- Email: [email protected]
Relaxation phenomena and induced polarization
Geoexplorotion, Elsevier Science
22 (1984) 107-127 Publishers B.V., Amsterdam
RELAXATION
PHENOMENA
107 - Printed
in The Netherlands
AND INDUCED POLARIZATION
JAMES R. WAIT Department (U.S.A.) (Accepted
of Electrical for publication
Engineering, October
University of Arizona,
Tucson,
Arizona
85721
16, 1983)
ABSTRACT Wait, J.R., 1984. 107-127.
Relaxation
phenomena
and induced
polarization.
Geoexploration,
22:
We deal with the general subject of the linear system behavior of material media that have dispersive electrical properties, In this review we deal with phenomenological theory which begins with a general statement of Ohm’s law that relates the macroscopic electric field and the current density at a point in the medium. The complex constant of proportionality is the complex resistivity that forms the basis of the discussion. Various ways to represent the frequency dependence of the complex resistivity are described. The corresponding time domain or transient response characteristics are also considered in the context of the induced polarization properties of the medium. We point out that it is often fruitful to represent the dispersion as a superposition of basic relaxation processes. In fact, if such distributions are sufficiently broad, the effective complex resistivity varies in magnitude in almost a linear manner with the logarithm of frequency. The corresponding phase is approximately a constant that is proportional to the slope of the amplitude curve on log frequency basis. This same slope parameter is also descriptive of the transient decay of the system when the linear response is plotted on a log time base. This general behavior seems to be a general property of such composite media as textured rock, mineralized material and biological suspensions where we can expect that the scale of the dispersive mechanisms is quite broad.
INTRODUCTION
An experimentally observed fact is that the current density and the electric field are proportional to each other in most conductive materials. This is Ohm’s law! Such a statement presupposes that the medium properties are averaged in some sense so we are dealing with the macroscopic state of matter. In discussing the phenomenon of induced electrical polarization in geological and biological media, this will be our first step. For background information see Wait (1959), Matijevic (1974), Tareev (1975), Grant et al. (1978). Without loss of generality we will assume that the time factor is exp(iwt) where w is the angular frequency. Thus Ohm’s law may be written
0016-7142/84/$03.00
0 1984 Elsevier Science
Publishers
B.V.
108
2(h)
p(b)
= IT(b)
(1)
is the current density, where_ p(h) is the resistivity of the medium, ah) and E (iw) is the electric field. In writing eq. 1 we have made the following assumptions. (1) p does not depend on the magnitude of 3 i.e., non-linear effects are negligible or non existent. (2) p, wh$h San be a function of position, does not depend on the direction of J or E; i.e., we are only dealing with isotropic regions. (3) p is a single valued function of iw and defined in such a manner that causality is preserved (de$cribed mor$ fully below). As indicated above E(h) and J(b) are vectors while their orthogonal components are complex phasors. Thus, for example, the real physical quantity eX(t) (i.e., the x component) is taken to be: e, (t) = Re. E, (iw)eiwt = IE,I cos(ot
+ &)
(2)
where = phase of E,
&
Then clearly eq. 1 signifies that: IExl = IpJxl = IpIlJxl
(3)
and @X = $Jx-@ where GX =phaseof
Jx
(4)
and -a
= phaseofp. The complex
p(h)
= [o(h)
resistivity,
as defined,
+ ie(iw)w]-’
where u( iw) and c(h) are the real at the angular frequency w. Note defining u and E to be real functions Alternatively we may regard the as the complex conductivity 6. Thus: C(h)
= [p(io)]-’
= o(iw) + if(io)o
can be written
in the form: (5)
conductivity and the real permittivity that there is no loss of generality in for real w. reciprocal of the complex resistivity p (6)
109
Or, in some instances, ;(iw) given by: ;(iw)
= [iup(i’
we may wish to deal with a complex
permittivity
= e(h) + (iw)-‘o(i0)
(7)
There is really no fundamental reason to choose the complex p rather than the complex 6 or 2. But as a matter of convention and in accord with current usage we shall employ the complex p. RATIONAL
APPROXIMATIONS
A convenient
representation N
p(h)
= IpleTi@ = C A, n=O
for the complex
(Wait, 1959) is:
1
an + iw
where n extends over all integers from cients. As a practical matter N is chosen ly represent p over the desired range of selected to best represent the given data. of w, we note that: Re.p(iw)
resistivity
(8) 0 to iV and An and QI, are coeffito be sufficiently large to adequatew. The coefficients An and QI, are On the basis of eq. 8 for real value
A, &
= g n=O
n
and Im.p(io)
= - 5
n=O
An
(10)
w a; + cd=
The preceding two equations demonstrate that Re.p is an even function and 1m.p is an odd function of w. Furthermore, on noting that lpl’ = (Re.p)’
+ (Im.p)2
and @ = -arc tan [ Im.p/Re.p] , we see that Ipl is an even function is an odd function of w . Thus, for all real w ,
p(h)
of w
while Q
= p*(-iw)
(114
= p(-io)
(lib)
or p*(b)
where the asterisk denotes a complex conjugate. In writing eq. 8 as being a suitable or adequate representation of eq. 5 for real w we are permitted to take N to be infinite. Or instead of a discrete sum over n, an integral representation may be used. Thus, for example, we can write
110
7 G(a) --& da
P(~w) =
(12)
0
where (Y is real and encompasses distribution function.
all values from 0 to 00. The G (cu) is a real
EXTENSION TO TIME DOMAIN
We now may generalize our formalism to arbitrary time variation. Thus, for example, we may stipulate that the current density (x component) is given by: j,(t)
= JoS(t)u(t)
(13)
where S(t) is the source signal waveform, Jo is a constant t > 0, = 0 for t < 0. By Fourier synthesis we can write: Jx(iw)eiwt
and u (t) = 1 for
dw
(14)
where &(iw)
=
Irnj,(t)eeiwtdt -Ca
(15) = Jo i
S(t)e-‘wf
dt
0
Thus, we regard Jx (iw) as the spectral content of the function j, (t). In Laplace transform notation we would write eqs. 14 and 15 in the form: j,(t)
= 1-l J,(s)
(16)
= Xjx((t) = LJ,S(t)
(17)
and Jx(s)
where s is formally identified with iw. But now we continue complex plane; thus, for example, eq. 16 is given by: j,(t)
=
-& JC”wJx(s)est
ds
c-i-
where c is some small positive quantity. In accordance with Ohm’s law, given by eq. 1, the electric e, (t) is related to the causative source j, (t) by:
s into the
(18)
field response
111
ex(t)
=
_frn
k
The Laplace transform e,(t)
Jx(ia)eiwt
p(b)
dw
equivalent
(19)
is
@)I
= L-l [p(s)&
(20)
= JCJ -l [P(S)R b)l where R(s)
(21)
= JZs(t)
THE IDEAL
DIELECTRIC
MODEL
In an ideal lossy dielectric, which is not physically realizable, o(h) = u and e(h) = E are constants for all frequencies. Likewise a(s) = u and E(S) = E for all complex s. Thus: 1 P(S) = u + ES
(22)
In this case, ex(t)
= Jod:-’
[
iA R(s)]
On noting that the “impulse L-1
p(s)
=
1-l
l
u+a = i exp(-(u/e)t)
response”
= 1 L-1 E
(23) is given by :
1
s+u/e
u(t)
(24)
we may write eq. 23 in the form of a convolution ex(t) Indeed that:
1 t = Jo S(T) exp[-(u/e) e s0
(t - T)]d7
integral:
(25)
if ,S(t) = t,, 6(t - to ), where 6 (t - t,, ) is a unit impulse at t = &,, we see
e%(t) = Jo 4
exp[-(u/e)(t-tO)]u(t-to)
(26)
112
This has the form of a decaying exponential following the application of the source impulse current. The form is sketched in Fig. 1 where the time constant is E/U seconds.
t e,(t)
~ t=t,
t
Fig. 1. The exponential decay of the electric field in an ideal lossy dielectric following the cessation of a steady current.
In the case of a step function would specify that S(t) = u(t-tl) ex(t)
1 t = JO exp[-(a/e) s e f, = JQ f
[I-
excitation of the ideal lossy dielectric we in eq. 13. Then we see from eq. 25 that:
(t-T)]dT
(27)
e- w4wf]u(~-~l)
(23)
The build up of e, (t) to its final value is shown in Fig. 2.
__---__---
----/----
e, (t)
Fig. 2. Exponential build up of the electric field in an ideal lossy dielectric following the application of a step function current.
GENERALLY
DISPERSIVE
CASE
In dealing with a generally convenient to write:
dispersive
complex
resistivity
function,
it is
113
P(iW) =
P,
+
Jr&
(PO- P,)
dT
(29)
0
which is really equivalent to eq. 12. Here p, designates the high frequency limit and p. designates the low frequency limit of the function p(h). The precise physical meaning of these limiting values is not clear at this stage. Suffice it to say that the following limits hold:
p(iw) =
ii0
PO
in a mathematical sense where p, and p. are real parameters. The function f(7) as used in eq. 29 is the distribution of the relaxation times. It is normalized such that: m
s
f(T)dT
= 1
(39)
0
In some cases it is desirable function. Thus we write: p(i0)
= p’(0)
to use yet another
- ip”(w)
form for the distribution
(31)
where
P’(W) =
P,
+
(PO- Pw)
J {(JT* dT 1
0
and
P”(W) = (PO- P,)W
7
7 f(T) 1
+
w2T2
d7
(33)
0
For real frequencies w, p’ and -p” are the real and imaginary parts of p (iw) as can be ascertained from eq. 29. In the case of a single relaxation time f(r) = 6(7-TV) and we have:
P’(W)=
(POPm +
Pco)
1 + cd*r* 0
(34)
and
p”(U) =
(PO- Pm)W~o 1 + w*r*0
(35)
114
As pointed out by Cole and Cole (1941) in a different context, the points (p’,p”) lie on a semi-circle with the center on the p’ axis and they intersect this axis at p’ = p,, and p’ = p,. This follows from the equality: EP”(W)12 = [PO -
P’(W)1b’(w) - PC01
(36)
which can easily be ascertained from eqs. 34 and 35. The so-called Cole-Cole plot is obtained when p”(w) is plotted versus p’(w) for various angular frequencies. An example is chosen for the parameters p. = 1.2 units and p, = 0.2 units. The resulting semi-circle is sketched in Fig. 3 for w ranging from 0 to 00 as designated by the dimensionless parameter ~7~.
p”(
Fig. 3. The complex resistivity using the Cole-Cole plot for the ideal Debye model (i.e., single relaxation time).
This full semi-circle locus is characteristic of a single relaxation time. When there is a distribution of relaxation times it is possible to derive the following inequality that holds for any real positive value of o :
[P”(~)12 < [PO- P’(W)1b’(o) - Ll The proof of this statement
(37)
follows from the definitions
32 and 33.
THE COLE-COLE SPECIAL FORM
The special Cole-Cole
form (1941)
for the dispersion
amounts
to writing:
1 P(iO)
= Pea + (PO - P,)
1 + (iLJro)k
(33)
where k is an empirically adjusted non-integer. In the limit where k-+1 we retrieve the Debye form (Frohlich, 1949) given by eqs. 34 and 35. Another way of writing eq, 38 is (Pelton et al., 1978): p(iw)
= p.
1 [ 1 -m.
l-
1 1 + (iWTO)k11
(39)
115
where m0 =-
PO - Pm
PO
is sometimes called the “peak chargeability” of the region. To illustrate the properties of the Cole-Cole equation given by eq. 38 we plot p’ and p” as a function of ln(wrO). Such an example is shown in Fig. 4 where we choose p. = 1.2 and p, = 0.2. The values of h = 1 and l/2 are chosen. Of course k = 1 corresponds to the Debye form whereas the case k = l/2 allows for a broader dispersion effect.
-6
-4
-2
0 In(w%
2
4
6
1
Fig. 4. The real and imaginary part,s~omplex k = % and the Debye model (i.e., k = 1).
resistivity
for the Cole-Cole
model
for
The Cole-Cole plot for the case h = l/2 is shown in Fig. 5. It is a circular arc with its center below the p‘ axis. To prove this statement we consider the two phasors: Ptio)
PO -
- P,
=
(PO -
PJ
1
+
P@) = b0 - Pee) 1 -
(l;To)k 1 1
+
fiWTo)k
1
(41)
These are shown by the two straight line segments in Fig. 5 that subtend an angle 0 at a point P on the curve. Now the ratio of eq. 41 to eq. 40 is: PO -
P(iW)
P(iW)
-
= +
(i~7~)~ = (~7~)~ei (-8
Pm
where n-0 = nk/2. As indicated the angle 8, in radians, does not depend the position of P on the arc (i.e., it does not depend on 0).
on
116
f f”
0.5 W-
P
I
@
0~
0.5
r40
1.0
;
1.5
PO r’
-
Fig. 5. The complex resistivity for an Argand plot showing that the Cole-Cole form is an arc of a circle.
Probably the reason the Cole-Cole equation (with adjustable po, p, and k) is so versatile is the inherent control over the distribution of the relaxation times (Tareev, 1975; Pelton et al., 1978). For example we may rewrite eqs. 32 and 33 in the form:
P’(U) =
(42)
PC.3
and
p”(0) = (PO- p,)w
J 7F(:=ryyr)
(43)
7=0
where F(ln7)
= 7 f(7)
is the logarithmic distribution function.
In(7/To
)
-
Fig. 6. The d~tribution of relaxation times for the Cole-Cole model.
Following the analysis of Bottcher equivalent problem, we may write:
and Bordewijk
(1978)
for an
117
F(ln7)
sin ~(12-1)
= 1 2n
cosh[lz ln(TO/T)] - cosn(h -1)
1
(44)
This quantity is symmetric about T = T,, which is called the critical relaxation time. As indicated in Fig. 6, it becomes increasingly broad for decreasing values of k. TRANSIENT
RESPONSE
OF THE COLE-COLE
Using Laplace transform function takes the form:
notation
FORM
(i.e., iw + s) the complex
resistivity
1 P(S) = P, + (PO-PJ The response tained from: A(t)
=
(45)
1 + (S70)k
of the electric
field for a step-function
current
density
1 L-l p(s)s_’
is ob-
(46)
PO
This is normalized context of induced M(t)
such that A(=) = 1. The more relevant polarization, is:
= [l -A(t)]
in the
u(t)
1 ----
=J-l
quantity,
1 P(S)
[ S
PO
s
1
(47)
Using the general Cole-Cole form we see that: (46) The special case k = 1, which is the Debye form leads readily to M(t)
-
PO-pme-1 PO
l
PO - PC.3 = _ eetjTou(t)
s + (l/To)
PO
(49)
which is the simple exponential relaxation as illustrated in Fig. 1. Another special case is k = l/2, which is the Warburg form and is only a bit more complicated. Then we find that:
M(t) =
PO PO
J-’
[$
[s + (:irfhl] $$
0
(56) = p”~opm ef/%erfc[(t/70)115]u(t)
118
where we have used formula 52 (p. 211) in Roberts and Kaufman (1966, p. 211). This result is quite different from a single exponential. For example, we note that, for small x : e”‘erfc(x)
= ex’ [++I&
+~---.)I
Thus : M(t)
2
M(0)
[1-(J-f&)n]u(t)
(52)
for t << TV. On the other hand, for large t, we exploit behavior, valid for x >>> 1, expressed by: eXa erfc(x)
1 1 xn”2
1 1.3 1 - 2x2 + 22x4-(
= l/(xn%)
for
sz M(O)
+
23 x6
.I’ 1
x --f m
This tells us that the asymptotic M(t)
1.3-5
the asymptotic
(53)
tail is given by:
$ [(y _ ; (yq3’*+ **.]
(54)
which certainly has a much slower decay than an exponential. When the exponent k in the Cole-Cole form is any number between 0 and 1, we are faced with the inverse transform given by eq. 48. A general evaluation is not simple (e.g., see Lee, 1981). However, the limiting forms can be obtained rather simply. For example, we may write:
M(f) = M(O)&_’
1
s~+~ )I 1 k s70
(55)
km u(t) When k = 1 (i.e., Debye form) this reduces to exponential note that P(m+l) = m!. For small t we see that:
(56) decay when we
119
M(t)
1
1
E M(0)
l-
t
___
k
-
r&+1)
(57)
( 70 11
In the case k = ‘/z (i.e., Warburg form) this is consistent with eq. 52 if we note that I’(3/2) = n%/2. To deal asymptotically with the case of large t, we proceed as follows: M(t)
(STOP
= M(0) x-’
s(I + @TOP) m
=
c
n/r(o)
(-l)yTO)Wn+l)
~-1
#n+l)
-
1)
n=O
Es
5
M(0)
(-1)”
n=O
1 I-(1 - k(n+l))
Writing out the first three terms:
1
+
l?(l-3h)
70
0t
3k
(59)
*+* 3
we see that the decay is slower according to the smallness of Itz.The special case /z = l/z (i.e., Warburg form) is consistent with eq. 54 if we note that r(S) = n’h and I’(-%) = -2n”h, and of course I’(0) + =J. The result for M(t) given by eq. 59 can be used to predict the general asymptotic behavior for large t for any k. Here we need to take note of the properties of the Gamma function. For any v it is defined by:
0
which is finite when v > 0. Also from the definition
it follows that:
r(v+l) = e(v) ww-4
=
&
In the case of an integer (V = n)
r(n) = (n-l)! for n > 0. For further
properties
of Gamma functions
see Goldman
(1966).
120 TRANSIENT
RESPONSE FOR LOG UNIFORM DISTRIBUTION
The Cole-Cole dispersion is remarkably useful and it can often closely to experimental data over a broad range of frequencies. form is to write as before:
be fitted Another
1
p(iw)
= P, +
(PO
-
PA
J
F(ln7)
7=0
but following F(ln7)
Shuey and Johnson
l+iwr
(61)
d (lnr)
(1973), we now choose
= F.
for l/b < 7 < l/a clearly :
and = 0 outside
p(iw) = p, + (p. - p,)
f” llb
this range, where
F”
(1 + iw7)7
0 < a < b < 00. Then
d7
(62) = P, + (p. - P,)F~ ln
Now as w + 0 the right hand side of eq. 62 should reduce to po. Thus F. = (In b/a)-‘. Compacting the notation we write: p(iw)
= p, +52ln
(63)
where 1 Q = (PO - Pm) In b/a If we restrict p(iw)
the frequency
range such that
b S w % a it is evident
z p, + S2(lnb - lnw) - is2 $-
that: (65)
This indicates p’(o) has a slope of -52 as a function of lnw while p”(o) is constant. This behavior for frequency and time domain data on rock samples was noted by Wait (1959). To illustrate the above features of the dispersion for this model we write: dp(io)/d(lnw)
dp(iw)
= w -
do
Thus we note that if b % w S a
1
1
b + io
a + iw
---
(66)
121
(67)
which is consistent with eq. 65 above. Working with eq. 66 we may readily deduce M(t). Thus: M(t)
= M(0)
the decay response
function
1
L-l a
=
[l ---$ i
M(O)
e-at; e-b: dt]
(68)
u(t)
a
where we have used the inverse transform: -at _ ,-&t L-1
In
s+
b
-
e
Another
(69)
u(t)
s+a
t
form of eq. 68 is: 1
M(t)
= M(0) [Ei(-bt)
- Ei(-at)]
In 4
(76)
where: Ei(-x)
= - i
e-’ G du
(71)
X
is the exponential
integral. Here we have also noted that:
m e-at _ ,a J 0
dt = In 5 t
The slope of M(t)
as a function
dM(t) = t dM(t) = d In t
-M(O)
dt
dM(t) dln t
z -M(O)
of In t is a useful parameter.
- ’
te-at
_
It is given by
,-bt)
(73)
In i
This reduces to a constant -
(72)
a
-
1
In t
in the case where l/b << t Q l/a. Then:
= _ap,’
(74)
122
where 52 as defined by eq. 64 is the slope of the p(k) vs. In w curve. Sometimes the curvature of the decay response is of interest (Wait, 1959). This is defined by: (oewaf - beSbt)
M,(t) =
(75)
a Clearly M,(t)
2 0
when
t P -
1
In b a
b-a
KRAMERS-KRONIG
RELATIONS
We wish to derive the relationship between p’(o) and p”(w) for real values of w. To begin with we consider a complex integral representation for the in the Zower half of the function p(h) - p, that is regular (analytic) complex w plane. We choose a closed contour as indicated in Fig. 7 and apply Cauchy’s integral formula to write, for a selected real value L, the following: p(G)
1
- ,um=
2ni
+
[J semi-circle radius R j+r
s
+
w--R
of
s
+
semi-circle radius r
Here it is assumed that the function
of
p(b)
s
-R
w=&r
b(io)
-PA
w-l3
1
dw
(76)
is regular in the lower half plane.
I
Fig. 7. The contour relations.
in the complex
Also we define p, such that
w plane that is used to derive
the Kramers-Kronig
123
where R is the radius of the large semi-circle. The first integral in eq. 76 thus goes to zero in this limit. To deal with integration around the small semi-circle of radius r we introduce the substitution = &+reio
0
where
O<$
Now we may write: p(G)
=
f-’ _ j--- “;‘;‘-
pm+ &[-
G+r
-m
+
n p(iw)-p,
s
rei+
0
where, of course, latter integral. In the limit t-0
p(G)
= p, +
dw
(irei@) d$
rei@ cancels we obtain
---&[ ni(p(iL)
1
(771
in the numerator
and denominator
of the
the form:
-p,)
-
Jwp’~!-Gpw do]
(78)
-co
where the integral deduce that:
is the Cauchy
principal
value. After reshuffling
dw = p,--
1 in
+- p(h) f_m o-ij
eq. 78 we
do
(79)
The latter form follows from the identity: +oO
dw f _m w--G
= 0
(80)
If we now write: p(h)
= p’(w) - ip”(0)
where p’ and p” are real, eq. 79 can be split into real and imaginary to yield: p’(G) and
= p_+;
+- p”(W) m wp”(W) f dw = p, + z f _m 0-G lr 0 cd2 - ((2)’
do
parts
(81)
124
(82) In writing the second forms the even/odd and
P’(W) = p’(-a)
p”(w)
properties
were used:
= -p”(-w)
that hold for real w . The integral forms specified by eqs. 81 and 82 are the Kramers-Kronig relations (Bottcher and Bordewijk, 1978). These tell us that p’(o) and p”(w) cannot be independently specified. In fact a knowledge of one, for all real w, will determine the other. However, we shall bear in mind here that p(io) has been assumed regular in the lower half of the complex w plane and the lower and upper limits p. and p, are well defined. Actually there is a very similar argument that demonstrates the equivalence between the amplitude and phase dispersions. For example we may write: p(L)
= Ip(iw)l e-i*(w)
(83)
for real frequencies where Q(w) is the function a(w) is defined according to:
phase.
Then
the
“attenuation”
= p. e- a(w)
Ip(
(84)
for all real frequencies. Also Q(w) -+ 0 at both w + 00 we must have:
It is clear that Q(o) = -@(--w) and a(~) = a(-~). limits w + 0 and 00. By definition a(O) = 0 but at
= p. e-OL(-)
P,
or equivalently a(-~)
=
(85) :
lnPo/P,
=
a,
Now it proves to be convenient F(iw)
= In -
to introduce
a complex
function:
PO =
a(w)
+
i@(u)
p(iw)
that is actually valid everywhere in the complex axis of w, (Yand + have the properties indicated. In analogy to eqs. 81 and 82 we may write: a(w) and:
2 = (Y, - n
= f
0
iw plane. But on the real
w@(a) w2
-
(;;)2
dw
(87)
125
These show explicitly that the “attenuation”, as a function of frequency can be determined from a knowledge of the phase as a function of (all) frequencies. The converse is also true - the phase can be determined uniquely from the attenuation rate for this class of systems. This, of course, is a well known fact but the relationships are usually discussed in a different context. Goldman (1966) gives a useful summary. The phase integral theorem follows from eq. 88 by simply setting w = 0, whence:
If we change given by:
the variable
to In w this becomes
the phase area theorem
as
m s w=o
(a(w) d(ln w) = (Y,
(90)
That is, the area under the phase curve, plotted vs. log of frequency is a constant. The integral formula for the phase function given by eq. 88 can be converted to a more useful form by a simple change of variable. For example, we set w/L3 = eU so that: du - 1 f IT 0 Following yield :
an integration
by parts,
where we have used the standard
(91)
-?-du sinh u the two integrals
can be combined
result that
du = -ln[cothl~~] s sinh u Another integral:
form
to
of eq. 92 is obtained
(93) if we add and subtract
the known
+-
s -0u
In coth
I
t
du = ; I
(94)
126
Then we obtain: @(cl) = -
;( $) -b /I [- -(s))
1ncothj;l
du
(95)
c where :
is
the slope of the attenuation
function
on a In o base at the frequency
w = &. In fact the peaked nature of the In coth i I
I
function, as depicted in
Fig, 8, indicates that the integral on the R.H.S. of eq. 95 may be neglected if a(w) is not varying rapidly in the vicinity of w = &. This leads to the simple formula:
(96) which relates the phase at a frequency ij to the slope of the attenuation curve on a In w base at the same frequency 6~. In the more general case, one should retain the integral term on the R.H.S. of eq. 95. Or in many cases it may be convenient to work directly with eq. 92 which is amenable to numerical integration.
6-
Fig. 8. The function
CONCLUDING
In cothlul21
where u =lnlw/&l.
REMARKS
As we have indicated there are many possible ways to represent the linear system behavior of material media with dispersive electrical properties. Only in very limited situations could we assume a ~equency-~dependent conductivity and permittivity. However, an example might be saline water
127
solutions free of impurities where the conductivity would be a constant of the order of mhos/m and the permittivity would be near 80 relative to free space (Frohlich, 1949). In nearly all other cases of composite media such as textured rock, mineralized material, and biological suspensions, the constitutive properties are highly dispersive particularly in the audio and radio frequency range. Here it is often fruitful to represent the dispersion as a superposition of relaxation processes. If such distributions are sufficiently broad the effective complex resistivity exhibits a magnitude which varies almost linearly with the logarithm of the frequency. The corresponding phase is approximately a constant that is nearly proportional to the slope of the amplitude curve. ACKNOWLEDGEMENTS
I am grateful to Joann Main who typed Debroux for his helpful comments.
the manuscript
and Patrick
REFERENCES Bottcher, C.J.F. and Bordewijk, P., 1978. Theory of Electric Polarization. Elsevier, New York, N.Y. Cole, K.S. and Cole, R.H., 1941. Dispersion and absorption in dielectrics. J. Chem. Phys., 9: 341-351. Daniel, V.V., 1967. Dielectric Relaxation. Academic Press, New York, N.Y. Frohlich, H., 1949. Theory of Dielectrics. Clarendon Press, Oxford. Fuller, B.D. and Ward, S.H., 1970. Linear system description of electrical parameters of rocks. IEEE Trans., vol. GE 8, pp. 7-17. Goldman, S., 1966. Laplace Transform Theory and Electrical Transients. Dover Pubhcations, Inc., New York, N.Y. Grant, E.H., Sheppard, R.J. and South, G.P., 1978. Dielectric Behavior of Biological Molecules in Solution. Clarendon Press, Oxford. Lee, T., 1981. The Cole-Cole model in time domain induced polarization. Geophysics, 46: 860-868. Matijevic, E. (Editor), 1974. Surface and Colloid Science, Vol. 7. Electra-Kinetic Phenomena. John Wiley, New York, N.Y. Pelton, W.H., Ward, S.W., Hallof, P.G., Sill, W.R. and Nelson, P.H., 1978. Mineral discrimination and removal of inductive coupling with multifrequency IP. Geophysics, 43: 588-609. Roberts, G.E. and Kaufman, H., 1966. Table of Laplace Transforms. W.B. Saunders, London. Shuey, R.T. and Johnson, M., 1973. On the phenomenology of electrical relaxation in rocks. Geophysics, 38 (1) 37-48. Tareev, B., 1976. Physics of Dielectric Materials. MIR Publishers, Moscow. Wait, J.R. (Editor), 1969. Overvoltage Research and Geophysical Applications. Pergamon Press, Oxford. Wait, J.R., 1982. GeoElectromagnetism. Academic Press, New York, N.Y. Zonge, K.L., Sauck, W.A. and Sumner, J.S., 1972. Comparison of time, frequency, and phase measurements in induced polarization. Geophys. Prospect., 20: 626-648.
22 (1984) 107-127 Publishers B.V., Amsterdam
RELAXATION
PHENOMENA
107 - Printed
in The Netherlands
AND INDUCED POLARIZATION
JAMES R. WAIT Department (U.S.A.) (Accepted
of Electrical for publication
Engineering, October
University of Arizona,
Tucson,
Arizona
85721
16, 1983)
ABSTRACT Wait, J.R., 1984. 107-127.
Relaxation
phenomena
and induced
polarization.
Geoexploration,
22:
We deal with the general subject of the linear system behavior of material media that have dispersive electrical properties, In this review we deal with phenomenological theory which begins with a general statement of Ohm’s law that relates the macroscopic electric field and the current density at a point in the medium. The complex constant of proportionality is the complex resistivity that forms the basis of the discussion. Various ways to represent the frequency dependence of the complex resistivity are described. The corresponding time domain or transient response characteristics are also considered in the context of the induced polarization properties of the medium. We point out that it is often fruitful to represent the dispersion as a superposition of basic relaxation processes. In fact, if such distributions are sufficiently broad, the effective complex resistivity varies in magnitude in almost a linear manner with the logarithm of frequency. The corresponding phase is approximately a constant that is proportional to the slope of the amplitude curve on log frequency basis. This same slope parameter is also descriptive of the transient decay of the system when the linear response is plotted on a log time base. This general behavior seems to be a general property of such composite media as textured rock, mineralized material and biological suspensions where we can expect that the scale of the dispersive mechanisms is quite broad.
INTRODUCTION
An experimentally observed fact is that the current density and the electric field are proportional to each other in most conductive materials. This is Ohm’s law! Such a statement presupposes that the medium properties are averaged in some sense so we are dealing with the macroscopic state of matter. In discussing the phenomenon of induced electrical polarization in geological and biological media, this will be our first step. For background information see Wait (1959), Matijevic (1974), Tareev (1975), Grant et al. (1978). Without loss of generality we will assume that the time factor is exp(iwt) where w is the angular frequency. Thus Ohm’s law may be written
0016-7142/84/$03.00
0 1984 Elsevier Science
Publishers
B.V.
108
2(h)
p(b)
= IT(b)
(1)
is the current density, where_ p(h) is the resistivity of the medium, ah) and E (iw) is the electric field. In writing eq. 1 we have made the following assumptions. (1) p does not depend on the magnitude of 3 i.e., non-linear effects are negligible or non existent. (2) p, wh$h San be a function of position, does not depend on the direction of J or E; i.e., we are only dealing with isotropic regions. (3) p is a single valued function of iw and defined in such a manner that causality is preserved (de$cribed mor$ fully below). As indicated above E(h) and J(b) are vectors while their orthogonal components are complex phasors. Thus, for example, the real physical quantity eX(t) (i.e., the x component) is taken to be: e, (t) = Re. E, (iw)eiwt = IE,I cos(ot
+ &)
(2)
where = phase of E,
&
Then clearly eq. 1 signifies that: IExl = IpJxl = IpIlJxl
(3)
and @X = $Jx-@ where GX =phaseof
Jx
(4)
and -a
= phaseofp. The complex
p(h)
= [o(h)
resistivity,
as defined,
+ ie(iw)w]-’
where u( iw) and c(h) are the real at the angular frequency w. Note defining u and E to be real functions Alternatively we may regard the as the complex conductivity 6. Thus: C(h)
= [p(io)]-’
= o(iw) + if(io)o
can be written
in the form: (5)
conductivity and the real permittivity that there is no loss of generality in for real w. reciprocal of the complex resistivity p (6)
109
Or, in some instances, ;(iw) given by: ;(iw)
= [iup(i’
we may wish to deal with a complex
permittivity
= e(h) + (iw)-‘o(i0)
(7)
There is really no fundamental reason to choose the complex p rather than the complex 6 or 2. But as a matter of convention and in accord with current usage we shall employ the complex p. RATIONAL
APPROXIMATIONS
A convenient
representation N
p(h)
= IpleTi@ = C A, n=O
for the complex
(Wait, 1959) is:
1
an + iw
where n extends over all integers from cients. As a practical matter N is chosen ly represent p over the desired range of selected to best represent the given data. of w, we note that: Re.p(iw)
resistivity
(8) 0 to iV and An and QI, are coeffito be sufficiently large to adequatew. The coefficients An and QI, are On the basis of eq. 8 for real value
A, &
= g n=O
n
and Im.p(io)
= - 5
n=O
An
(10)
w a; + cd=
The preceding two equations demonstrate that Re.p is an even function and 1m.p is an odd function of w. Furthermore, on noting that lpl’ = (Re.p)’
+ (Im.p)2
and @ = -arc tan [ Im.p/Re.p] , we see that Ipl is an even function is an odd function of w . Thus, for all real w ,
p(h)
of w
while Q
= p*(-iw)
(114
= p(-io)
(lib)
or p*(b)
where the asterisk denotes a complex conjugate. In writing eq. 8 as being a suitable or adequate representation of eq. 5 for real w we are permitted to take N to be infinite. Or instead of a discrete sum over n, an integral representation may be used. Thus, for example, we can write
110
7 G(a) --& da
P(~w) =
(12)
0
where (Y is real and encompasses distribution function.
all values from 0 to 00. The G (cu) is a real
EXTENSION TO TIME DOMAIN
We now may generalize our formalism to arbitrary time variation. Thus, for example, we may stipulate that the current density (x component) is given by: j,(t)
= JoS(t)u(t)
(13)
where S(t) is the source signal waveform, Jo is a constant t > 0, = 0 for t < 0. By Fourier synthesis we can write: Jx(iw)eiwt
and u (t) = 1 for
dw
(14)
where &(iw)
=
Irnj,(t)eeiwtdt -Ca
(15) = Jo i
S(t)e-‘wf
dt
0
Thus, we regard Jx (iw) as the spectral content of the function j, (t). In Laplace transform notation we would write eqs. 14 and 15 in the form: j,(t)
= 1-l J,(s)
(16)
= Xjx((t) = LJ,S(t)
(17)
and Jx(s)
where s is formally identified with iw. But now we continue complex plane; thus, for example, eq. 16 is given by: j,(t)
=
-& JC”wJx(s)est
ds
c-i-
where c is some small positive quantity. In accordance with Ohm’s law, given by eq. 1, the electric e, (t) is related to the causative source j, (t) by:
s into the
(18)
field response
111
ex(t)
=
_frn
k
The Laplace transform e,(t)
Jx(ia)eiwt
p(b)
dw
equivalent
(19)
is
@)I
= L-l [p(s)&
(20)
= JCJ -l [P(S)R b)l where R(s)
(21)
= JZs(t)
THE IDEAL
DIELECTRIC
MODEL
In an ideal lossy dielectric, which is not physically realizable, o(h) = u and e(h) = E are constants for all frequencies. Likewise a(s) = u and E(S) = E for all complex s. Thus: 1 P(S) = u + ES
(22)
In this case, ex(t)
= Jod:-’
[
iA R(s)]
On noting that the “impulse L-1
p(s)
=
1-l
l
u+a = i exp(-(u/e)t)
response”
= 1 L-1 E
(23) is given by :
1
s+u/e
u(t)
(24)
we may write eq. 23 in the form of a convolution ex(t) Indeed that:
1 t = Jo S(T) exp[-(u/e) e s0
(t - T)]d7
integral:
(25)
if ,S(t) = t,, 6(t - to ), where 6 (t - t,, ) is a unit impulse at t = &,, we see
e%(t) = Jo 4
exp[-(u/e)(t-tO)]u(t-to)
(26)
112
This has the form of a decaying exponential following the application of the source impulse current. The form is sketched in Fig. 1 where the time constant is E/U seconds.
t e,(t)
~ t=t,
t
Fig. 1. The exponential decay of the electric field in an ideal lossy dielectric following the cessation of a steady current.
In the case of a step function would specify that S(t) = u(t-tl) ex(t)
1 t = JO exp[-(a/e) s e f, = JQ f
[I-
excitation of the ideal lossy dielectric we in eq. 13. Then we see from eq. 25 that:
(t-T)]dT
(27)
e- w4wf]u(~-~l)
(23)
The build up of e, (t) to its final value is shown in Fig. 2.
__---__---
----/----
e, (t)
Fig. 2. Exponential build up of the electric field in an ideal lossy dielectric following the application of a step function current.
GENERALLY
DISPERSIVE
CASE
In dealing with a generally convenient to write:
dispersive
complex
resistivity
function,
it is
113
P(iW) =
P,
+
Jr&
(PO- P,)
dT
(29)
0
which is really equivalent to eq. 12. Here p, designates the high frequency limit and p. designates the low frequency limit of the function p(h). The precise physical meaning of these limiting values is not clear at this stage. Suffice it to say that the following limits hold:
p(iw) =
ii0
PO
in a mathematical sense where p, and p. are real parameters. The function f(7) as used in eq. 29 is the distribution of the relaxation times. It is normalized such that: m
s
f(T)dT
= 1
(39)
0
In some cases it is desirable function. Thus we write: p(i0)
= p’(0)
to use yet another
- ip”(w)
form for the distribution
(31)
where
P’(W) =
P,
+
(PO- Pw)
J {(JT* dT 1
0
and
P”(W) = (PO- P,)W
7
7 f(T) 1
+
w2T2
d7
(33)
0
For real frequencies w, p’ and -p” are the real and imaginary parts of p (iw) as can be ascertained from eq. 29. In the case of a single relaxation time f(r) = 6(7-TV) and we have:
P’(W)=
(POPm +
Pco)
1 + cd*r* 0
(34)
and
p”(U) =
(PO- Pm)W~o 1 + w*r*0
(35)
114
As pointed out by Cole and Cole (1941) in a different context, the points (p’,p”) lie on a semi-circle with the center on the p’ axis and they intersect this axis at p’ = p,, and p’ = p,. This follows from the equality: EP”(W)12 = [PO -
P’(W)1b’(w) - PC01
(36)
which can easily be ascertained from eqs. 34 and 35. The so-called Cole-Cole plot is obtained when p”(w) is plotted versus p’(w) for various angular frequencies. An example is chosen for the parameters p. = 1.2 units and p, = 0.2 units. The resulting semi-circle is sketched in Fig. 3 for w ranging from 0 to 00 as designated by the dimensionless parameter ~7~.
p”(
Fig. 3. The complex resistivity using the Cole-Cole plot for the ideal Debye model (i.e., single relaxation time).
This full semi-circle locus is characteristic of a single relaxation time. When there is a distribution of relaxation times it is possible to derive the following inequality that holds for any real positive value of o :
[P”(~)12 < [PO- P’(W)1b’(o) - Ll The proof of this statement
(37)
follows from the definitions
32 and 33.
THE COLE-COLE SPECIAL FORM
The special Cole-Cole
form (1941)
for the dispersion
amounts
to writing:
1 P(iO)
= Pea + (PO - P,)
1 + (iLJro)k
(33)
where k is an empirically adjusted non-integer. In the limit where k-+1 we retrieve the Debye form (Frohlich, 1949) given by eqs. 34 and 35. Another way of writing eq, 38 is (Pelton et al., 1978): p(iw)
= p.
1 [ 1 -m.
l-
1 1 + (iWTO)k11
(39)
115
where m0 =-
PO - Pm
PO
is sometimes called the “peak chargeability” of the region. To illustrate the properties of the Cole-Cole equation given by eq. 38 we plot p’ and p” as a function of ln(wrO). Such an example is shown in Fig. 4 where we choose p. = 1.2 and p, = 0.2. The values of h = 1 and l/2 are chosen. Of course k = 1 corresponds to the Debye form whereas the case k = l/2 allows for a broader dispersion effect.
-6
-4
-2
0 In(w%
2
4
6
1
Fig. 4. The real and imaginary part,s~omplex k = % and the Debye model (i.e., k = 1).
resistivity
for the Cole-Cole
model
for
The Cole-Cole plot for the case h = l/2 is shown in Fig. 5. It is a circular arc with its center below the p‘ axis. To prove this statement we consider the two phasors: Ptio)
PO -
- P,
=
(PO -
PJ
1
+
P@) = b0 - Pee) 1 -
(l;To)k 1 1
+
fiWTo)k
1
(41)
These are shown by the two straight line segments in Fig. 5 that subtend an angle 0 at a point P on the curve. Now the ratio of eq. 41 to eq. 40 is: PO -
P(iW)
P(iW)
-
= +
(i~7~)~ = (~7~)~ei (-8
Pm
where n-0 = nk/2. As indicated the angle 8, in radians, does not depend the position of P on the arc (i.e., it does not depend on 0).
on
116
f f”
0.5 W-
P
I
@
0~
0.5
r40
1.0
;
1.5
PO r’
-
Fig. 5. The complex resistivity for an Argand plot showing that the Cole-Cole form is an arc of a circle.
Probably the reason the Cole-Cole equation (with adjustable po, p, and k) is so versatile is the inherent control over the distribution of the relaxation times (Tareev, 1975; Pelton et al., 1978). For example we may rewrite eqs. 32 and 33 in the form:
P’(U) =
(42)
PC.3
and
p”(0) = (PO- p,)w
J 7F(:=ryyr)
(43)
7=0
where F(ln7)
= 7 f(7)
is the logarithmic distribution function.
In(7/To
)
-
Fig. 6. The d~tribution of relaxation times for the Cole-Cole model.
Following the analysis of Bottcher equivalent problem, we may write:
and Bordewijk
(1978)
for an
117
F(ln7)
sin ~(12-1)
= 1 2n
cosh[lz ln(TO/T)] - cosn(h -1)
1
(44)
This quantity is symmetric about T = T,, which is called the critical relaxation time. As indicated in Fig. 6, it becomes increasingly broad for decreasing values of k. TRANSIENT
RESPONSE
OF THE COLE-COLE
Using Laplace transform function takes the form:
notation
FORM
(i.e., iw + s) the complex
resistivity
1 P(S) = P, + (PO-PJ The response tained from: A(t)
=
(45)
1 + (S70)k
of the electric
field for a step-function
current
density
1 L-l p(s)s_’
is ob-
(46)
PO
This is normalized context of induced M(t)
such that A(=) = 1. The more relevant polarization, is:
= [l -A(t)]
in the
u(t)
1 ----
=J-l
quantity,
1 P(S)
[ S
PO
s
1
(47)
Using the general Cole-Cole form we see that: (46) The special case k = 1, which is the Debye form leads readily to M(t)
-
PO-pme-1 PO
l
PO - PC.3 = _ eetjTou(t)
s + (l/To)
PO
(49)
which is the simple exponential relaxation as illustrated in Fig. 1. Another special case is k = l/2, which is the Warburg form and is only a bit more complicated. Then we find that:
M(t) =
PO PO
J-’
[$
[s + (:irfhl] $$
0
(56) = p”~opm ef/%erfc[(t/70)115]u(t)
118
where we have used formula 52 (p. 211) in Roberts and Kaufman (1966, p. 211). This result is quite different from a single exponential. For example, we note that, for small x : e”‘erfc(x)
= ex’ [++I&
+~---.)I
Thus : M(t)
2
M(0)
[1-(J-f&)n]u(t)
(52)
for t << TV. On the other hand, for large t, we exploit behavior, valid for x >>> 1, expressed by: eXa erfc(x)
1 1 xn”2
1 1.3 1 - 2x2 + 22x4-(
= l/(xn%)
for
sz M(O)
+
23 x6
.I’ 1
x --f m
This tells us that the asymptotic M(t)
1.3-5
the asymptotic
(53)
tail is given by:
$ [(y _ ; (yq3’*+ **.]
(54)
which certainly has a much slower decay than an exponential. When the exponent k in the Cole-Cole form is any number between 0 and 1, we are faced with the inverse transform given by eq. 48. A general evaluation is not simple (e.g., see Lee, 1981). However, the limiting forms can be obtained rather simply. For example, we may write:
M(f) = M(O)&_’
1
s~+~ )I 1 k s70
(55)
km u(t) When k = 1 (i.e., Debye form) this reduces to exponential note that P(m+l) = m!. For small t we see that:
(56) decay when we
119
M(t)
1
1
E M(0)
l-
t
___
k
-
r&+1)
(57)
( 70 11
In the case k = ‘/z (i.e., Warburg form) this is consistent with eq. 52 if we note that I’(3/2) = n%/2. To deal asymptotically with the case of large t, we proceed as follows: M(t)
(STOP
= M(0) x-’
s(I + @TOP) m
=
c
n/r(o)
(-l)yTO)Wn+l)
~-1
#n+l)
-
1)
n=O
Es
5
M(0)
(-1)”
n=O
1 I-(1 - k(n+l))
Writing out the first three terms:
1
+
l?(l-3h)
70
0t
3k
(59)
*+* 3
we see that the decay is slower according to the smallness of Itz.The special case /z = l/z (i.e., Warburg form) is consistent with eq. 54 if we note that r(S) = n’h and I’(-%) = -2n”h, and of course I’(0) + =J. The result for M(t) given by eq. 59 can be used to predict the general asymptotic behavior for large t for any k. Here we need to take note of the properties of the Gamma function. For any v it is defined by:
0
which is finite when v > 0. Also from the definition
it follows that:
r(v+l) = e(v) ww-4
=
&
In the case of an integer (V = n)
r(n) = (n-l)! for n > 0. For further
properties
of Gamma functions
see Goldman
(1966).
120 TRANSIENT
RESPONSE FOR LOG UNIFORM DISTRIBUTION
The Cole-Cole dispersion is remarkably useful and it can often closely to experimental data over a broad range of frequencies. form is to write as before:
be fitted Another
1
p(iw)
= P, +
(PO
-
PA
J
F(ln7)
7=0
but following F(ln7)
Shuey and Johnson
l+iwr
(61)
d (lnr)
(1973), we now choose
= F.
for l/b < 7 < l/a clearly :
and = 0 outside
p(iw) = p, + (p. - p,)
f” llb
this range, where
F”
(1 + iw7)7
0 < a < b < 00. Then
d7
(62) = P, + (p. - P,)F~ ln
Now as w + 0 the right hand side of eq. 62 should reduce to po. Thus F. = (In b/a)-‘. Compacting the notation we write: p(iw)
= p, +52ln
(63)
where 1 Q = (PO - Pm) In b/a If we restrict p(iw)
the frequency
range such that
b S w % a it is evident
z p, + S2(lnb - lnw) - is2 $-
that: (65)
This indicates p’(o) has a slope of -52 as a function of lnw while p”(o) is constant. This behavior for frequency and time domain data on rock samples was noted by Wait (1959). To illustrate the above features of the dispersion for this model we write: dp(io)/d(lnw)
dp(iw)
= w -
do
Thus we note that if b % w S a
1
1
b + io
a + iw
---
(66)
121
(67)
which is consistent with eq. 65 above. Working with eq. 66 we may readily deduce M(t). Thus: M(t)
= M(0)
the decay response
function
1
L-l a
=
[l ---$ i
M(O)
e-at; e-b: dt]
(68)
u(t)
a
where we have used the inverse transform: -at _ ,-&t L-1
In
s+
b
-
e
Another
(69)
u(t)
s+a
t
form of eq. 68 is: 1
M(t)
= M(0) [Ei(-bt)
- Ei(-at)]
In 4
(76)
where: Ei(-x)
= - i
e-’ G du
(71)
X
is the exponential
integral. Here we have also noted that:
m e-at _ ,a J 0
dt = In 5 t
The slope of M(t)
as a function
dM(t) = t dM(t) = d In t
-M(O)
dt
dM(t) dln t
z -M(O)
of In t is a useful parameter.
- ’
te-at
_
It is given by
,-bt)
(73)
In i
This reduces to a constant -
(72)
a
-
1
In t
in the case where l/b << t Q l/a. Then:
= _ap,’
(74)
122
where 52 as defined by eq. 64 is the slope of the p(k) vs. In w curve. Sometimes the curvature of the decay response is of interest (Wait, 1959). This is defined by: (oewaf - beSbt)
M,(t) =
(75)
a Clearly M,(t)
2 0
when
t P -
1
In b a
b-a
KRAMERS-KRONIG
RELATIONS
We wish to derive the relationship between p’(o) and p”(w) for real values of w. To begin with we consider a complex integral representation for the in the Zower half of the function p(h) - p, that is regular (analytic) complex w plane. We choose a closed contour as indicated in Fig. 7 and apply Cauchy’s integral formula to write, for a selected real value L, the following: p(G)
1
- ,um=
2ni
+
[J semi-circle radius R j+r
s
+
w--R
of
s
+
semi-circle radius r
Here it is assumed that the function
of
p(b)
s
-R
w=&r
b(io)
-PA
w-l3
1
dw
(76)
is regular in the lower half plane.
I
Fig. 7. The contour relations.
in the complex
Also we define p, such that
w plane that is used to derive
the Kramers-Kronig
123
where R is the radius of the large semi-circle. The first integral in eq. 76 thus goes to zero in this limit. To deal with integration around the small semi-circle of radius r we introduce the substitution = &+reio
0
where
O<$
Now we may write: p(G)
=
f-’ _ j--- “;‘;‘-
pm+ &[-
G+r
-m
+
n p(iw)-p,
s
rei+
0
where, of course, latter integral. In the limit t-0
p(G)
= p, +
dw
(irei@) d$
rei@ cancels we obtain
---&[ ni(p(iL)
1
(771
in the numerator
and denominator
of the
the form:
-p,)
-
Jwp’~!-Gpw do]
(78)
-co
where the integral deduce that:
is the Cauchy
principal
value. After reshuffling
dw = p,--
1 in
+- p(h) f_m o-ij
eq. 78 we
do
(79)
The latter form follows from the identity: +oO
dw f _m w--G
= 0
(80)
If we now write: p(h)
= p’(w) - ip”(0)
where p’ and p” are real, eq. 79 can be split into real and imaginary to yield: p’(G) and
= p_+;
+- p”(W) m wp”(W) f dw = p, + z f _m 0-G lr 0 cd2 - ((2)’
do
parts
(81)
124
(82) In writing the second forms the even/odd and
P’(W) = p’(-a)
p”(w)
properties
were used:
= -p”(-w)
that hold for real w . The integral forms specified by eqs. 81 and 82 are the Kramers-Kronig relations (Bottcher and Bordewijk, 1978). These tell us that p’(o) and p”(w) cannot be independently specified. In fact a knowledge of one, for all real w, will determine the other. However, we shall bear in mind here that p(io) has been assumed regular in the lower half of the complex w plane and the lower and upper limits p. and p, are well defined. Actually there is a very similar argument that demonstrates the equivalence between the amplitude and phase dispersions. For example we may write: p(L)
= Ip(iw)l e-i*(w)
(83)
for real frequencies where Q(w) is the function a(w) is defined according to:
phase.
Then
the
“attenuation”
= p. e- a(w)
Ip(
(84)
for all real frequencies. Also Q(w) -+ 0 at both w + 00 we must have:
It is clear that Q(o) = -@(--w) and a(~) = a(-~). limits w + 0 and 00. By definition a(O) = 0 but at
= p. e-OL(-)
P,
or equivalently a(-~)
=
(85) :
lnPo/P,
=
a,
Now it proves to be convenient F(iw)
= In -
to introduce
a complex
function:
PO =
a(w)
+
i@(u)
p(iw)
that is actually valid everywhere in the complex axis of w, (Yand + have the properties indicated. In analogy to eqs. 81 and 82 we may write: a(w) and:
2 = (Y, - n
= f
0
iw plane. But on the real
w@(a) w2
-
(;;)2
dw
(87)
125
These show explicitly that the “attenuation”, as a function of frequency can be determined from a knowledge of the phase as a function of (all) frequencies. The converse is also true - the phase can be determined uniquely from the attenuation rate for this class of systems. This, of course, is a well known fact but the relationships are usually discussed in a different context. Goldman (1966) gives a useful summary. The phase integral theorem follows from eq. 88 by simply setting w = 0, whence:
If we change given by:
the variable
to In w this becomes
the phase area theorem
as
m s w=o
(a(w) d(ln w) = (Y,
(90)
That is, the area under the phase curve, plotted vs. log of frequency is a constant. The integral formula for the phase function given by eq. 88 can be converted to a more useful form by a simple change of variable. For example, we set w/L3 = eU so that: du - 1 f IT 0 Following yield :
an integration
by parts,
where we have used the standard
(91)
-?-du sinh u the two integrals
can be combined
result that
du = -ln[cothl~~] s sinh u Another integral:
form
to
of eq. 92 is obtained
(93) if we add and subtract
the known
+-
s -0u
In coth
I
t
du = ; I
(94)
126
Then we obtain: @(cl) = -
;( $) -b /I [- -(s))
1ncothj;l
du
(95)
c where :
is
the slope of the attenuation
function
on a In o base at the frequency
w = &. In fact the peaked nature of the In coth i I
I
function, as depicted in
Fig, 8, indicates that the integral on the R.H.S. of eq. 95 may be neglected if a(w) is not varying rapidly in the vicinity of w = &. This leads to the simple formula:
(96) which relates the phase at a frequency ij to the slope of the attenuation curve on a In w base at the same frequency 6~. In the more general case, one should retain the integral term on the R.H.S. of eq. 95. Or in many cases it may be convenient to work directly with eq. 92 which is amenable to numerical integration.
6-
Fig. 8. The function
CONCLUDING
In cothlul21
where u =lnlw/&l.
REMARKS
As we have indicated there are many possible ways to represent the linear system behavior of material media with dispersive electrical properties. Only in very limited situations could we assume a ~equency-~dependent conductivity and permittivity. However, an example might be saline water
127
solutions free of impurities where the conductivity would be a constant of the order of mhos/m and the permittivity would be near 80 relative to free space (Frohlich, 1949). In nearly all other cases of composite media such as textured rock, mineralized material, and biological suspensions, the constitutive properties are highly dispersive particularly in the audio and radio frequency range. Here it is often fruitful to represent the dispersion as a superposition of relaxation processes. If such distributions are sufficiently broad the effective complex resistivity exhibits a magnitude which varies almost linearly with the logarithm of the frequency. The corresponding phase is approximately a constant that is nearly proportional to the slope of the amplitude curve. ACKNOWLEDGEMENTS
I am grateful to Joann Main who typed Debroux for his helpful comments.
the manuscript
and Patrick
REFERENCES Bottcher, C.J.F. and Bordewijk, P., 1978. Theory of Electric Polarization. Elsevier, New York, N.Y. Cole, K.S. and Cole, R.H., 1941. Dispersion and absorption in dielectrics. J. Chem. Phys., 9: 341-351. Daniel, V.V., 1967. Dielectric Relaxation. Academic Press, New York, N.Y. Frohlich, H., 1949. Theory of Dielectrics. Clarendon Press, Oxford. Fuller, B.D. and Ward, S.H., 1970. Linear system description of electrical parameters of rocks. IEEE Trans., vol. GE 8, pp. 7-17. Goldman, S., 1966. Laplace Transform Theory and Electrical Transients. Dover Pubhcations, Inc., New York, N.Y. Grant, E.H., Sheppard, R.J. and South, G.P., 1978. Dielectric Behavior of Biological Molecules in Solution. Clarendon Press, Oxford. Lee, T., 1981. The Cole-Cole model in time domain induced polarization. Geophysics, 46: 860-868. Matijevic, E. (Editor), 1974. Surface and Colloid Science, Vol. 7. Electra-Kinetic Phenomena. John Wiley, New York, N.Y. Pelton, W.H., Ward, S.W., Hallof, P.G., Sill, W.R. and Nelson, P.H., 1978. Mineral discrimination and removal of inductive coupling with multifrequency IP. Geophysics, 43: 588-609. Roberts, G.E. and Kaufman, H., 1966. Table of Laplace Transforms. W.B. Saunders, London. Shuey, R.T. and Johnson, M., 1973. On the phenomenology of electrical relaxation in rocks. Geophysics, 38 (1) 37-48. Tareev, B., 1976. Physics of Dielectric Materials. MIR Publishers, Moscow. Wait, J.R. (Editor), 1969. Overvoltage Research and Geophysical Applications. Pergamon Press, Oxford. Wait, J.R., 1982. GeoElectromagnetism. Academic Press, New York, N.Y. Zonge, K.L., Sauck, W.A. and Sumner, J.S., 1972. Comparison of time, frequency, and phase measurements in induced polarization. Geophys. Prospect., 20: 626-648.