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ELECTRICAL RESISTIVITY METHODS
Most minerals are electrical insulators; that is, they resist the flow of electricity very effectively.
Quartz, feldspars, olivine, amphiboles and pyroxenes, the more common rock-forming minerals, are
similar to glass with respect to their electrical conductivity. Among igneous or metamorphic minerals,
only graphite, certain sulfides (those with metallic luster) , magnetite, come copper minerals and
elemental gold, silver, iron or copper are good electrical conductors. Calcite and dolomite are also
insulators, as are most halides and sulfates. Clays, on the other hand, can be excellent electrical
conductors.
Recall Equation 3 from Basic Electricity, the concepts of electrical resistance and electrical
resistivity. Consider a uniform cylinder of length "L" and cross-sectional area "A" with electrical
current flowing parallel to the length. The resistance "R" of this cylinder is directly proportional to "L"
and inversely proportional to "A". ρ is a basic physical property of the material from which the
cylinder is made. This quantity is termed electrical resistivity and is a fundamental physical
property (like density or hardness). Resistivity does not involve geometrical factors. The S.I. units
for resistivity are OHM-METERS. Those of you familiar with water quality may have used electrical
conductivity to characterize the dissolved solids (TDS) is a water sample. Units of conductivity are
the reciprocal of resistivity (conductivity = 1/ρ), formerly MOHS/M (pronounced like “Moe” of the 3
Stooges), now called Siemens. Fluids are generally characterized by conductivity while solids are
usually characterized by resistivity. This geophysics class shall use resistivity. You will have to be
able to convert from one to the other.
RESISTIVITY OF ROCKS and ARCHIE'S LAW
For rocks composed of non-conducting matrix minerals and saturated with water, an empirical
relationship known as Archie's Law is useful in analysis of electrical properties. Archie's Law is
commonly written
-m
(2-1)
ROCK
FLUID

=
A
where ρFLUID equals the electrical properties of the fluid in the pores, Φ is the porosity (ratio of void
volume/total volume), and A and m are constants that depend on the geometry of the pores. For many
rocks, A = about 1 and m = about 2. See Keller, G.V. (1982) for a broader discussion. Papers
discussing various A and m values for specific rocks (shaly sands, clean sandstones, etc.) have been
published in the journal Geophysics.
Note: there are several versions of Archie's Law that attempt to include the effects of partial
saturation (water-gas or water-oil) or mixed fluids in the pores, or, the air water mixes in the vadose
zone. We will not attempt to use these formulas in this course but it is necessary to point out the fact
that Equation 2-1 is not the only way in which Archie's Law is written.
The electrical resistivity of a fluid depends on the amount of ionic material in solution and on
the temperature of the liquid. Figure 2-1 is a nomogram that allows you to compute relationships
between temperature, TDS (equivalent NaCl salinity) and fluid resistivity - as well as the relationship
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between rock resistivity, fluid resistivity and "formation factor" (A *Φ ). There are methods for
calculating equivalent NaCl based on TDS involving a mixture of components but I do not have that at
my fingertips (please inform me if you come across this information and I will incorporate it into my
class notes). If you ever find such information is critical to your research, contact me and we should be
able to find it.
-m
NOTE: Based on Figure 2-1, Archie's Law can written
F
R
  m
W
where Φ is
porosity. This is the same as equation 2.1 with A = 1.
FIGURE 2.1: Nomogram relating resistivity, formation factor, salinity and temperature. Any of
these parameters may be determined if the other three are known. After Meidev (1970), who
developed this chart to interpret electrical prospecting for hot and saline geothermal fluids.
Field Techniques
Resistivity can be measured by planting 2 sets of electrodes into the earth. Via one set, a
measured electrical current is transmitted into the ground. A second set is used to monitor changes in
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the potential induced by this current. The general form of equations relating resistivity to current and
induced voltage is
(2-2)
V
=G( )
I
where "G" is determined by the geometry of the array (array: spatial deployment of electrodes). V
and I are measured using a voltmeter (or potentiometer) and ammeter respectively. The arrays we most
frequently use are collinear - that is, the 4 electrodes lie in a straight line. There are configurations that
are not collinear but I use those only when the situation so demands.
Wenner array: the Wenner array geometry is depicted in Figure 2-2. Throughout this discussion,
electrodes A and B refer to current electrodes (usually metal stakes) and electrodes M and N refer to
potential electrodes (usually porous pots).
In the Wenner array, distances AM = MN = NB = "a-spacing".
Suppose the earth were a uniform, homogeneous, isotropic half-space (from this point, all "uniform"
structures are homogeneous and isotropic). It is not difficult to compute the potential field set up by
this array relating the resistivity of the earth to the array geometry
(2-3)
V
a = 2 a ( )
I
Consider next an earth that is not uniform, but is instead composed of a single uniform layer
over a uniform half-space. If the aperture (electrode spacing) if the array is small compared to the
thickness of the layer, the array does not "see" the underlying half-space and measured the true
resistivity of the layer. If, on the other hand, the array is very large with respect to the thickness of the
surficial layer, the array samples the half-space rather than the layer and measures the true resistivity of
the half-space. If the array dimension is about the same as the layer thickness, an intermediate value is
measured. Equation 2-3, however, is calculated based on the response of a uniform earth. Hence, we
do not call this value the resistivity of the earth but, rather, the APPARENT RESISTIVITY (ρa).
Resistivity is calculated as if the earth were a homogeneous, isotropic half-space, using Equation 2-3.
We interrogate the earth by increasing the a-spacing and measuring apparent resistivity at
various spacings. Data are plotted on LOG-LOG graph paper, apparent resistivity on the vertical
axis as a function of electrode spacing. For the Wenner array, electrode a-spacing is usually used as
the independent (horizontal, or X-axis) variable. These data define a SOUNDING CURVE. An
electrical sounding is a survey in which we interrogate more and more deeply by measuring apparent
resistivity ρa as a function of electrode spacing AB/2 or "a".
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Figure 2-2: Several of the
more popular arrays used in
electrical resistivity
prospecting. “I” indicates
current electrodes, “V”
indicates potential electrodes.
From Edwards (1977).
Interpretation will be discussed
shortly. Edwards (1977)
developed the concept of
EFFECTIVE DEPTH, ZE,
the interval within the
subsurface of a homogeneous earth that contributes 50% of the signal. For the Wenner array, the
center of this effective depth is given by
(2-4)
Z E = 0.519
a
(Table 2 of Edwards, 1977) where "a" or “L” is the distance between electrodes (Figure 2-2). In other
words, if the L-spacing for a given measurement is 10. meters, 50% of the signal is controlled by a
zone centered about 5.2 m below the surface, and the effective depth zone extends from 0.5 ZE to 1.6
ZE (from about 2.6 m to about 8.3 m) (Figure 2-3). In order to probe the earth, we begin with short
a-spacings, make the measurements of I and V, and expand the array by increasing "a" in a systematic
manner. Data are plotted on the log-log graph (ρa as a function of a for Wenner array, as a function of
AB/2, or L, for Schlumberger) generating the sounding curve (Figure 2-4).
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Figure 2-3: region
of the subsurface
contributing 50%
to an electrical
resistivity signal
(after Edwards,
1977).
Figure 2-4: Sounding curve from field
measurements made near the King Road
Landfill about 1987. Values of the origin
are 0.1 meter and 0.1 Ohm-meter.
In the field, a typical Wenner array
a-spacing for the first measurement is 2.0
meters. This means each potential
electrode is place 1.0 m from the sounding
point and each current electrode is placed
3.0 m from the sounding point. It is
generally a good idea to obtain 5 evenly
spaced data points per log cycle. This
means a = 2.0, 3.2, 5.0, 7.9, 13, 20, 32, 50,
79, 130, and so on until sufficient
sampling depth has been achieved (or
until you run out of wire). Note that these
values represent 100.7, 100.9, 101.1, 101.3
and so on. For the Wenner array, the
distance from the sounding point to each
potential electrode is a/2 and the distance
to each current electrode is 1.5*a. The
decision as to "How large an 'a' is
enough?" (how far should we go?)
depends on the specific geologic problem. This will be discussed below.
Important: sounding curves are ALWAYS plotted on log-log paper on which cycles form
squares. Each cycle on the vertical axis is equal in length to each cycle on the horizontal axis.
Take care when using Excel as the program usually sets log axes and scales that do not conform
to the criterion.
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SCHLUMBERGER ARRAY
The Schlumberger array resembles the Wenner array (Figure 2-2). The main difference in terms of
deployment geometry is the distance between potential electrodes (MN) is not held to half the distance
between the current electrodes (AB) as in the case of the Wenner array. Apparent resistivity
calculations are slightly more complex:
2
(2-5)
2
AB
MN
(
) -(
)
V
2
( )
a = 2
MN
I
and the data plotted are apparent resistivity as a function of AB/2. We usually employ a symmetric
Schlumberger array, one in which electrodes are symmetrical about the sounding point. There are
variations (see Parasnis, pp. 189-190 for an example) on the Schlumberger array that can be used for
PROFILING (seeking lateral rather than vertical variations in electrical properties). One restriction
on use of the Schlumberger array: electrode separation AB must be at least 5 times separation MN;
(2-6)
AB > 5 * MN
for Equation 2-5 to be valid (again, from potential field theory "It can be shown that - - " but you will
have to accept this ex cathedra rather than have me go through the mathematical proof). Effective
depth midrange ZE is about
(2-7)
ZE/L = 0.190
(Edwards, 1977) where "L" is distance AB, and the 50% sample zone extends from 0.5 to 1.6 ZE. Note
that with the Wenner array I defined ZE in terms of dimension "a", which is 0.66 * AB/2, or one-third
of AB/(L). Thus, effective depth for a given AB in the Wenner case is 0.519/3, or .173*L. The
Schlumberger provides, for a given spacing of current electrodes, about 10% greater interrogation
depth than that provided by the Wenner array.
As with the Wenner array, we begin a sounding with a short AB/2 and expand in "log" steps. I
begin with AB/2 = 2.5 m and MN/2 = 0.5 m (remember Equation 2-6). At 5 data points per log cycle,
the array expands as follows: 2.5, 4.0, 6.3, 10, 16, 25, 40, etc. (note that these are 100.4, 100.6, 100.8, 101,
101.2 and so on). Potential electrodes MN are moved only when potential drops become too small to
measure with sufficient precision. In a typical survey, it may not be necessary to increase the MN/2
distance until AB/2 is 10. meters. At this point, we measure (V/I) for both the old MN/2 value (0.5 m)
and for the new MN/2 (10/5, or 2 m). This procedure permits us to detect near-surface heterogeneities,
something not available to us with the Wenner array.
There are three significant advantages for using the Schlumberger rather than the Wenner array.
First, the Schlumberger array has a slightly greater interrogation depth. Second, since the resistivity is
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being sampled between points MN, the lateral resolution is better for the Schlumberger array.
(IMPORTANT: I have encountered engineers who think the AB distance determines lateral
resolution; they are wrong. I have not found the source of this widespread error but would appreciate
your informing me if you find this in print somewhere. A bit of analysis along the line of "voltage
drops across resistors in series" should allow you to understand why my statement regarding MN is
true.) Third, because MN and AB can be changed independently, lateral variations between the MN
electrodes are detected when the Schlumberger array is used. Because both AB and MN must be
moved simultaneously in making a Wenner sounding, a Wenner user cannot determine if details of a
curve are controlled by variations as a function of depth or by lateral variations in electrical properties.
Some investigators like to do "profiling" - that is, selecting a specific array spread (AB, MN are
constant) and moving the array from point to point. Although there is a time and a place for Wenner or
Schlumberger profiling, this method can be difficult to properly interpret, one reason (I think) the
USEPA does not like electrical resistivity as much as electromagnetic (EM) when prospecting for
contaminated ground water. Shallow resistivity variations not linked with conditions at depths of
interest can obscure signals sought. Investigators who profile without due respect for shallow
resistivity variations are likely to fail. Lateral variations in resistivity are best detected by
performing soundings along a profile.
INTERPRETATION OF SOUNDING CURVES
Figure 2-5 shows Schlumberger sounding curves for a single layer over a half-space, for a range
of conditions (half-space much higher resistivity to half-space much lower resistivity – compared to
surface layer).. Note that the layer thickness is 1 m for both cases. These curves begin to depart from
the horizontal "homogeneous" line just a bit to the left of AB/2 = 1.0 m. Note that these curves do not
"level out" until AB/2 is 10 to 100 times the layer thickness for this resistivity contrast (note
comments on profiling, previous page). Often we cannot extend a line far enough for the curve to level
off; however, we can still determine the resistivity contrast (provided it is not extreme) by noting the
slope of the curve. Figure 2-5 is a set of "2-layer" (actually, layer over half-space) MASTER
CURVES. By comparing these curves with a set of field data, we can determine the resistivity of the
layer (it is equal to the horizontal line value prior to the point where the curve turns up or down) and
the contrast between that layer and the underlying material - from which we then calculate the
resistivity of the half-space.
Example: the resistivity of the top layer is 100 ohm-meters. We note by comparing slopes with master
curves that the underlying material is of lower resistivity and that the slope matches the 0.2 curve. This
means the ratio ρ2/ρ1 = 0.2, or, the underlying material has a resistivity of 20 ohm-meters.
IMPORTANT: TO USE MASTER CURVES, YOUR FIELD DATA MUST BE PLOTTED ON
A GRAPH WITH PRECISELY THE SAME SCALES AS THE MASTER CURVES - ELSE
THE DEPTH INDEX WILL YIELD INCORRECT RESULTS !!
On curve matching, the point where layer thickness h = AB/2 is called the DEPTH INDEX. I use a
set of these master curves on a transparent sheet - once I find a match, I can read the thickness of the
layer by noting where this (h = 1) line lies with respect to my data. This value, read from the plot of
my field data, gives me the thickness of the top layer.
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Figure 2-5: master curves for
Schlumberger soundings,
single layer over half-space.
I cannot recall the origin of
this scanned image.
MORE COMPLEX
MODELS
Few earth structures can be
modeled as a single layer
over a half- space. One
more typical situation occurs
when the unsaturated zone
overlies a porous unconfined
aquifer, which in turn
overlies low- porosity
bedrock. In this case, the
top layer has a higher
resistivity than the middle
layer, which has a lower
resistivity than the bedrock.
Figure 2-6-A shows the
H-type curve as representing
this condition. An A- type
curve shows two layers over
the halfspace with increasing
resistivity with depth, and
the K-type curve shows a
high-resistivity layer
between a low-resistivity
layer and half-space. The Q-type curve shows two layers over a half-space with resistivity decreasing
with depth. You will not be expected to recall from memory A-, H-, Q- or K- type curves. You
should, however, be able to inspect a curve and draw conclusions regarding the minimum number of
electrical layers and the general trend of high/low relative resistivities. Figure 2-7 shows further
complexity: 3 layers over a half-space (4 electrical units). Later, we will use computer programs to
model the electrical response of a layered earth and to interpret by inversion a Schlumberger sounding
curve.
Just as the depth index offers a clue to the thickness of the top layer, inflection points offer
clues to the depth to contacts between electrical units further down. For example, in Figure 2-6, note
that the K- and H-type curves bottom out or peak at AB/2 distances about equal to the depth at which
resistivity changes occur.
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Figure 2-6: Sounding curves for 2 layers of a half-space. Another image borrowed from an unknown
source.
Figure 2-7: 3 examples of 3
horizontal layers over a
halfspace.
ONE FINAL WARNING:
potential fields are not
unique. That is, there are
numerous electrical
resistivity structures that
could yield any specific
sounding curve. The best
solution is the least
complex (Occam's Razor:
do not introduce
complexity that is not
mandated by the data)
sufficient to explain all
observations. Thus, upon
offering an interpretation for
a sounding curve, you state
that your interpretation is
consistent with the data, not that it is THE model representing reality.
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We have licensed software that inverts sounding curves and dipole-dipole data to generate a model
showing rock resistivity (not apparent resistivity) as a function of depth (for soundings) or depth and
horizontal position (for dipole-dipole profiles). This software is expensive and the license involves a
hardware key – a USB device that must be inserted into a port before the program will run and remain
in the port while the software is being used. If we lose the key we lose the license and will have to
purchase another, and these are not cheap. The use of these tools will be demonstrated once we have
data to interpret.
ELECTROMAGNETIC METHODS
Physical basis: Complex relationships known as Maxwell's Equations describe electromagnetic waves.
An electrical current induces a magnetic field (the basis for electromagnets), and a variable
magnetic field induces an electrical field. Note that the static magnetic field associated with DC
resistivity will not induce a secondary electrical field.
Electromagnetic geophysical exploration, or EM, exploits the induction of electrical fields in
the earth by magnetic fields on the surface to determine whether materials at depth are good or poor
conductors of electricity. If a good conductor exists at depth, a changing magnetic field will induce
currents to flow, which in turn generate a secondary magnetic field. If materials at depth resist
electrical currents, no secondary magnetic field is produced. Detectors are often designed to detect the
electrical field generated by the secondary magnetic field. Confused? You are not alone. Few
geologists or civil engineers have taken the upper-division class in classic electromagnetic theory (the
only undergraduate physics class that requires extensive use of vector calculus - calculus 5 or 6)
recommended to those who wish to master the theoretical aspects of EM methods.
EM devices consist of a transmitter and a receiver. The transmitter sends out an
electromagnetic wave of specified frequency and amplitude. The receiver detects secondary fields of
identical frequency and a black box converts this signal into numbers for us. There are two values that
can be measured: the amplitude, or strength, of the secondary wave, and, the phase relationship
between the primary and secondary wave. This phase is called quadrature (sometimes it is just called
"phase") and is measured as an angle. Quadrature (some claim) can tell us something about the nature
of a buried conductor (clay or leachate?) but many investigators remain unconvinced. In any event this
will not be of concern to you since the instrument we have does not measure quadrature.
Instrumentation: EM34-3XL from Geonics, LTD, Ontario, Canada. The transmitter operates at 3
distinct frequencies: high frequency for shallow penetration, low frequency for deep penetration and an
intermediate frequency. The transmitter and receiver are separated 10 meters for shallow exploration,
20 m for intermediate and 40 m for deep penetration. Quantity measured: apparent conductivity in
millimhos per meter (mmho/m). The SI unit for conductivity is the Siemen - which is equivalent to
the mho. Please note that this is apparent, not true, conductivity. Many users of this device seem
unaware of the difference between apparent and true conductivity. You, however, having been
introduced to the concept of apparent resistivity will not commit this sin. The receiver and transmitter
are connected by a cable of appropriate length. Settings on the transmitter for depth (10 m, 20 m, 40
m) must be matched by detector settings. The coils must be parallel and "facing" the same direction (a
large dot on each coil marks a side; if the dot on the transmitter faces north, the receiver coil dot must
also be on the north side of the coil). The cable provides separation control and a left-hand dial on the
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receiver is used to fine-tune the distance. If the distance needle is in the green area, the conductivity
needle shows apparent conductivity in mmhos/m. The receiver box also includes a range dial - if a
reading is off the scale or if the needle hardly moves, the range dial must be used to increase of
decrease the sensitivity of the receiving electronics.
MAKING A MEASUREMENT: this requires 2 persons, the transmitter and the receiver. The
transmitter simply holds his coil steady while the receiver adjusts his distance to bring the needle into
the green. Once the receiver reads and records the conductivity value, he signals the transmitter and
they move on to the next station. Measurements can be made using a horizontal dipole, a vertical
dipole, or, both. A horizontal dipole measurement requires vertical coils in line, a vertical dipole
requires the coils to lie flat on the ground. The vertical dipole is said to have better depth resolution
but I have found it difficult to obtain highly repeatable measurements with the vertical dipole. Small
alignment errors (transmitter, receiver, or both) do not seriously impact horizontal dipole
measurements but similar errors generate large variations in vertical dipole readings. An attached
figure shows the relative response with depth for both horizontal and vertical dipole orientations in a
homogeneous earth.
Qualitative interpretation of EM readings: remember that high conductivities correspond to low
resistivities and that you are measuring apparent conductivity. Readings from a series of 10 m, 20 m
and 40 m coil spacings can be plotted in a pseudosection much like dipole-dipole readings, with the
horizontal position of the plotted apparent resistivity midway between the coils and the vertical
distance from the relative response curves and coil separation. Contour values at linear intervals rather
than logarithmic intervals. If a profile consists of measurements made using a single intercoil
separation, values can be plotted like SP readings, apparent conductivity as a function of distance along
profile. When a number of profiles in parallel have been run, data can be plotted and contoured in plan
view. This approach near a landfill sometimes yields a leachate map. As with resistivity, clays can
obscure the signal.
Advantages of EM: EM is rapid and does not require electrodes to be placed in the ground. EM can
be run over paved surfaces and frozen ground. EM generates a lot of numbers quickly. Most
important, the USEPA often specifies EM rather than resistivity measurements as part of remedial
investigations because, as a reconnaissance method, EM sometimes locates plumes more cheaply and
rapidly than possible with resistivity.
Disadvantages of EM: EM is more susceptible to cultural noise than resistivity and is more difficult to
isolate from cultural noise. For example, if I know the location of a pipeline, I can minimize its
influence on electrical measurements by setting up my array at right angles to the pipeline. An EM
measurement is not as easily modified. EM has less lateral and vertical resolution than resistivity. The
signal is influenced by a larger subsurface zone than are resistivity measurements, and because there
are only 3 limited 'spacings' in 2 orientations, only 6 measurements per field point are possible. This
means the unknowns (layer thicknesses and conductivities) that can be deduced from the EM34 are 6 at
most. The EM34 is also limited in its depth of interrogation to perhaps 30 meters - the R-60 resistivity
system has been used to interrogate to almost 1 km and could have seen even more deeply if more wire
had been available. Because of the limited resolution and susceptibility to cultural noise, the EM34 has
sometimes failed to detect plumes clearly mapped by resistivity techniques. Perhaps the greatest
disadvantage of EM is that persons unfamiliar with the apparatus sometimes succeed in collecting
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numbers that the USEPA considers valid and meaningful. I have encountered field technicians who
operated the EM34 and who had no idea what they were doing, what the numbers might indicate, how
cultural factors might influence the measurements.
EM CONDUCTIVITY SOUNDINGS - QUANTITATIVE INTERPRETATION
Calculating layer thickness from EM-34 Data: If the earth consists of a single layer over a uniform
half-space, the EM-34 readings can be used to calculate the thickness of the layer, the true conductivity
of the layer and the true conductivity of the half- space. Let C10, C20 and C40 represent the apparent
conductivities at 10, 20 and 40 meters separation. Use these reading to calculate the ratio R
(3-1)
C
40 - C 20
R=
C 20 - C 10
and compare this value to the appropriate curve from the figure (Appendix I) titled TWO LAYER
CALCULATION. Find the ratio along the vertical axis, note where the appropriate (vertical or
horizontal dipole?) curve has that value, and read depth Z from the horizontal axis. Now that you
know Z, the ratio Z/S (layer thickness/coil separation) can be used to calculate the response function
R(Z/S) which, for a horizontal dipole, is
(3-2)
2
Z
Z
Z
R H ( ) = (2 ) + 1 - (2 )
S
S
S
The conductivities of the layer and half-space (C1 and C2) are found by calculating
(3-3)
( C 40 - C 20 )
( C2 - C1 ) =
Z
Z
(R( ) - R(
))
40
20
and then plugging this into
(3-4)
Z
C 1 = C 40 - ( C 2 - C 1 ) R H ( )
40
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Obviously, C2 = C1 + (C2 - C1) (Equation 3-4 minus Equation 3-3).
MORE COMPLEX EARTH MODELS
The equation relating coil separation S (in meters) and apparent conductivity AC(S) (in mmhos/m) in
an earth consisting of 2 layers over a halfspace is
(3-5)
AC ( S )  C1 (1  R1 )  C 2 ( R1  R2 )  C 3 R2
where C1, C2 and C3 are true conductivities of layers 1, 2 and the halfspace. Response functions R1
and R2 are found by calculating
Z
Z1
1 2
R1 = ((2 ) + 1) - 2
S
S
(3-6)
Z 2 + 1) - 2 Z 2
=
((
2
)
R2
S
S
2
We have 3 equations (3-5), one each for AC(10), AC(20) and AC(40), our 3 observed
quantities. We have 5 unknowns (Z1, Z2, C1, C2 and C3). If 2 of these variables are known, the other 3
can be determined.
Assume we know Z1 and Z2 from other sources (wells, seismic refraction). Begin by solving
Equation 3-3 for C1 in terms of AC(10) and the other factors, known and unknown
(3-7)
C1 =
(C (10 ) - C 2 ( R1 - R 2 ) + C 3 R 2 )
1  R1
where "R" values are calculated from Equation 3-6 for the known values Z1 and Z2. A bit more
algebraic substitution (do you recall how to solve systems of simultaneous equations?) yields unique
values for C1, C2 and C3.
For students familiar with matrix algebra, solving systems of equations such as this should be
trivial. Just substitute the values for apparent conductivity at S = 10, 20 and 40 m along with the R1
and R2 for S = 10, 20 and 40 m into Equation 1 to obtain 3 equations in 3 unknowns.
It is also possible to set C3 (low, probably 1 or 2 mmho/m) and Z1, then solving for 3 equations
14
in C1, C2 and Z2. The algebra is slightly more complex but could be well worth the effort if both depth
to non-conductive bedrock and C2 (aquifer conductivity) vary. This would make an excellent
homework problem.
On converting from conductivity to resistivity: Electrical conductivity expressed in millimhos/meter or
millisiements/meter (1 mmho = 1 mSi) can be converted to the equivalent electrical resistivity by (1)
converting the units from "milli" - multiply conductivity by a well-selected "1" (1 mho/1000 mmhos)
yields the conductivity in mhos/meter - then take the reciprocal, changing mhos/meter to ohm-meter.
FORWARD MODELS - EM34
McNeill (1980) includes in TN-6 response functions for both vertical and horizontal dipoles as
a function of depth:
(3-8)
(3-9)
4z
 H (z)  2 
(4 z 2  1)1 / 2
(3-10)
(3-11)
4z
V ( z ) 
(4 z 2  1) 3 / 2
1
RV ( z ) 
(4 z 2  1)1 / 2
RH ( z )  (4 z 2  1)1 / 2  2 z
where z is the depth divided by the intercoil spacing (10, 20 or 40 meters). Equations 3-8 and 3-9
give the relative influence of current flow as a function of depth, while equations 3-10 and 3-11
represent cumulative response of all material deeper than depth z.
Equations 3-10 and 3-11 are very useful in calculating the apparent conductivity of a
hypothetical layered earth (Figure EM-1). Let z1 and z2 be depths to the bottoms of the top and second
layer respectively. Let C1, C2 and C3 be the true conductivities of the top, middle and bottom
(actually, bottomless) layers respectively. Apparent conductivity CA for a given separation S and
orientation D is
(3-12)
CA( S , D)  C1 (1  RD ( z1 ))  C2 ( RD ( z1 )  RD ( z2 ))  C3 ( RD ( z2 ))
15
First, calculate response functions
R for the appropriate D (dipole
orientation - H or V) and z (recall that z =
d/S) using equation 3-10 for V or 3-11
for H. Note that the sum of all
coefficients to be multiplied by the
various "true" conductivities C built from
response functions R(z) for a given
calculation must equal 1. Multiply
these depth response coefficients with
layer conductivities and add the products.
Equation 3-12 can be expanded and
used for many layers! If there were 4
layers, C3 would be multiplied by R(z2)R(z3) and C4 would be multiplied by
R(z3).
Figure EM-1: cartoon illustrating
hypothetical 3-layered earth from which 6
apparent conductivities (10, 20 and 40
meter separations, V and H) can be
calculated using equations 3-10, 3-11 and
3-12. In forward models, additional
layers can be included.
Suppose that you make 3 measurements
(10 m, 20 m and 40 m separation, either
horizontal or vertical dipole) with the EM34. Suppose furthermore that you know the depths to the
bottoms of the first layer (zone of aeration) and second layer (water-saturated sediment) overlying
bedrock (C3), as depicted in Figure EM-1, by some independent method such as seismic refraction.
Equation 3-12 can be used to set up a series of 3 simultaneous equations that can be most easily solved
using matrix algebra.
Inverting measurements to map variations in layer thicknesses (assuming no lateral changes in layer
conductivities) should be possible but the mathematics is not as straightforward. If there is but one
layer over the half-space, equation 3-5 becomes
(3 – 13)
AC ( S )  C1 (1  R1 )  C 2 ( R1 )
McNeill (1983) tells us that, if we measure apparent conductivity at 10, 20 and 40 m spacings (all
either horizontal or vertical) we have 3 equations
CA(10)  C1  (C 2  C1 ) R( 10z )
(3 – 14)
16
CA( 20)  C1  (C 2  C1 ) R( 20z )
CA(40)  C1  (C 2  C1 ) R( 40z )
These terms can be rearranged into
(3 – 15)
CA(40)  CA( 20) R( 40z )  R( 20Z )

CA( 20)  CA(10) R( 20z )  R( 10z )
and
(3 – 16)
CA(40)  CA( 20)
 C 2  C1
z
z
R( 40 )  R( 20 )
The right-hand side of equation 3 - 15 is plotted in Figure EM-2. Calculate the ratio based on
measured quantities (left-hand side of equation 3 – 15) and you have z! Equation 3 – 10 or 3 – 11 now
yields R(40) and R(20) so you can use equation 3 – 16 to obtain C2 – C1, and then you can use equation
3 – 14 to obtain C1!
17
2
Figure EM - 2
1.8
1.6
1.4
Ratio
1.2
1
0.8
Horizonta
l
Vertical
0.6
0.4
0.2
0
1
10
100
Z (Meters)
Figure EM-2: For a single layer over a half-space, divide the quantity CA(40)-CA(20) by the quantity
CA(20)-CA(10) and find the resulting ratio on the appropriate curve. Then read the value of Z
(thickness of layer) directly from the graph.
To Do: Review your spreadsheet skills by calculating a series of graphs, one showing response as a
function of depth, the other cumulative response as a function of depth, for both horizontal and vertical
dipoles and all 3 EM34-3XL spacings. Hint: equations 3 – 8 through 3 – 11.
18
REFERENCES
Edwards, L.S. (1977), A modified pseudosection for resistivity and IP; Geophysics, 42, 1020-1036.
EM34-3 Operating Instructions, Geonics Limited, 1987.
Ernstson, K. and H.U. Scherer (1986), Self-potential variations with time and their relation to
hydrogeologic and meteorologic parameters; Geophysics, 51, 1967-1977.
Keller, G.V. (1982), Electrical Properties of Rocks and Minerals in the Handbook of Physical
Properties of Rocks, CRC Press (R.S. Carmichael, Editor), pp. 217-293.
McNeill, J.D., Survey interpretation techniques EM38; Technical Note TN- 9, Geonics Limited, 1984.
McNeill, J.D., Electrical conductivity of soils and rocks; Technical Note TN-5, Geonics Limited, 1980.
McNeill, J.D., EM34-3 Surveying interpretation techniques; Technical Note TN-8, Geonics Limited,
1983.
Meidav, T. (1970), Application of electrical resistivity and gravimetry in deep geothermal exploration;
in Proc. U.N. Symposium on Development and Utilization of Geothermal Resources (Pisa);
Geothermics, Special Issue 2, pp. 303-310.
Orellana, Ernesto and H.M. Mooney (1966), Master tables and curves for vertical electrical sounding
over layered structures: Madrid Interciencia, 150 pp., 66 tables.
Ram Babu, H.V. and D. Atchuta Rao (1988), A rapid graphical method for the interpretation of the
self-potential anomaly over a two-dimensional inclined sheet of finite depth extent; Geophysics, 53, pp.
1126-1128.
Van Nostrand, Robert G. and K.L. Cook (1966), Interpretation of Resistivity Data; U.S. Geol. Surv.
Professional Paper 499, 310 pp.
Zohdy, A.A.R., G.P. Eaton and D.R. Mabey (1974), Applications of Surface Geophysics to
Ground-water Investigations; Techniques of Water-Resources Investigations of the United States
Geological Survey, Chapter D-1.