On the determination of the apparent frequency-effect in the frequency-domain induced polarization method

Geoexploration, 14 (1976) 81-91 j Elsevier Scientific Publishing Company,

81 Amsterdam

- Printed

in The Netherlands

ON THE DETERMINATION OF THE APPARENT FREQUENCY-EFFECT IN THE FREQUENCY-DOMAIN INDUCED POLARIZATION METHOD

D. PATELLA

and D. SCHIAVONE

Observatory for Geophysics and Cosmical Physics, University of Bari (Italy) Institute of Geodesy and Geophysics, (Received

November

University of Bari (Italy)

25, 1974; accepted

August

15, 1975)

ABSTRACT Patella, D. and Schiavone, in the frequency-domain

D., 1976. On the determination Induced Polarization method.

of the apparent Geoexploration,

frequency-effect 14: 81-91.

In the variable-frequency technique of the Induced Polarization (IP) method, the frequency-effect parameter, defined as: f, = (pdc - pm)/pdc, cannot actually be directly obtained since the resistivity p m at infinite frequency cannot be measured. Thus some indirect procedures are needed to evaluate f,. We propose a possible solution through the study of the correlation between frequency-domain and transient IP measurements, based on the application of the Laplace transform theory. From this a simple integral relation between the complex frequency-dependent resistivity and the time-dependent chargeability is at first obtained. Then the analysis of the properties of IP transients, which indicates that the chargeability can be expressed as a sum of n exponential decay terms with different initial amplitudes Aj, allows to obtain an explicit form of the complex frequency-dependent resistivity, from which it follows that the frequency-effect is simply the sum of the coefficients AI. This result allows to calculate f, in several different ways; the simplest consists in the measurements of the real or in-phase component of the complex resistivity at a discrete number of frequencies of the applied current.

INTRODUCTION

In the frequency-domain or variable-frequency technique of the Induced Polarization method the amount of polarization in a uniform and linearly polarizable medium is detected as a decrease in the resistivity when the frequency of the applied current is increased (Marshall and Madden, 1959; Wait, 1959; Hallof, 1964). This dispersion effect has been observed for various earth materials and mainly for mineralized media where it is very pronounced; at high frequencies, the resistivity, for all materials, approaches asymptotically a constant real value (Wait, 1959). Phase-shifts between the applied sinusoidal currents and the corresponding generated electric fields, have been also observed (Wait, 1959; Madden and Cantwell, 1967).

82

These experimental evidences led to conceive of a general phenomenon for earth materials for which the resistivity is a complex function of frequency. On these bases Wait (1959) has shown that the following relation, for the case of sinusoidal currents in media with complex resistivities, can be written: E(iw )exp(iw t) = p (iw )J exp(iw t)

(1)

where: J is the amplitude of the source current density; p (iw ) is the complex resistivity; E(io ) is the complex frequency-dependent amplitude of the electric field; and exp(iw t) is a factor representing the oscillating behaviour of the fields with angular frequency w . Elq. 1 can be more concisely rewritten as: E(L)

= p(io)J(iw)

(2)

DEFINITION OF FREQUENCY-EFFECT

Suppose now we can always know the resistivity very high and very low frequencies and put:

of a polarizable

medium

(3)

p. =fmop (iw 1 being p. the d.c. resistivity

and:

Pm = lim p(iw) &2-L=

(4)

being p_ the resistivity at the highest frequencies. We define the following theoretical parameter for a polarizable when studying its frequency behaviour: frequency-effect

medium,

fe:

fe = (PO -Pm MJd

= I-P-

/PO

(5)”

As can be seen, the frequency-effect is a dimensionless defined by the knowledge of the two extreme resistivity CORRELATION

at

WITH TRANSIENT MEASUREMENTS

parameter completely values, p o and p _.

(TEE-DOMAIN)

Let us suppose that the uniform and linearly polarizable medium is electrically studied by the application of a source current in the form of a step function, whose density is given by: J(t) = Ju(t) *Note that the definition of the frequency-effect, here given by eq. 5, differs from that currently adopted and given by: f, = (pO - poo)/pm = p. /p,, - 1 (see Wait, 1959; Hallof, 1964).

(f-5)

83

where : u(t) = 0

fort<

0

(7)

u(t) = 1

for t > 0

(8)

In other words let us consider the charging cycle of the time-domain technique. For this we have sho)vn in a previous paper (Patella, 1972) that the following generalized Ohm’s law can be written: E(t) = PO [l - m(t)!J(t) where:

m(t)

(9)

p. is again the d.c. resistivity of the medium; E(t) is the electric field response, measured as an increasing transient; and m(t) is a function of the time defined in the interval [ O,m j as follows: for t = 0

(0 < m, < 1)

= m,

(10)

O
forO<

t
(11)

m(t) + 0

fort+

00

(12)

m. is the well known chargeability

The parameter Putting: P(t)

= PO [1

--

(Seigel, 1959).

m(t)1

eq. 9 can be rewritten

(13) as:

E(t) = P (t)J(t)

(14)

From eq. 14 and bearing in mind eqs. 13, 10 and 12, the following values are obtained for E(t): E(0)

=;iim+ E(t) =;;:+[p(t)J(t)]

= J ;iT+p(t)

E(m)

= /;i~ E(t) =;;i~

= J/i=

[p(t)J(t)]

=po(l

p(t) = POJ

-mo)J

limiting (15) (16)

From eqs. 15 and 16 it is easily obtained: m, = [:i%

E(t) ,:F+

E(t)]

/!Fm

E(t)

(17)

which gives the way for obtaining the chargeability of a polarizable medium when studying the charging cycle of the time-domain technique. Applying now the Laplace transform integral to E(t) and J(t) of eq. 14, it is obtained :

L [E(t)1 =

s E(t)exp(iwt)dt

= E(iw)

(18)

= J(io)

(19)

0

and:

L [J(t)1 =

s 0

J(t)exp(-iot)dt

84

Putting eqs. 18 and 19 into eq. 2, we have: L

[E(t)1 = p&I

L [J(t)1

(20)

Since the Laplace transform L [J(t)]

of the step-function

(6) is:

= J/io

(21)

then we can write: L

Jp(iw)liw

[E(t)1 step =

(22)

Eq. 22 represents in the frequency-domain the electric field response of the polarizable medium to a step function current density with amplitude J, while the response in the time-domain to the same step function excitation is p(t)J. Hence eq. 22 must be related to p(t) J through the Laplace transform, i.e.: p(iw)/iw

=

p(t)exp(-iwt)dt

s

(23)

0

From the initial and final value theorems and with eqs. 3, 4 and 22, we have: lii=E(t)

= LfiimeitiL [E(t)] = JJFo

which confirms Fme+E(t) +

PC0 =po

theory

= p. J

(24)

the result given by eq. 16, and:

= lim ioL[E(t)] cd+-

Comparing

p(iu)

of the Laplace transform

= J lim p(iw) cd+-

= pmJ

(25)

eq. 25 with eq. 15, it is obtained:

U--o)

(26)

If in eq. 17 the values of the above limits are substituted, mo = (POJ-

P,J)/Po

J= (PO

-

P,)/PO

Bearing in mind eq. 5, the following

we have:

= 1 -P,/Po

important

(27) result is obtained:

fe =mo

(28)

i.e., the theoretical frequency-effect parameter of a polarizable medium, measured by the frequency-domain technique, is equal to the maximum theoretical chargeability of the same medium, measured during the charging cycle of the time-domain technique. FREQUENCY-DOMAIN MEDIUM

MEASUREMENTS

IN THE CASE

OF AN INHOMOGENEOUS

When dealing by the frequency-domain technique with an inhomogeneous medium, that is the real field situation, an apparent frequency-effect fe,= can

85

be defined

according

to the following

expression:

(29)

ft?,a= 1 - Pm,alPo,a

being po,a the d.c. apparent resistivity and P_,~ the apparent resistivity at infinite frequency. Thus the apparent frequency-effect can be considered as the true frequencyeffect of a hypothetical homogeneous medium for which the same values of are measured. Poo,a and ~o,a According to eq. 26, the apparent resistivity ,D_,~ can be defined as: Pm,a = Po,a (1 -n&a)

(30)

being m, ,a the apparent chargeability of the charging cycle of the time-domain technique. Putting eq. 30 into eq. 29, it is easily obtained:

fe,a = m,,,

(31)

Thus the apparent frequency-effect is equal to the apparent chargeability. As a consequence, also for an inhomogeneous medium, the frequency-domain and the charging cycle of the time-domain are theoretically equivalent for studying its polarization properties. From eq. 29 it follows that the apparent frequency-effect can be determined once the values of p. ,D and pm,= have been obtained. While there is no problem for the evaluation of p. ,*, the measurement of pm,= is practically impossible. Thus some indirect procedures are needed to evaluate fe,a. PRACTICAL

DETERMINATION

Preliminary

considerations

Let us consider rewritten as: pa(t) = P~,~U -

OF THE APPARENT

FREQUENCY-EFFECT

eq. 13 that in the case of an inhomogeneous

m,(t)1

medium,

can be (32)

From laboratory and field experiments it has been recognized by many authors (Wait, 1959; Keller, 1959; Roussel, 1962; Ryapolova, 1968; Ogilvy and Kuzmina, 1972; Bertin and Loeb, 1974) that the function m=(t), appearing in eq. 32, can be represented by a sum of exponential terms of the type: n

m,(t)

= c

Aiexp(-+)

j=l

with Aj 2 0 and aj > 0 for every j.

(33)

From oq, 3% iit;is easily rmbl;ained :

87

Putting: t(iw)

= 1

(40)

-Pa(i41p0,a

from eq. 39 we have: n ;(iw)

=

n

c

Ap2/(ui

+ a”)

j=l

f

i c

Aj~jti/(ai + u2)

(41)

j=l

Thus : n

Re[[(io)]

= C Aj ti2/(aj + w2)

(42)

j=l

and : n

IF [g(io)]

= C Ajujw/(aT + w2)

(43)

j=l

Practically:

Re[t(io )I = 1 - Reb,(ia

)I /PO,== 1 - bn(iW )I

ho,0

in-phase

(44)

while:

~~[~(i~)l=--I~[~,(i~)Il~,,,

= --[P~(~u)I~~~-~~-~~~IP~U

(45)

Thus, from eq. 44 and eq. 42, a set of measurements of the in-phase component of the apparent resistivity p,(iw ) at different frequencies makes possible the simplest determination of the sum of the coefficients Ai at once, i.e., the apparent frequency-effect (see eq. 38). Other approaches for the determination of fe,a are based on the out-of-phase component, or phase-angle, or magnitude analysis of p,(iw ), but in these cases a more complicated mathematics is necessary. Determination of the apparent frequency-effect phase component

from the analysis of the in-

The real component of the function .$(io) can be determined at each frequency by measuring firstly the d.c. apparent resistivity p o ,a and secondly the in-phase component of the frequency dependent apparent resistivity Pa(iw ) (see eq. 44). Let us consider now eq. 42 where Re[.$(io)], (= Re(w) for simplicity), can be rewritten as follows:

88

Re(w)

=

;Aj

t

;r j(a;+w2)

(u; + 02) P+j

j-l

I

1

In the ratio of eq. 46 the numerator, N=

Alw2n

(46)

1

N, can be expanded

in the form:

+Aloc,,lw2(“-1)+Alcu,,Iw2(n-2)+...+

+Alcrl n-_2~4 +A2in

+A+l,n-m-lo2

+A2a2

+A2ct2 n-2u4

, 1~

+

2(n-f)+A2a,

202(n

+A2cu2,n-1~2

2)+...+

+

+......... .... .... ....+ +A,oZn

+Anan

+ Anan n-2 a4

lo2(n-‘)+A~~n,2~2(R~1)+. + Anan,,n--1 ti2

= (CAj)a2”+(CAjaj,l) j=l

+

(

g

=

W”n-1’+(;:Aj(ll,2)W2(n-2~+...+ j-1

j=l

Ajaj,nl)

j=l

. .+

ti4 + (5

Ajaj,n-1)

a2

(47)

j=l

where aj, h are the coefficients resulting from the expansion the numerator of eq. 46 in descending powers of w2 . Putting now:

of the products

= ?k

in

(48)

j=l

eq. 47 reduces to: n N

=

( 1w2n+y,w2(n-1)+...+~n-2w4 CAj

+Yn-lo2

(49)

j=l

Accordingly D=u?~

the denominator,

+Plw 2( Jn--11&(n--21+.

D, of eq. 46 can be expanded . . +fln+02

Thus from eq. 46 and with eqs. 49 and 50 we have:

+/jn

as follows: (50)

89

+

CJJ4

Yn-2

-on

m,Re(o)04

+Y~-~w’

,Pn-1Re(w)w2

--PJ?e(o)=

= Re(w)02n

(51)

Eq. 51 can be considered

as an equation

of the 2n unknowns:

~Aj,71,.--,7n-1,-~~,...i-Pn j=l We are only interested in the search for the unknown EAi, which is, as previously stated, the apparent frequency-effect. To this end it is sufficient to solve a system of 2n equations of the type (51) obtained by choosing 2n values of the angular frequency w and by measuring the corresponding values of Re(w ). The solution is given by: &Aj

= A’/A

(52)

j=l A and A are respectively:

where the determinants

Re(w,)w;”

( Re(w*,w;”

A’=

...

z(n-I)

Re(w2)w;('*--')

...

w; Re(w,)w: w; Re(w,)w;

Ww,) Re(w,)

.................

......... ......

.........

.......

.................

......... ......

.........

.......

.................

...............

z(n-I) wz,,

R~(w,,)w:~-~‘)

Re(w,,,)wi,, . -

zn

wz

j ... i

, ... / zn ~ w2n I

Re(w,)w;(" ‘)

.......

zn WI

a=

w2

2(11 I)

.........

” Re(w 2n )“2n

/

wr

WI

z(n

1)

w;wl)

Rdw

.

2

I

)w I

. . &,,

(n--1)

Re(w2)w:(”

..w.

.

‘)

.

2

W;

(53)

Re(w,,,)

RE(w, )w f

Re(w, 1

Re(w2)w,Z

Re(wZ

)

, .

II..... .. z(n-I)

w2tl

(54) ............. ......

........

.....

...................

........

......

Re(wd+,,

2(n

~1)

...

w&,

W~Z,I)whl R4w2,,)

90

For the practical application of (52) some considerations are needed. At first the point out that the method here proposed is based on the starting hypothesis that the function m,(t) can be expressed by a sum of n exponential terms, each of them related to a different polarization phenomenon. Such hypothesis has been largely verified on many types of mineralized or unmineralized rock formations and it has been found, as a general rule, that field and laboratory transients are well interpolated by three or four exponential decay terms at most. Hence it follows that, for the determination of the apparent frequency-effect, it is sufficient to make eight measurements of the in-phase component of pJiw ) at eight different frequencies. Secondly, we draw the attention on the fact that, in principle, the determination of the apparent frequency-effect by the above method does not depend on the frequency range used for the measurements. Practically, the frequency range O.Ol+lO c/set appears to be the optimum band for investigating I.P. phenomenon for the following main reasons (Ness, 1959): (a) above 10 c/set electromagnetic coupling of cables may disturb the measurements by adding spurious anomalies; (b) below 0.01 c/set natural earth currents may obscure the response from the internal I.P. sources; (c) within this range the greatest variations of Pa(io ) are generally experienced. CONCLUDING

REMARKS

From the foregoing analysis it results that, even if the maximum apparent frequency-effect, as defined by eq. 29, is not practically measurable, it can be, however, indirectly determined. It seems to us that the procedure here suggested does probably constitute, to our knowledge, the first attempt in the evaluation of the frequency-effect parameter according to its definition. In fact the frequency-effect, commonly adopted by other I.P. workers, is determined by measuring the resistivity at two different and finite frequencies in the range O.Ol+lO c/set; it is clear that in this way only a rough approximation of the maximum frequency-effect is in general obtained. ACKNOWLEDGEMENTS

The authors wish to express their gratitude to Professor D.S. Parasnis for his comments on the manuscript. This study has been performed with financial aid from the National Research Council (C.N.R. - Comitato per le Scienze Geologiche e Minerarie - Contract No. 73.00121.05).

91

REFERENCES Bertin, J. and Loeb, J., 1974. Traitement “a la main” et sur ordinateur des transitoires en Polarisation Provoqube. Geophys. Prospect., 22: 93-106. Hallof, P.G., 1964. A comparison of the various parameters employed in the variablefrequency induced polarization method. Geophysics, 29: 425-433. Keller, G.V., 1959. Analysis of some electrical transients measurements on igneous, sedimentary and metamorphic rocks. In: J.R. Wait (Editor), Overvoltage Research and Geophysical Applications. Pergamon, London, pp. 92-111. Madden, T.R. and Cantwell, T., 1967. Induced Polarization, a review. In: The SEG Mining Geophysics Volume, Editorial Committee, Mining Geophysics, Vol. II, Theory. SEG, Tulsa, Okla., pp. 373-399. Marshall, D.J. and Madden, T.R., 1959. Induced Polarization, a study of its causes. Geophysics, 24: 790-816. Ness, N.F., 1959. Analysis of the frequency response data. In: J.R. Wait (Editor), Overvoltage Research and Geophysical Applications. Pergamon, London, pp. 84-91. Ogilvy, A.A. and Kuzmina, E.N., 1972. Hydrogeologic and engineering geologic possibilities for employing the method of induced potentials. Geophysics, 37: 839-861. Patella, D., 1972. An interpretation theory for Induced Polarization vertical sounding (time-domain). Geophys. Prospect., 20: 561-579. Roussel, J., 1962. Etude sur modeles rdduits des phenomenes de polarisation provoquee. Ann. Geophys., 18: 360-371. Ryapolova, V.A., 1968. Methodical instructions on the method of Induced Potentials application in engineering geological investigations. Moscow ONTI-TSNIIS (in Russian). Seigel, H.O., 1959. Mathematical formulation and type curves for Induced Polarization. Geophysics, 24: 547-565. Wait, J.R., 1959. The variable-frequency method. In: J.R. Wait (Editor), Overvoltage Research and Geophysical Applications. Pergamon, London, pp. 29-49.