARD

pa and ()8 have reference reference to the apparent apparent resistivity resistivity and to the phase phase in which Pa relative to level Zl, zr, the level of the bottom bottom of the sea; sea; that that is is to say, the theparamrelative parameters of an MT-sounding MT-sounding that that could be be performed performed on the sea sea bottom bottom if drained. eters CONCLUSIOK. FIELDS APPLICATION FOR CONCLUSION. FIELDS OF PRACTICAL PRACTICAL APPLICATION FOR THE THE MAGNETOMAGNETOTELLURIC METHOD TELLURIC METHOD It follows from the above that that the ideal way to apply apply the magneto-telluric magneto-telluric It consists in performing performing an NIT-sounding MT-sounding as as described. When When the subsoil method consists is approximatively approximatively tabular, tabular, the harmonic harmonic analysis of the telluric telluric and magnetic magnetic is possible to conduct a careful quantitative quantitative interpretation interpretation which diagrams makes it possible gives us us the thickness and the resistivity resistivity of the various strata. strata. gives The The periods higher than than one one second second are exactly exactly adapted to the study study of large sedimentary beds beds and their petroliferous petroliferous structure. structure. Besides, Besides, their their recording does does sedimentary involve serious serious technical difficulties. difficulties. not involve The study of the shortest periods, less less than than one second, second, seems seems technically technically The diflicuit at the present stage stage of of the art. However, However, it is is less less urgent urgent in the light light of of difficuit needs in geophysical prospecting. It It should eventually eventually allow present needs allow us us to adapt adapt magneto-telluric method to various applications applications requiring the magneto-telluric requiring detail detail of the kind kind involved in civil civil engineering engineering studies, in mineral mineral prospecting, and search involved and in in the search underground water. water. for underground want to draw attention attention to the fact that that an isolated MT-sounding We want NIT-sounding carried unknown area can present information on in the center of a large unknown information similar similar to that that by a wildcat wildcat well in a large scale scale reconnaissance. For given by For instance, instance, the measurement of the number number of kilometers kilometers thickness of sediment sediment in the center of a large ment problem which cannot cannot be solved even partially basin presents a problem partially by by any any geophysimethod up until until now. The The magneto-telluric magneto-telluric method method should be able to solve cal method the problem by use problem by use of only only a single station. station. The The discarding discarding of the base base station, station, which is is indispensable indispensable in in the telluric telluric methmethod, gives the operator operator more freedom of movement movement and and improves improves the organizaorganization survey. He tion of his survey. He is no longer compelled compelled to proceed slowly. slowly. He He can afford to operate in a more rational rational way way by by setting setting up his initial initial stations stations at at some distance distance from each other. Later Later on, he can locate stations stations with with a closer closer spacing, but but only only to the extent extent required required by by continuity. continuity. Consequently, out the survey layout survey of a large sedimentary sedimentary basin by by Consequently, one can lay performing at the start a small number of MT-soundings far performing start number MT-soundings far removed removed from from one another, another, but but with with a great great depth depth of investigation. investigation. In In the second second step, one will will intercalate stations closer together, and at these he will perform intercalate closer together, at these will perform MT-soundings l\fT-soundings with with a more moderate moderate depth depth of penetration. penetration. Finally, Finally, the the continuity continuity between between the stations stations will will be assured assured either either by by soundings with with a relatively relatively small small depth depth of ininvestigation, vestigation, or, once in a while, while, by by simple, simple, quick quick determinations determinations of the apparent apparent resistivity resistivity summarily summarily evaluated evaluated through through a very very simplified simplified analysis analysis of the magmagneto-telluric data. It is unnecessary to add that the magneto-telluric neto-telluric data. It that magneto-telluric method method will will be particularly particularly appreciated appreciated every every time time that that a deep petroliferous petroliferous structure structure

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B.ISIC TIIEOR1TIIEORI- OF OF THE .\LlG.YF:TO-TELLCRIC .II.~G.~~TO-TELLI'RIC .IIETHOD B.L')fC .\[ETHOD

635

appears the surface. surface. In In this this case appears in complete complete disharmony disharmony with with the the structure structure on the it is essential great depth depth of investigation. investigation. essential for the the prospector prospector to penetrate penetrate to a great If approximately stratified, stratified, the the If it it should should happen happen that that the earth earth is not not even even approximately quantitative interpretation of the must be just just about about ruled ruled out. out. quantitative interpretation the MT-soundings MT-soundings must A knowledge variation as a function function of knowledge of the the apparent apparent resistivities resistivities and and of their their variation the nevertheless, certain certain indicaindicathe period period and and of the direction direction of the the line line provides, provides, nevertheless, tions though they they are are to a large large tions which which can be diagnostic diagnostic in special special cases, even even though degree degree qualitative. qualitative. .kCKSOWLEDGhZESTS .\CKXOWLEDGME"-;TS

.,

I am very Mrs. F. F. 13. B. Riek, Riek, Jr., ]r., to very glad glad to express express my my heartiest heartiest thanks thanks to Mrs. Dr. and Dr. Dr. R. R. L. Caldwell, Field Dr. Robert Robert '-an ITan Xostrand, Sostrand, Dr. Dr. Milton Milton B. B. Dobrin Dobrin and L. (‘ aldwell, Field Research Magnolia and to Mr. Mr. R. E. Selzer, Research Laboratories, Laboratories, Magnolia Petroleum Petroleum Company, Company, and Selzer, physicist physicist at the Institut Institut de Physique Physique du Globe Globe of Paris. Paris. Mrs. the translation translation of my my French French Mrs. Riek, Riek, a professional professional translator, translator, undertook undertook the text Drs. \-an '-an Sostrand, X ostrand, text into into English English for the the Magnolia Magnolia Petroleum Petroleum Company. Company. Drs. Dobrin, competence and and judgment. judgment. Dobrin, and and Caldwell Caldwell edited edited the the translation translation with with competence My the proofs. proofs. My friend friend Mr. Mr. Selzer Selzer gave gave me valuable valuable help help in correction correction of the The theoretical theoretical work work reported reported in this this paper paper was done The done some some time time ago and and has mentioned in applications applications for patents patents which which have been mentioned have been been made made in several several countries to protect protect the the new new prospecting prospecting met method countries hod involved. involved. Because Because of the the popotential practical practical applications applications I have have had had to postpone postpone any tential any publication publication related related to magneto-telluric phenomena magneto-telluric phenomena for for many many years. years. Meanwhile, the Russian Russian scientist scientist Tikhonov, Tikhonov, and Meanwhile, the and the the Japanese Japanese scientists scientists Kate, Kikuchi Kikuchi and and Rikitake Rikitake had had also recognized recognized the Kato, the existence existence of such an effect. effect. To my my knowledge, knowledge, they they have have not not pointed pointed out out the the possibility To possibility disclosed disclosed by by my my work of applying applying these results results to practical practical geophysical geophysical exploration. work exploration. They They have, have, however, paid paid attention attention to their their possible possible use for investigating investigating the however, the electrical electrical conconductivities very deep deep regions regions in the the Earth's Earth’s crust. crust. duct ivities of very It is therefore therefore a real pleasure pleasure for me to give give these scientists It scientists proper proper credit credit and and subject: the papers papers that, that, to my my kno\\-ledge, knowledge, they they have have published to list the published on 1 t his subject: Kate, Y. Y. and and Kikuchi, Kikuchi, T. T. (1950) (1950) Sci. Sci. Rep. Rep. T6hokll TGhokzr CniT. tyf~fz’. Ser. I’ , Geoplzysics, Geophysics, Kato, T", \‘Ol. 2, p. p. 139. 139. Vol. 2, Zrrsf. Tokyo Tok>w Clzil'., I.H;T’., 1701. Rikitake, T. T. (1950) (1950) RIIIl. R~tll. Earthquake Earlkquake Res., Illst. Rikitake, Vol. 28, 28, 11. p. 45, 219. Also Also (1951) (rgjr) '-01. 1701. 29, pp. pp. 61, 271. Tikhonov, A.1\'. A. N. (1950) (1950) Dokl. Dokl. Akad. Sal&k S_S.S.R. S.S.S.R. 73, 295. Tikhonov, Akad. "Yauk

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