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Introduction to Geophysics
The DC Resistivity Method
Modelling & Inversion
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1
History
In 1912 Conrad
Schlumberger, using
very basic equipment,
recorded the first map
of equipotential curves
in his Normandy estate
near Caen.
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http://www.slb.com/about/history.aspx
2
Applications
•
•
•
•
•
•
•
•
•
•
•
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investigation of lithological underground structures,
estimation of depth, thickness and properties of aquifers,
mapping of preferential pathways of groundwater flow,
determination of the thickness of the weathered zone
detection of fractures and faults in crystalline rock,
localization and delineation of the horizontal extent of dumped materials,
estimation of depth and thickness of landfills,
detection of inhomogeneities within a waste dump,
mapping contamination plumes,
monitoring of temporal changes in subsurface electrical properties,
detection of underground cavities,
classification of cohesive and non-cohesive material in dikes, levees, and dams.
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Goal
•
Derive general formula for the “apparent resistivity”:
Apparent resistivity
Potential difference (Voltage)
Resistance
U 
 a  K  R  K   ,
I 
e.g. K  2 a (Wenner)
Electrode spacing
Geometric factor
Source current
& applications
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DC Resistivity Method
DC Resistivity survey across a circular mound, thought
to contain Irish archaeological burial chambers.
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geotomosoft.com
5
DC Resistivity Method
DC Resistivity survey mapping lithology of near-surface soil materials to examine
possible contamination from fertilisers and pesticides.
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Loke, 2000
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DC Resistivity Method
DC Resistivity survey mapping a dolerite dyke causing a prominent high resistivity
zone surrounded by shales.
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Loke, 2000
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DC Resistivity Method
DC Resistivity survey mapping caves within a limestone bedrock. A known air-filled cave
causes a high resistivity anomaly near the centre. In the course of this survey a new cave
was discovered. Causing a high resistivity anomaly near the bottom left corner.
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Loke, 2000
8
Conduction of electricity in rocks
• Electrical conduction in rocks occurs in three ways
ELECTRONIC
ELECTROLYTIC
DIELECTRIC
by motion of electrons
by movement of ions
by displacement of electrons and
positively-charged atomic nuclei
• Most rocks are partial ELECTRONIC conductors
• However, the bulk conductivity of rocks in the upper few km of the
crust is mainly due to ELECTROLYTIC conduction
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Ohm’s Law (1827)
Circuit theory: U = 𝑅 ∙ 𝐼
General form: 𝐄 = 𝜌 ∙ 𝐉
𝐿
𝑅=𝜌
𝐴
𝐿
⟹𝑈 =𝜌 ∙𝐼
𝐴
𝑉 =Ω∙𝐴
𝑉
𝑚
𝐴
= Ω𝑚 ∙ 𝑚2
𝑈
𝐼
⟹ =𝜌
𝐿
𝐴
𝐄 is the electric field intensity vector in V/m
𝐉 is the current density vector in A/m2
𝜌 is the electrical resistivity in Ω m (s in S/m)
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Ohm’s Law (1827)
DC Resistivity Method
“equivalent circuit”
-
+
I
completely described
by Ohm’s law
𝑹 =
𝑼
resistance
𝑰
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Outline
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•
The electric field and potential
•
Potential and current flow in the subsurface
•
DC resistivity measurements
•
Common electrode configurations
•
Resistivity sounding and profiling
•
Modelling and inversion
12
The electric field
• Electric fields may be generated by charges.
• A stationary charge Q+ creates an electric field E.
• The electric field intensity E is defined as the force F exerted by Q+ on a test
charge q0 at position r.
Fel r 
Er  
q0
𝑉𝑜𝑙𝑡𝑠
𝑁𝑒𝑤𝑡𝑜𝑛
=
𝑚𝑒𝑡𝑟𝑒 𝐶𝑜𝑢𝑙𝑜𝑚𝑏
The magnitude is given by Coulomb’s law
Q
1 𝑄+
𝐸 𝑟 =
4𝜋 𝜀0 𝑟 2
F 

 c.f. gravity : g  grav 
m0 

(i) A test charge does not alter the electric field – it is an idealized quantity whose
physical properties are assumed to be negligible except for the property being studied.
(ii) Note that electric fields also exist without charges (for example as light which is an electromagnetic wave)
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The electric field
• In accordance with Coulomb’s law, the E-field falls off as the
square of the distance of the test-charge.
• Adding or removing charges changes the E-field.
• The E-field is a vector field as it varies from point to point
(A vector has a direction and a magnitude).
Electric dipole field of opposite charges.
The E-field is visualized by “field lines”
which are tangential to the E-field vectors.
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The electric potential
• Instead of the electric field – a vector quantity – we can use a
scalar quantity to describe and calculate DC electric
phenomena.: the electric potential.
• The electric potential at a point is equal to the electric potential
energy of a charged particle divided by its charge.
• Because it is a scalar, it simplifies many calculations.
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15
KEA230
The electric potential
Va
a
Vb
+
+
b
E
y
x


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d
Static (DC, zero-frequency) electric fields are conservative:
The work done in moving a charge from point a to point b depends
only on the start and end points, not on the path taken
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The electric potential
• Relationship between: Work ~ E-Field ~ Electric Potential
Wab  Fx d  q0 E x d
and
Wab  Vb  Va  q0
• The work done in moving a test charge q0 from point a to point b
can be defined in terms of scalar potentials Va and Vb.
• Therefore, the electric field can be given in terms of a
Vb  Va V
Ex  

Potential Difference:………………………………………
d
x
W  Nm or J, V 
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J
or Volt,
C
  V
m
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The electric potential
b
b
b
Wab  d W  F  d  q0 E  d   Vb  Va   q0



a
a
a
(Jackson, 1999)
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The electric potential
• The simple example given on the previous slides can be generalised to give
E  V
 V ˆ V ˆ V ˆ 
 
i
j
k 
y
z 
 x
“Steepest ascent”
• That is, the electric field is the gradient of the potential function (V)
• The potential V obeys Laplace’s Equation
 V0
2
  2V  2V  2V 
  2  2  2   0
y
z 
 x
“Average bending”
• This is the same equation which governs magnetic and gravitational
(potential) fields!
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KEA230
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•
The electric field and potential
•
Potential and current flow in the subsurface
•
DC resistivity measurements
•
Common electrode configurations
•
Resistivity sounding and profiling
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Current flow due to a single electrode
Potential about a single electrode on the surface of the earth
• The most simple solution can be obtained for a uniform half-space with
resistivity “Rho” for a single point-source electrode on the surface, with
current of “I” Ampere.
• Because of this special setup, we have a radial symmetry with respect to the
point source. This solution is the basis for all subsequent derivations.
*A half-space is a simplified model of the
local earth – it subdivides the “space” into
two halves, where the upper half is air
and the lower half consist of the uniform
property under investigation.
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Lowrie, 2007, p. 212
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Current flow due to a single electrode
Potential about a single electrode on the surface of the earth
Solution for a single point electrode
𝐼
𝐄 = 𝜌𝐉 = 𝜌
𝐫
2𝜋𝑟 2
𝜕
𝐄 = −𝛻𝑉 = − 𝑉𝐫
𝜕𝑟
𝜕
𝐼
𝑉 = −𝜌
𝜕𝑟
2𝜋𝑟 2
𝐼
𝑉=𝜌
2𝜋𝑟
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Lowrie, 2007, p. 212; c.f. Telford, 1991, p.523
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Current flow due to a single electrode
Potential about a single electrode on the surface of the earth
• The potential V about a single (point) electrode on the surface of a
homogeneous half-space:
I
Vr  
2r
ρ: half-space resistivity
I: electrode source current
r: radial distance from source
I
Surface of half-space*
r

“Equipotential” (surface of equal potential)
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Current flow due to a single electrode
Potential about a single electrode on the surface of the earth
• The E-field anywhere within the half-space can be calculated using the
equation on the previous slide.
Electric field = gradient of the potential
𝐄 = −𝛻V =
𝐼𝜌
𝐫
2
2𝜋𝑟
• The current density J within the half-space can then be calculated using
Ohm’s law.
General form of Ohm’s Law
1
𝐼
𝐉= 𝐄=
𝐫
𝜌
2𝜋𝑟 2
Note: Current flows parallel to E
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Current flow due to a single electrode
Potential about a single electrode on the surface of the earth
T. Boyd, Colorado School of
Mines
current
potential
Equipotential
•
•
•
Potential decreases away from the electrode
Current flows perpendicular to the equipotential contours
Logistically difficult to set up in practice: 2nd current electrode at ‘infinity’
(in practice, 10 times spacing of potential electrodes)
•
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 Use two current electrodes closer together
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Current flow due to two electrodes
Potential about two electrodes on the surface of the earth
• Because potential is a scalar, the total potential due to two current
electrodes C1 and C2 can be calculated by adding the potentials for
electrodes carrying current +I (C1) and -I (C2) respectively.
• The potential at any point P within the half-space is then:
I   I 
V P  

2r1
2r2
r1 and r2 are the distances of
point P from electrodes C1 and
C2 respectively
• Once the potential at any point is known, the current density can be
calculated using exactly the same procedure as for a single electrode
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Current flow due to two electrodes
Potential about two electrodes on the surface of the earth
C2
C1
Equipotential
lines
T. Boyd, Colorado School of Mines
In a half-space current always
flows perpendicular to the
equipotential contours
Current
streamlines
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Current flow due to two electrodes
Potential about two electrodes on the surface of the earth
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Knödel et al, 2007, p. 205
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•
The electric field and potential
•
Potential and current flow in the subsurface
•
DC resistivity measurements
•
Common electrode configurations
•
Resistivity sounding and profiling
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DC Resistivity measurements
Equipment
resistivity meter and battery
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cables (1.26 km) with 64 electrodes
Revil et al, 2012
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DC Resistivity measurements
Equipment
Contact resistance between the stainless steel electrodes
and the ground is decreased by adding salty water (right)
Cable layout (above) and reels for cable (right)
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Revil et al, 2012
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DC Resistivity measurements
Apparent resistivity
•
•
The theoretical discourse so far solved current flow and the electric potential in
the subsurface for a homogeneous ground (half-space).
With these preparations we can now solve for the apparent resistivity for a known
source current, measured potential (voltage) and the assumption of a
homogeneous half-space.
Apparent resistivity 𝜌𝑎 is defined as the resistivity of homogeneous
ground that would give the same voltage-current relationship as
measured over a inhomogeneous ground*.

Apparent resistivity is a useful data transformation to provide a ‘normalised’ data
set which accounts for system configuration.

Only for a homogeneous ground the apparent resistivity equals the true resistivity.

The exact form of the formula for the apparent resistivity depends on the relative
positions of the electrodes.
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KEA230
DC Resistivity measurements
Apparent resistivity
1)
inject current into a half-space through current electrodes C1 & C2
2)
estimate the resistivity by measuring the potential difference
across a pair of potential electrodes P1 and P2
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(Telford et al., 1990)
𝑅𝑉𝑀 = ∞
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DC Resistivity measurements
Apparent resistivity
•
•
•
Potential difference V between P1 and P2 is measured:
V = (Potential at P1) - (Potential at P2) = V1-V2
Assumption of Geometry !
𝑅 = −∆𝑉 𝐼 Ohm’s Law
 


V1  VP C   VP C 
1 1
1 2

 

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r𝑅
2
𝑉𝑀
=∞

r3
r4
(Telford et al., 1990)
r1
 

V2  VP C   VP C 
2 1
2 2

 

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DC Resistivity measurements
Apparent resistivity
𝜌𝐼
𝜌𝐼
𝜌𝐼
𝑉1 =
−
=
2𝜋𝑎 2𝜋 2𝑎
4𝜋𝑎
Rearrange to give
𝜌𝐼
⇒ ∆𝑉 = 𝑉1 − 𝑉2 =
2𝜋𝑎
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𝜌𝐼
𝜌𝐼
𝜌𝐼
−
=−
2𝜋 2𝑎
2𝜋𝑎
4𝜋𝑎
2πa = K = geometric factor
(depends on electrode array)
∆𝑉
⇒ 𝜌 = 2𝜋𝑎
=𝑘∙𝑅
𝐼
(Telford et al., 1990)
𝑉=
𝜌𝐼
2𝜋𝑟
𝑉2 =
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DC Resistivity measurements
Apparent resistivity
•
The general formula for the resistivity  of a uniformly resistive,
homogenous Earth is: 𝜌 = 𝑘 ∙ 𝑅 = 𝑘 ∙ ∆𝑉 𝐼
•
In an inhomogeneous Earth, it is the apparent resistivity a
Example:
apparent resistivity
curve over a twolayered earth
r1 = 250 Wm
25m
r2 = 50 Wm
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DC Resistivity measurements
Apparent resistivity
DC Resistivity (Wenner), Fingal, Tasmania
• The basic equipment consists of 2 (active) current and 2 (passive)
potential electrodes and a recording instrument, the Terrameter.
The Terrameter is designed to
measure the resistance of the
ground −∆𝑉 𝐼 with high accuracy.
This is done by balancing the
internal resistor of the instrument
so that it completely nullifies any
current flow within the potential
electrodes.
Terrameter and Wenner electrode array
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DC Resistivity measurements
Apparent resistivity
Observed field data:
• during measurement record the
resistance values for the
associated “a”–spacing
• calculate the apparent resistivities
(spread sheet)
⇒ 𝜌 = 2𝜋𝑎 = 𝑘 ∙ 𝑅
• These observed (raw) data are
input to the modelling program.
The apparent resistivity is the resistivity of the homogeneous half-space which
would produce the observed instrument response for a given electrode spacing.
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DC Resistivity measurements
Apparent resistivity
a ~ 600 Wm
a ~ 1000 Wm
(asymptotic)
a ~ 50 Wm
600 Wm
50
Wm
Estimated
Layer
boundaries
1000 Wm
DC Resistivity (Wenner), Fingal, Tasmania
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39
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•
The electric field and potential
•
Potential and current flow in the subsurface
•
DC resistivity measurements
•
Common electrode configurations
•
Resistivity sounding and profiling
40
Common electrode configurations
General formula for 4-point electrode layout
𝜌𝐼 1 1
𝑉𝑃2 =
−
2𝜋 𝑟3 𝑟4
𝜌𝐼 1 1
𝑉𝑃1 =
−
2𝜋 𝑟1 𝑟2
𝜌𝐼
𝑉=
2𝜋
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1 1
1 1
−
−
−
𝑟1 𝑟2
𝑟3 𝑟4
𝜌 = 2𝜋
𝑉
𝐼
1
1 1
1 1
−
−
−
𝑟1 𝑟2
𝑟3 𝑟4
Lowrie, 2007, p. 213
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Common electrode configurations
1) Wenner array
K  2a
Simple setup but cumbersome in practice
2) Schlumberger array
K
  AB 2  MN 2 
4 
MN


A and B: current electrodes
M and N: potential electrodes
A,B move; M,N fixed
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Convenient as only 2 electrodes move
Knödel et al, 2007, p. 210
42
Common electrode configurations
3) Pole-dipole array
ab
K  2
ba
Good for mapping of confined conductors
4) Dipole-dipole array
K   n n  1n  2  a
minimum coupling between
current and potential wires
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Knödel et al, 2007, p. 210
43
Common electrode configurations
5) Pole-Pole array
K  2a
a
gives wide horizontal coverage and good depth
coverage however suffers from poor resolution
6) In boreholes
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Knödel et al, 2007, p. 210
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Common electrode configurations
Sensitivity patterns
Wenner
Schlumberger
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Knödel et al, 2007, p. 211; cf. Loke, 2000, pp.10
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Common electrode configurations
Sensitivity patterns
Pole-dipole
Dipole-dipole
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Knödel et al, 2007, p. 211; cf. Loke, 2000, pp.10
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Common electrode configurations
Sensitivity patterns
Pole-Pole
Borehole
(dip-dip)
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Knödel et al, 2007, p. 211; cf. Loke, 2000, pp.10
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Common electrode configurations
Median depth of investigation (𝒛𝒆 ). 𝑳 is the length of the array.
Array Type
𝒛𝒆 𝑳
Wenner
~0.173
Schlumberger
~0.191
Dipole-Dipole
~0.203
• These median depths are strictly only valid for a homogeneous earth
model
• This depth does not depend on the measured apparent resistivity or the
resistivity of the homogeneous earth model.
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Loke 2000, p.13
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Common electrode configurations
Comparison of most common arrays
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Reynolds, 2011, p.298
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•
The electric field and potential
•
Potential and current flow in the subsurface
•
DC resistivity measurements
•
Common electrode configurations
•
Resistivity sounding and profiling
50
DC Resistivity measurements
Resistivity profiling (CST)
► PROFILING: determine the lateral distribution of resistivity
(in an area)
C
P1
C1
C
P12
C
PC
1
22
C
PC
12 2
C
PC
12 2
C
PC21 2
C
PC21 2
PC212
PC2 2
C2
(z)
In profiling, the electrode separations remain fixed,
and the whole array is moved along the survey line
• Any configuration can be used, although
•
•
Wenner is logistically easy, and
Dipole-Dipole is highly sensitive to lateral resistivity variations
• The shape of a resistivity profile over a particular structure depends
very strongly on the electrode array used
• Profiling is sometimes referred to as CST (constant separation traversing)
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DC Resistivity measurements
Resistivity profiling (CST)
Telford, 1999
 Profiles over a vertical contact using different arrays.
 Discontinuities in the vicinity of the contact.
 Profile shape dependent on electrode layout.
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DC Resistivity measurements
Resistivity profiling example
Zhody, Eaton, Mabe, U.S.G.S., 1974
Shorter spacing: Shallow features
enhanced
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Larger spacing: Shallow features
suppressed
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DC Resistivity measurements
Resistivity Sounding
► SOUNDING: determine the variation in resistivity with depth
(below a given point)
• The electrode array is expanded about a fixed central point
• Soundings can be carried out using:
n
Schlumberger
n
Wenner
n
Dipole-Dipole
o The Schlumberger array is easiest logistically, as (in theory) only the
current electrodes need be moved
o For other arrays (e.g. Wenner) the potential electrodes must be moved
each time the current electrodes are expanded
• Sounding is sometimes called VES (vertical electric sounding)
• Remember assumption of uniform Earth when converting to 𝝆𝒂
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54
DC Resistivity measurements
Resistivity Sounding
C1
P1 C1
P1
P2
C2 P2
C2
(z)
Fixed central point
•
Objective is to determine the variation in resistivity with depth below
the midpoint of the array
•
•
Electrode array is expanded about a fixed central point
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This example shows a Wenner array
55
DC Resistivity measurements
Resistivity Sounding
Moving the current electrodes wider apart increases the depth of investigation
Close spacing 
shallow investigation
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(Kunetz, 1966)
Wider spacing 
deeper investigation
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DC Resistivity measurements
Resistivity Sounding
Current-Depth vs Electrode-Separation
C1
C2
L
For an electrode spacing L = 2z1, roughly half the current in the
ground passes below depth z1
i.e. for at least 50% of the current to flow at a depth z1, the current
electrode separation needs to be at least twice the depth
For a pair of electrodes on the surface of a half-space, note the
proportion of current flowing below a given depth, z1
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Reynolds, 2011, p. 294
DC Resistivity measurements
Resistivity sounding example
van Overmeeren, (Geophysics, V46, 1981)
App. resistivity from measured data and interpretation models
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58
DC Resistivity measurements
(Lowrie, 1997)
Type Curves for Layered Earth
(For more than 3 layers the letters H, A, K, and Q are used in combination)
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DC Resistivity measurements
Layered earth parameters
vertical current
T = Transverse Resistance
T is the sum of the individual layer thicknesses
multiplied by the layer resistivities
S = Longitudinal Conductance
S is the sum of the individual layer thicknesses
divided by the layer resistivities
horizontal current
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(Zohdy, 1980, USGS)
60
DC Resistivity measurements
Layered earth parameters
vertical current
T = Transverse Resistance
T is the sum of the individual layer thicknesses
multiplied by the layer resistivities
S = Longitudinal Conductance
Sometimes
all
that
can
be
S is the sum of the individual layer thicknesses
determined are S or Tdivided
for multilayer
by the layer resistivities
electrical sounding curves
horizontal current
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(Zohdy, 1980, USGS)
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DC Resistivity measurements
Equivalence
• Note that the interpretation of a multilayer sounding curve generally is not
unique. This means that a given electrical sounding curve can correspond to a
variety of subsurface distributions of layer thicknesses and resistivities.
• For example, consider two three-layer models of the K-type where the product
of thickness and resistivity of the middle layer in both model is the same (same
TRANSVERSE RESISTANCE) – upper and lower layer in both models have the
same resistivity. In this case the apparent resistivity curve for both models are
indistinguishable; this is known as T-Equivalence.
(Likewise for curves of the H-type which can be subject to S-equivalence when
the middle layers for two models have the same LONGITUDINAL CONDUCTANCE.)
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62
DC Resistivity measurements
Equivalence
K: (T-equivalence)
H: (S-equivalence)
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(Zohdy, 1980, USGS)
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DC Resistivity measurements
Resolution
• The resistivity method is more sensitive to the presence of nearsurface layers than to those at depth. The deeper a layer is, the
thicker it has to be in order to be resolved.
• In general, a model layer must have a thickness at least 10% of its
depth of burial in order to be resolved by a resistivity sounding
• The ratio of the layer thickness to the depth of the top of the layer is
called the relative thickness (RT)
• Layers with a small relative thickness produce little effect on the
shape of a resistivity sounding curve, and are said to be suppressed
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64
How visible is the layer
at greater depths?
DC Resistivity measurements
Resolution
10 W m background & 1 m thick, 100 Wm layer at a depth of 1 m (RT = 1), 10 m (RT = 0.1), 20 m (RT = 0.05)
10 Wm
10 Wm
100 Wm
10 Wm
RT=1
10 Wm
100 Wm
10 Wm
RT=0.1
10 Wm
RT=0.05
100 Wm
DC Resistivity measurements
Resolution
• As a general rule, the resistivity method cannot
resolve thin layers (RT < 0.1).
• The method can resolve considerable detail near
the surface, but can only see bulk zones at depth.
• Usually, soundings can resolve no more than six to
eight layers.
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DC Resistivity measurements
Suppression
• Even layers with a large relative thickness can produce
little observable effect on observed apparent resistivities
if the resistivity contrast between the layers is too small
• This is another example of suppression, as there may be
no hint of the layer in the field data.
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DC Resistivity measurements
Suppression
In this example, the
middle layer of resistivity
13 Wm is suppressed
10 Wm
13 Wm
100 Wm
RT = 1
10 Wm
No middle
layer
100 Wm
Forward and Inverse modelling
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Geophysical modelling
Escript Finite Element Solver (UQ Geocomp)
3D Current-Dipole Potential
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Geophysical modelling
• A geophysical model is a simplified concept of how a physical property is
distributed within the Earth – it is a translation of geological formation
characteristics into geophysical parameters (e.g. electrical resistivity).
• Geophysical methods enable us to make deductions about buried structure.
We observe readings of a physical quantity (like apparent resistivity, ρa),
then use the underlying physics to deduce the buried structure that is
causing the observed readings.
• We generally use computer modelling to help us deduce structure. The
program calculates a structure (described by the model parameters) that
results in a good match of modelled values to our observed data.
• Observed data: are the readings we take in the field. They may be written
down manually, or logged by a field computer.
• Modelled values: are the values of the same physical quantity that the
computer calculates . They are sometimes called ’response’, ‘synthetic’ or
‘calculated’ values.
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Forward and Inverse modelling
Forward modelling
Forward algorithm
• Model parameters (1D)
• layer thicknesses
• layer resistivities
• (electrode spacings)
• Numerical
implementation
• ‘Brute force’ integration
• Digital filters
Model + ‘manual’
fit to data
• Trial & error
• adjust thicknesses
• adjust resistivities
• adjust number of layers
Assumption
of structure:
1D, 2D, 3D
Inverse modelling
• Observed data
• apparent resistivities
• (electrode spacings)
Inverse algorithm
(needs forward step)
• Numerical
implementation
• Marquardt inversion
• Steepest descent
Best fit model
• Simultaneous
computation of layers &
resistivities (subject to
data uncertainty)
Forward modelling
•
•
•
Assumption of 1D structure
Layer thicknesses and resistivities are known
Theoretical response calculates apparent resistivities
Ω𝑚
𝑚
ρ1 = 250
ρ2 = 1000
d1 = 1.5
d2 = 2
ρ3 = 500
d3 = 8
ρ4 = 7.5
d4 = 3
ρ5 = 10000
d5 = ∞
Forward modelling
0.1
1
10
100
Electrode Spacings (m)
100
Wenner
Ω𝑚
Apparent Resistivity (Ohm·m)
1000
1000
𝑚
10
ρ1 = 250
ρ2 = 1000
d1 = 1.5
d2 = 2
ρ3 = 500
d3 = 8
ρ4 = 7.5
d4 = 3
ρ5 = 10000
Theoretical response (Ghosh, 1971a)
d5 = ∞
Inverse modelling
0.1
1
10
100
Electrode Spacings (m)
100
Wenner
Starts with measurement
10
?
Apparent Resistivity (Ohm·m)
1000
1000
Inverse modelling
0.1
1
10
100
Electrode Spacings (m)
𝑁
𝑚𝑖𝑛! 𝑒 =
𝑛=1
𝜌𝑎𝑜𝑏𝑠
− 𝜌𝑎𝑚𝑜𝑑
𝑛
2
100
𝑛
𝑢𝑛
Wenner
Ω𝑚
Apparent Resistivity (Ohm·m)
1000
1000
10
ρ1 = 250
𝑚
ρ2 = 1000
d1 = 1.5
d2 = 2
ρ3 = 500
d3 = 8
ρ4 = 7.5
ρ5 = 10000
?
Reconstruction of 1D subsurface structure via Least Squares inversion
d4 = 3
d5 = ∞
Least-squares inversion
• The inversion process attempts to find the layered earth model
which provides the best fit to the observed data.
• A popular way of measuring this fit is using a least-squares
method.
• The inversion process minimises the function
𝑁
𝑒=
𝑛=1
𝜌𝑎𝑜𝑏𝑠
𝜌𝑎𝑚𝑜𝑑
𝑛
𝑛
𝜌𝑎𝑜𝑏𝑠
− 𝜌𝑎𝑚𝑜𝑑
𝑛
2
𝑛
𝑢𝑛
Minimise sum of
(obs – mod)2 values
observed apparent resistivity at n-th spacing with uncertainty 𝑢𝑛
modelled apparent resistivity at n-th spacing
for N different electrode spacings
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Least-squares inversion
Observed data
Minimisation is a standard
mathematical procedure
(requires derivatives of e)
𝑁
𝑚𝑖𝑛! 𝑒 =
𝜌𝑎𝑜𝑏𝑠
Modelled data
− 𝜌𝑎𝑚𝑜𝑑
𝑛
Uncertainty
(noise, errors)
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𝑛
𝑢𝑛
𝑛=1
calculated
apparent resistivity
2
electrode spacing
(Wenner, layered half-space)
kernel function: controlled by layer
thicknesses and resistivities.
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Least-squares inversion
Observed data
Minimal
“Error Squares”
Modelled data
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Least-squares inversion
Modelled data
Observed data
sum of (obs – mod)2
is minimal
Uncertainty
(noise, errors)
RMS
The misfit between the observed and modelled (i.e. computer calculated)
data is often expressed in terms of a percent root mean square (%RMS) error
it serves as a measure of how far on average the misfit departs from zero, where the
individual differences (residuals) are mapped into a single number of statistical value.
observed data
•%𝑅𝑀𝑆 = 100 ∙
modelled data
1
𝑁
𝑁
𝑛=1
𝑜𝑏𝑠 − 𝜌𝑚𝑜𝑑
𝜌𝑎
𝑎
𝑛
𝑛
𝑚𝑜𝑑
𝜌𝑎
𝑛
2
n-th data point
number of data
A good result will have a low %RMS, and will be geologically reasonable.
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Caution
• A geologic section differs from a geo-electric section when the
boundaries between geologic layers do not coincide with the boundaries
between layers characterized by different resistivities.
• Thus, the electric boundaries separating layers of different resistivities
may or may not coincide with boundaries separating layers of different
geologic age or different lithologic composition.
• For example, when the salinity of ground water in a given type of rock
varies with depth, several geoelectric layers may be distinguished within
a lithologically homogeneous rock. In the opposite situation layers of
different lithologies or ages,or both, may have the same resistivity and
thus form a single geoelectric layer.
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Inverse algorithm
estimate of
starting model
Each cycle through
the inversion process
is called an iteration
Calculate response
of modelled values
(forward step)
Misfit large
Alter model
parameters so as
to reduce misfit
Compare
observed and
model values —
calculate misfit
Misfit small
Inversion process
is complete:
Output final model
Inverse example
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Inverse example – observed data
Wenner resistivity sounding data from a
palaeochannel at Fingal (South Esk River Valley)
We start with observed (raw) data
~ H-type
Apparent resistivities
(Wenner sounding)
Data point n with
uncertainty un
Software = Rinvert
Inverse example – starting model
•A starting model can be ‘guessed’ from the observed apparent
resistivities  with experience, the depth of interfaces between
layers can be estimated directly from the a against a curve
a at small spacings ~ 600 Wm
~ H-type
600 Wm
a increasing at
large spacings
Estimated
Layer
boundaries
a decreases
to ~ 50 Wm at
mid-spacings
50
Wm
Starting
model for
inversion
1000 Wm
Inverse example – initial response
• An inappropriate starting model can result in the inversion process
failing to converge to a best-fit final model, with a nonsense output
• The response of the starting model should be reasonably close to the
observed data and of the same general shape
Starting model response
Observed data
Inverse example - inversion results
550 Wm / 0.5m
30Wm / 50.5m
>105 Wm
Inverse example - inversion results
 This resistivity soundings has been interpreted in terms of a
layered Earth model (one-dimensional, 1D).
 This is a fast, simple way of determining structure.
 NB: the layered-Earth model carries the assumption that the
subsurface resistivity varies only with depth
 The surface of the earth is assumed to be perfectly horizontal
 On a local scale, the layered earth is a useful approximation to
many common geological situations, e.g. a weathered layer
overlying fresh bedrock, or layers of flat-lying sediments
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References
Zhody, A.A.R., Eaton, G.P., and Mabey, D.R., “Application of surface geophysics to groundwater investigations“, In:
Techniques of Water Resources Investigations, section 2, USGS, 1974, pp. 5—64 ( http://pubs.usgs.gov/twri/twri2d1/html/pdf.html )
Merrick, N.P., and Poezd, E., “RINVERT for Windows Software for the Interpretation of Resistivity Soundings“, 1997, 28,
Exploration Geophysics, pp. 110—113
Ghosh, 1971a, “The Application Of Linear Filter Theory To The Direct Interpretation Of Geoelectrical Resistivity Sounding
Measurements”: Geophysical Prospecting, Volume 19, Issue 2, pp.192–217
Loke M.H.: “Electrical imaging surveys for environmental and engineering studies”, 2000, Lecture Notes
Knödel K., Lange G., Voig H.J.: “Environmental Geology, Handbook of Field Methods and Case Studies”, 2007, Springer
Telford, W.M, Geldart, L.P., Sheriff, R.E.: “Applied Geophysics”, 1991, Cambridge University Press
Reynolds, J.M., "An Introduction to Applied and Environmental Geophysics", 2011, John Wiley & Sons
Lowrie W., Fundamentals of Geophysics, 2007 , Cambridge University Press
Revil A., Karaoulis M., Johnson T. , Kemna A.: “Review: Some low-frequency electrical methods for subsurface
characterization and monitoring in hydrogeology”, Hydrogeology Journal, 2012, 20, pp.617
Philip Kearey, Michael Brooks and Ian Hill, “An Introduction to Geophysical Exploration”, 2002, Blackwell
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