resistivity methods

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Chapter 3

Resistivity Methods

Resistivity methods use a DC or very low frequency current injected into the ground through electrodes

(and are therefore galvanic methods). Very low frequency methods are generally preferred as periodically

changing the direction of current flow prevents the build up of ions near the electrodes which would affect the

applied potential. Generally, two electrodes are used to inject current into the ground and two other electrodes

are used to measure the potential difference. Other systems make use of only three electrodes, or use long linear

current sources (consistent ground contact is difficult with linear systems adding a source of error). Deviation

of the potential from that expected for a homogenous half-space can be used to deduce information about the

electrical properties of the subsurface. Since both the injected current and the resulting potential are measured,

they can be combined to obtain a value for the resistance (and hence resistivity) of the subsurface.

3.1 Equipotential Line Method

This is a fixed source method that allows the drawing (mapping) of equipotential lines at the Earths surface.

Field work is similar to the SP method, except that an artificial current source is used to generate the measured

potential.

x

y

+ !+I !I

P(x,y)

r1 r2

a

Figure 3.1: Geometry for the equipotential line method

For a homogeneous Earth the potential at the observation point is

VP =I

2

1

r1

1

r2

=

I

2

1

(x2 + y2)1/2

1

((a x)2 + y2)1/2

(3.1)

21


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Resistivity 22

If x = a/2, then VP = 0. The potential profile between the two current inducing electrodes is given by

VP =I

2

1

x

1

a x

(3.2)

+ !

current

equipotentials

A B

Figure 3.2: A schematic of the current paths and equipotential lines at the surface of a homogeneous half-space.Current is generated by the applied potential between electrodes A and B.

The zero potential lies half way between the electrodes. Potential measurements are generally carried out

in the middle third of the region between A and B. If there is a buried body with anomalous conductivity the

current flow will be diverted. Current flow will tend to concentrate in a more conductive (less resistive) region

and avoid a less conductive (more resistive) region. Current flows perpendicular to the equipotential lines (i.e.

along the potential gradient) which are also distorted by the presence of an anomalous body.

a) b)

Figure 3.3: Schematics of the current paths (dashed arrows) and equipotential lines (solid lines) in the vicinityof a body that is anomalously a) conductive and b) resistive.

3.2 Fixed Source Methods

In these methods there are two electrodes (denoted A,B or C1,C2) used to apply the potential which generates

the current flow are kept at fixed positions, while two electrodes (denoted M,N or P 1,P2) are moved about on

a grid measuring the potential difference between grid points.

For a homogeneous half-space the potential difference between M and N is

V = VM VN =I

2r1

I

2r3

I

2r2+

I

2r4(3.3)


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Resistivity 23

A

A B

V

M N

r1

r2 r3

r4

Figure 3.4: A schematic (in map view) of the geometry in the fixed source method. A potential is applied,between the fixed electrodes (A,B - also called the current electrodes) and the resulting current measured bythe ammeter. The potential between points on the surface is measured using the mobile electrodes (M,N - also

called the potential electrodes).

Since the potential, current and geometry are all observed we can solve for the resistivity

a =V

I2

1

r1+

1

r4

1

r2

1

r3

1

=V

IG (3.4)

where G is known as the geometric factor (some definitions of G do not include the factor of 2 ). Note the

subscript a that has been added to the resistivity. This formula gives the resistivity for a perfectly homoge-

neous, isotropic, semi-infinite half-space, in practice geologic settings will contain some sort of heterogeneity or

anisotropy (or both) and the resistivity obtained by equation (3.4) is not the true resistivity of the ground be-neath the probe, a is known as the apparent resistivity. Were the subsurface an ideal half-space, then a would

remain constant as the potential electrodes are moved; instead one will find that a varies and the standard

procedure is to assign the obtained value of apparent resistivity to the point halfway between M and N.

The fixed source method can be used to generate an (apparent) resistivity map of an area. However, the

size of the area is limited by restrictions on the separation of A and B; it is difficult to generate sufficient current

if the electrodes are far apart. Therefore, it is often necessary to relocate A and B in order to map the entire

area of interest. The problem then becomes one of ensuring overlap of area from adjacent apparent resistivity

surveys, often the obtained a values do not agree due to variations in ground properties and quality of contact

between the old and new current electrode positions.

3.3 Moving Source Methods

In moving source methods both the current electrodes and the potential electrodes are moved, in this case it is

important to keep the array geometry fixed. A number of different electrode arrays (or spreads) are used (see

below). One example is the Wenner array in which the distance between adjacent electrodes is equal.

A BM N

a a a

Figure 3.5: Geometry of the Wenner array


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Resistivity 24

For this array the potentials are

VM =I

2a

I

4a=

I

4a

VN = I

2a+

I

4a=

I

4a

VMN = VM VN =I

2a(3.5)

that is a =VMN

I2a (3.6)

There are two methods whereby the arrangement of electrodes can be altered to check for the level of

errors in the survey. The first is known as the check of reciprocity, the two electrode arrangements shown below

should give identical values ofVMN and hence a.

A M N B

M A B N

The second method is the check of superposition and makes use of three electrode arrangements, known as the

alpha, beta and gamma configurations.

A M N B

A B N M

A M B N

The potentials measured from these three configurations should obey

(VMN) = (VMN) + (VMN) (3.7)

The agreement should be within 3% to be considered acceptable.

Electrode Arrays In addition to the Wenner array discussed above, the most common array types are the

Schlumberger and dipole-dipole arrays. The standard layout of these arrays are shown in figure 3.6; however,

there are many variations possible. The geometric factors of the standard arrays are

GSchlum =

ba2 b

2

4 , a 5b (3.8)

Gdd = n(n + 1)(n + 2)a (3.9)

A BM N

b

aA B M N

a ana

n=5a) b)

Figure 3.6: Geometry of the a) Schlumberger and b) dipole-dipole arrays

Each array has advantages and disadvantages; for example, the Wenner array has the best vertical reso-

lution, the dipole-dipole array the best depth penetration and the Schlumberger array is considered the most


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Resistivity 25

suitable for electrical profiling. The resolution characteristics of the spreads are described by array sensitivity

kernels, which describe how sensitive the measured apparent resistivity is to a change in true resistivity at

a given location in the subsurface. The choice of which array to use will depend on your exploration goals,

location, time and manpower.

In addition to the choice of array geometry there are several choices for the employed methodology when

conducting moving source measurements.

Profiling Moving the array along a line keeping the geometry fixed, a is calculated and assigned to the point

halfway between M and N, thus a = f(x). Can be used to study lateral resistivity variations. Note that the

shape of the profile depends on the particular electrode spread used.

!1!2

x

!a

!1 !2

x

!a

Figure 3.7: Examples of resistivity profiles using a Wenner array, note that different arrays will result in differentprofiles. In both cases 2 < 1. More examples can be found in the suggested texts.

Mapping In this case the electrodes are moved over a 2-D grid so that one obtains a = f(x, y), array

geometry is again kept fixed. The resulting map can be used to investigate structures such as salt domes,

weathered rocks (quarries) and buried archeological sites (figure 3.8).

Sounding In electrical sounding the centre of the array centre is kept fixed and the spacing between electrodes

gradually increased (figure 3.9). This method is often called vertical electrical sounding (VES). However, al-

though depth of current penetration tends to increase with increasing electrode spacing the resulting resistivities

are not simply a function of depth a = f(a) = f(z). We will return to this method and consider it in more

detail below.


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Resistivity 26

Figure 3.8: Example of resistivity mapping from the Whistling Elk, South Dakota archaeological site. The upperfigure shows the result of the resistivity survey carried out over an area of 17,000 m2 requiring nearly 34,000measurements. The lower figure shows the archaeological interpretation. For more details see the University ofArkansas Archeo-Imaging Lab (http://www.cast.uark.edu/%7Ekkvamme/ArcheoImage/archeo-image.htm).


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Resistivity 27

A BM NO

h!1

!2!a !!1

h

A BM NO

!1

!2

!a !f(!1,!2)

a) b)

A BM NO

h!1

!2 !a !!2

c)

a

!a

!1

!2

d)

Figure 3.9: Schematic of electrical sounding over a two layer Earth with 1 < 2. The spacing between electrodesin the Wenner array is successively increased a), b), c) and the apparent resistivity plotted against the array

spacing d).

Pseudosections and Continuous Vertical Electrical Sounding Pseudosections (figure 3.10) are created

by making repeated profile traverses with progressively increasing electrode spacing. The a values from suc-

cessive profiles are aligned and ordered by spacing producing a 2-D grid in x a space which is contoured to

produce the pseudosection. Continuous vertical electrical sounding (CVES) also produces a section of apparent

resistivity in x a space however, in this case the image is obtained by repeating electrical sounding investiga-

tions, with successive soundings shifted horizontally. Both methods produce images that are appear similar to

a resistivity cross-section; however, it must be remembered that what is plotted is apparent resistivity and that

electrode space a does not strictly correspond to depth. The practicality of these methods are greatly increased

by the use of multichannel systems which allow multiple electrodes to be set up and simultaneously measured,

these methods also benefit from the use of computers in processing and inverting the data.

3D Side by side pseudosections or CVES lines can be used to build up a three dimensional model of the

subsurface resistivity (figure 3.11). This method is the most expensive and time consuming (in terms of both

human and computer activity).


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Resistivity 28

Figure 3.10: An example of resistivity pseudosection construction and modelling. The location is a faultedTriassic sequence in Staffordshire, UK. (A) 2-D model. (B) Computed apparent resistivity pseudosection. (C)Field data. (D) Geological interpretation. From Griffiths et al. (1990), via Reynolds (1997)


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Resistivity 29

a)

b)

Figure 3.11: An example of a 3D resistivity model. The model is displayed as a) a resistivity cube and b)with a cut-off of 27 m in order to highlight the high resistivity features. The location consists of sand andgravel lens within a glacial till, located in East Yorkshire, U.K. From Catt (2008).


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Resistivity 30

3.4 Electrical Sounding

As discussed above, in electrical sounding the centre of the array is kept fixed and the electrode spacing gradually

increased, the measurements are used to construct a curve a = f(a). In order to interpret this curve in terms

of the subsurface resistivity structure we must consider how the electric potential varies within the subsurface.We know that V(z) in a homogeneous, isotropic medium is a smooth, continuous function and this remains true

in horizontally stratified medium; that is, there are no jumps in potential in the subsurface. As a result the

measured apparent resistivity will also vary smoothly with increasing electrode spacing. The shape of the a

curve will depend on the geometry of the stratified layers. Although natural media need not be homogenous or

isotropic (or horizontally layered), electrical sounding curves are interpreted in terms of the equivalent sequence

of isotropic layers which is often sufficient to help resolve ambiguities from mapping or profiling results. In the

following sections we will develop the theory required to interpret sounding curves. In practice, purpose-built

computer codes are used to invert for the subsurface structure.

3.4.1 Image Theory

For relatively simple geometries we can use the method of images, which was introduced in the discussion of SP

methods, to determine the subsurface potential and interpret the sounding curve. In an infinite, homogeneous,

isotropic medium the potential associated with a point current source is (recall equation (2.8))

V =I

4r(3.10)

As we have seen before, if there is an interface separating two media (figure 3.12), one of which is a perfect

insulator (in this case 2 = ), then the surface acts analogously to a perfect mirror and the potential within

the conductive medium is the same as if there were two identical sources within a single, infinite medium of

resistivity 1.

+

+

h

h

Pr1

r2

!1

!2 =!

+I

IM (= +I)

Figure 3.12: A point current source (I) is reflected in the boundary with the perfect insulator producing anidentical mirror image current source (IM).

The potential at an observation point within the original medium is given by the sum of the potentials

from the original source and the mirror source (recall equation (2.12))

VP =I

4

1

r1+

1

r2

(3.11)


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Resistivity 31

If both of the media are at least somewhat conductive (i.e. 2 = ), then the interface does not behave

as a perfect mirror. Instead we have an imperfect mirror and the image source strength will have some value

KI where K= 1 is the reflection coefficient.

+

+

h

h

P1r1

r2 !1

!2

+I

+KI

Figure 3.13: A point current source (I) is partially reflected by the boundary between the two conductivemedia. On the near side of the interface this produces a mirror image current source of strength ( KI).

For a point (P1) within the same medium as the current source the total potential is the sum of the

potentials of the source and mirror as if they were in a single infinite medium with = 1.

VP1 =1I

4

1

r1+

K

r2

(3.12)

For a point (P2) in the other medium, the interface acts like a filter, and the potential is that as if there

was an altered source of strength IKI within an infinite medium with = 2. Using the reflection analogy;

if the boundary is a partial mirror that has reflected an amount of current KI then the current that passes

through the boundary must be (1 K)I.

+

h

!1

!2

+(1-K)I

P2

r3

Figure 3.14: A point current source (I) is partially reflected by the boundary between the two conductivemedia. On the far side of the interface this produces a filtered current source of strength ((1 K)I).

The potential on the far side of the interface is

VP2 =2I

4

1K

r3

(3.13)

Equations (3.12) and (3.13) hold everywhere on their respective sides of the boundary. For an observation

point on the interface (figure 3.15) the potentials given by these two equations must be equal, so

1I

4

1 + K

r

=

2I

4

1K

r

(3.14)


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Resistivity 32

+

+

h

h

P1

r

r

!1

!2

+I

+KI

+

h

!1

!2

+(1-K)I

P2

r

Figure 3.15: A point current source (I) is reflected in the boundary producing a mirror image current sourceof strength (KI) in an infinite medium with resistivity 1. Equivalently, the point source is filtered by theboundary producing a single current source of strength ((1 K)I) in an infinite medium with resistivity 2.Note that in this case r1 = r2 = r3 = r.

and we can solve for the reflection coefficient

K =2 1

2 + 1(3.15)

Note that if2 < 1, then 1 K 0 and the potential in medium 2 is larger than if there were simply

a source within an infinite medium of resistivity 2. Current preferentially flows into the less resistive medium

and thus there is more current in the more conductive medium 2 than would be expected from the given

source within an infinite medium with resistivity 2. The interface has in some sense amplified the potential in

medium 2. Conversely, since the current is draw out of medium 1 the potential there is lower than if it were an

infinite medium. If2 > 1, then 0 K 1. In this case the current flows preferentially in medium 1. The

interface acts like a partial mirror increasing the current flow and potential in medium 1. Conversely the more

resistive medium 2 is in a sense partially shielded from the current source, resulting is less current and a lower

than expected potential in medium 2. The possible range for the reflection coefficient is 1 K 1 and it

is zero when 2 = 1 (i.e. when there is in fact a single, infinite medium).

Having established the initial theory for the method of images let us apply it to a simple geological case

(figure 3.16). Suppose that the subsurface consists of two units separated by a horizontal interface. An upper

layer of thickness h and resistivity 1 and a lower unit of resistivity 2 which extends to infinite depth. A

point current source is introduced on the surface and the resulting potential measured at various points on the

surface.We have two interfaces to consider: the surface, with reflection coefficient K0 = 1 (the air is effectively

a perfect insulator); and the geologic boundary, with a reflective coefficient K1 as given by equation (3.15).

Reflections will occur due to both of these interfaces. Our current source (I) sits at the surface and will be

reflected in the geologic boundary resulting in an image source of strength K1I a depth h below the contact

(2h below the surface). Unfortunately, this is not the only image. The first image source will be reflected in

the surface resulting in a second image source of strength K0K1I = K1I at a height 2h above the surface.

This image will also be reflected in the geologic contact resulting in a third image source of strength K21I a

depth 3h below the contact (4h below the surface). Reflection of image source three in the surface results in

a fourth image source, also with strength K21I at a height 4h above the surface. We have encountered a hall

of mirrors phenomenon and the original source will, in theory, be reflected an infinite number of times. In


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Resistivity 33

h

P

r1

!1

!2

! =!+

+

+

+

+

r1

r

r2

r2

I

IK2

1

IK2

1

IK1

IK1

Figure 3.16: A point current source (I) is reflected in the geologic boundary and the surface creating a hall of

mirrors collection of image current sources.

practice we need not account for every image, since |K| < 1 the image strengths are successively reduced and

since potential is inversely proportional to distance from the source. The need for infinite images may be clearer

if one thinks about the current produced being reflected back and forth between the surface and the geologic

boundary with some current also passing through the geologic contact (but not the surface) at each reflection,

similar to the familiar situation from seismic reflection theory. Please note that this is an analogy only, and one

that is, in fact, physically misleading. The current is not a travelling wave bouncing between the surface and

the interface at z = h; however, the net potential due to the presence of the subsurface interface can be shown

to be mathematically equivalent to a series of reflections (for more on potential theory see below).

At a point on the surface, the potential due to this series of mirrored sources will be

VP =1I

2

1

r+

2K1r1

+2K21

r2+

2K31r3

+

(3.16)

where rn =

r2 + (2nh)2. Note that we are using the half-space potential equation (2.10); since all of the

produced current goes into the subsurface the effective strength of the source is doubled, as discussed above.

Collecting the terms depending on K1 we have,

VP =1I

21

r

+ 2

n=1

Kn1r2 + (2nh)2

= (r) (3.17)or

VP =1I

2

1

r+ 2S(r, K1, h)

= (r) (3.18)

Note that the terms in the summation go to zero as n since |K1| < 1.

For the Wenner array (figure 3.5) the measured potential difference

V = VM VN = (a) (2a) (2a) + (a)

V = 2[(a)(2a)] (3.19)

and if we recall the definition of apparent resistivity for this array (equation 3.6) we obtain

a =

2a

I

2

1I

2

1

a+ 2S(a, K1, h)

1

2a 2S(2a, K1, h)


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Resistivity 34

a = 1 [1 + 4aS(a, K1, h) 4aS(2a, K1, h)] (3.20)

We can use the fact that,

aS(a, K1, h) = a Kn1

a2 + (2nh)2 = Kn1

1 + (2nh/a)2 (3.21)

and aS(2a, K1, h) = a Kn1

(2a)2 + (2nh)2= Kn1

4 + (2nh/a)2(3.22)

to write our expression for apparent resistivity as

a = 1

1 + 4

Kn11 + (2nh/a)2

Kn1

4 + (2nh/a)2

a = 1 [1 + 4F(K1,h/a)] (3.23)

The function F can be computed for any range of electrode spacings and any given K1. For small electrode

spacing h/a 1, thus F 0 and a 1, as would be expected. For large electrode spacings h/a 1, thus

F 1/2

Kn1 . Because K21 < 1 the infinite sum can be approximated as

K21 =

11K1

1 which gives

a 2. As expected the apparent resistivity varies smoothly from 1 to 2 as the electrode spacing increases.

In fact this holds true for any array geometry, although the particular form of F will differ.

Clearly, as we plot measurements of a against a there will be many different possible curves depending

on the values of 1, 2 and h. It would be useful to consider all possible values of the physical parameters

in a compact way, this is done by constructing master curves. To construct the master curves we note that

equation (3.23) can be re-written in the form

a

1= f(a/h) (3.24)

Taking the log of both sides we get

log a log 1 = f(log a log h) (3.25)

which is of the form y p = f(x q). Field measurements are in terms of a and a which can be plotted on a

log-log plot. Therefore, we obtain a field curve

log a = f(log a) (3.26)

which is of the form y = f(x). The field curve and the theoretical curve (for the correct value of K1) will have

the same shape but will have their origin shifted by p and q (that is, by the log of1 and h). Master curves are

constructed from equation (3.23) which is evaluated for various values of K1 on log-log plots of a/1 vs a/h

(figure 3.17).

The field curve is compared with the master curves which are overlain using a transparency. Keeping the

field and master axes parallel, the transparency is shifted until a good match with a master curve is obtained

(this gives the value of K1). The point on the master curve where a/1 = a/h = 1 will correspond to a point

on the field curve giving the true values of 1 and h (since the master and field curve origins are shifted relative

to each other by these values). Knowing, 1 and K1 the value of2 can be calculated.


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Resistivity 35

!! !"#$ " "#$ ! !#$

!!

!"#$

"

"#$

!

%&'()*+,

%&'(!)

*!!

,

k = -1.0

k = -0.8

k = -0.6

k = -0.4

k = -0.2

k = 0.0

k = 0.2

k = 0.4

k = 0.6

k = 0.8

k = 1.0

Wenner Array 2 Layer Master Curves

Figure 3.17: Master curves plotted as log(a/1) on the vertical axis against log(a/h) on the horizontal. Notethat the master curves may also be labeled by the ratio 2/1. For further detail and examples see, e.g. Milsom(1989), Reynolds (1997).

3.4.2 Potential Theory

If there are more than two layers in the subsurface, then image theory becomes much more difficult to use.

Potential theory is used instead.

If J is the current density and E the electric field vector in a given medium, then Ohms law states

J = E =1

E (3.27)

where and are, respectively, the conductivity and resistivity of the medium. Since the electric field is

conservative

J =1

E =

1

(V) =

1

V (3.28)

Consider some volume (v) of bounded by a closed surface (A). If this volume does not contain any current

sources or sinks, then any current that flows into v must also flow back out, i.e. the net flux through A must

be zero. Using Gausss Theorem we haveA

J n dA =

v

J dv = 0 (3.29)

Since we can choose an arbitrary volume we must have J = 0 at all points (except at sources or sinks). Thus,

(1

V) =

1

( V) +V

1

= 0 (3.30)

If the conductivity is constant within a given region, then this expression simplifies to

2V = 0 (3.31)


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Resistivity 36

which is Laplaces Equation, the fundamental equation governing potential theory. Our goal is to solve Laplaces

Equation subject to the boundary conditions imposed by the particular situation of interest. For the potential

due to a point source at the surface of uniform, horizontal layers it is logical to work in the cylindrical coordinate

system (note that r will now denote the distance from the z axis, not distance from the source as above). For

a point source and laterally homogeneous layers the potential will be a function of r and z only, and Laplaces

Equation is2V

r2+

1

r

V

r+

2V

z2= 0 (3.32)

and the potential in each layer (Vi) will be of the form

Vi(r, z) =

0

Fi()e

z + Gi()e+z

J0(r) d (3.33)

where, is a integration variable, J0 is the order zero Bessel function of the first kind and Fi and Gi are

functions to be determined from the boundary conditions:

1. V 0 if r or z

2. For small (r2 + z2)1/2 we will have V = 1I2 (r2 + z2)1/2

3. At the surface (z = z0), V /z = 0 except at r = 0

4. The potential must be continuous across interfaces, Vi(zi) = Vi+1(zi), where zi is the depth of the interface.

5. The current density normal to an interface must be continuous, 1i

Viz

zi

= 1i+1

Vi+1z

zi

For example, condition (1) implies that all Bi() = 0.

The potential at the surface of a two layer Earth will be

VP(r) =1I

2

1

r+ 2

0

K1e2h

1K1e2hJ0(r) d

VP(r) =1I

2

1

r+ 2

0

()J0(r) d

(3.34)

where K1 is the reflection coefficient and () is known as the kernel function.

h1

h2

1

2

3

0 =

Figure 3.18: Geometry of a three layer Earth

For a three layer Earth we have K1 = (2 1)/(2 + 1) and K2 = (3 2)/(3 + 2) and the surface

potential is given by

VP(r) =1I

21

r

+ 2

0

K1e2h1 + K2e

2(h1+h2)

1K1e

2h1 K2e

2(h1+h2) + K1K2e

2h2

J0(r) dVP(r) =

1I

2

1

r+ 2S3(r, K1, K2, h1, h2)

(3.35)


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Resistivity 37

The expression for surface potential can be used with the geometry of the chosen array to derive an expression

for the expected apparent resistivity of the form

a = 1

1 + 4F3

K1, K2,

h1a

,h2a

(3.36)

which, as in the two-layer case, can be used to plot master curves of log(a/1) vs log(a/h1) for the various

combinations of (K1, K2, h1, h2).

As before, the master curves are constructed for a given set of parameters by transforming equation (3.36)

to the form shown in equations (3.24) and (3.25) which will have shifted origins relative to the field curve having

the form of equation (3.26). Curve matching is used as in the two layer case; the origin gives the values for

1 and h1 and the shape of the curve will depend on K1, K2, h2 which can be used to determine the electrical

properties of the subsurface. Note that in the two layer case the different curves are distinguished based on

variations in a single parameter (K1, or equivalently 2/1) and generally 20 curves are used to find a match.

For the three layer case the different master curves are distinguished by variations in three parameters (2/1,

3/2 and h2/h1) and it becomes necessary to consider 20 20 20 different master curves. Compilations of

such curves have been published and there are also programs that generate the master curves numerically.

Three layer master curves are generally divided into four classes, or types, as shown in figure 3.19. Identifi-

cation of the curve type gives an indication of the subsurface electrical structure. Field curves can be matched in

their entirety or through a bootstrap method using a series of two layer curves matched to successive portions of

the field curve. With more than three subsurface layers complete matching to master curves become impractical

and the bootstrap method or numerical fitting is required.

K type

H type Q type

A type

2 > 1 2 > 3

2 < 32 < 11 > 2 > 3

1 < 2 < 3

Figure 3.19: Schematic three layer master curves

Unfortunately it is not possible to uniquely determine three layer master curves. Three inherent difficulties

that you will encounter are detailed here.


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Resistivity 38

Practical Ambiguity The master curves for a three layer Earth depend on parameters that are ratios of

the desired subsurface properties, i.e. 2/1, 3/2 and h2/h1. This leads to a practical ambiguity in the

interpretation of the field curves since proportional increases in all three resistivities or both layer thicknesses

leave the ratios unchanged, e.g. h2/h1 = (2 h2)/(2 h1). Therefore, such proportional changes in subsurface

properties will not alter the field curve and it will not be possible to uniquely determine the geoelectrical

structure.

Equivalence Principle This principle applies to K and H type curves. For K type curves the second layer is

the most resistive and current tends to flow vertically through this layer. The effect of this layer on the current

is then quantified by its transverse resistance (recall equation 1.13) T = h22. If this product remains constant,

then the effect of the layer on the field curve is unchanged; e.g. (2, h2) = (40 m, 0.5 m) or (20 m, 1 m) both

result in T = 20 m2 and are therefore electrically equivalent. Similarly, for H type curves the current tends

to be concentrated and horizontal within the less resistive, second layer. The effect of this layer is quantifiedby its horizontal conductanceH = h2/2 (the inverse of equation 1.16) and beds which have equal ratios of h2

and 2 can not be distinguished by inspection of the field curve. Numerical modelling may be able to limit the

range of possibilities, as can a priori geological knowledge.

Principle of Suppression In general, thin layers will have small effects and are difficult to detect unless

their resistivities are much greater (or much less) than the surrounding layers. Such layers are particularly

problematic in A and Q type curves as the width of the central plateau (see figure 3.19) depends on the

thickness of the middle layer. The suppression is further exacerbated if the resistivity contrast between the top

and middle layers is similar to the contrast between the top and bottom layers (recall that may vary by orders

of magnitude in the subsurface). Again, numerical modelling and a priori geological knowledge can be useful.

resistivity methods

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