THE ELIMINATION OF THE INFINITE POLE IN THE POLE-DIPOLE
RESISTIVITY ARRAY
A Thesis submitted to The Graduate Studies and Research in partial fulfilment
of the requirements for a degree in Master of Science from the
Department of Geological Sciences,
University of Saskatchewan
Saskatoon, Saskatchewan
By
Xiao Xia Liang
c©Copyright Xiao Xia Liang, December 2015. All rights reserved.
Permission to Use
In submission of this thesis as a partial fulfilment for a postgraduate degree from the Uni-
versity of Saskatchewan, I agree for this University’s libraries to make copies of this thesis
for any academic related reviews or inspections. Due recognition is given to the University
of Saskatchewan and myself.
Any use in part or in full of this thesis for capital gain is not permitted without my written
permission. Academic or scholarly use of this thesis in part or in full will require permission
from the supervising professor, or from the University of Saskatchewan, Geological Sciences
department head.
Head of the Department of Geological Sciences
University of Saskatchewan
114 Science Place
Saskatoon, Saskatchewan
S7N 5E2
i
Abstract
The pole-dipole (PD) electrical resistivity array is used as part of a geophysical technique,
which can be used in exploration for mineral, geothermal and hydrological resources. Fur-
thermore, it can be used in archaeological investigations. The PD array is useful in obtaining
large depths of investigation, but the array requires one pole to be planted at a greater
distance from the other electrodes. Placing the infinite pole is time consuming and costly,
especially in rough terrains. It can also be a liability in populated areas. Previous data in-
version software assumes the infinite pole to be planted at infinity. Therefore, the field data
collected has to mimic the assumptions of the inversion software.
Some recent inversion programs use all 4-electrode positions for data inversion. In this case,
is a good approximation of infinity for the infinite pole still necessary? Is the depth of inves-
tigation still the same or relevant with a non-infinite, infinite pole? Is the resolution of the
cross-sections retained?
To answer the above questions a Generalized Electrode Array (GEA) 1 dimensional (1D)
inversion program is used. Different sets of field data were collected with non-infinite pole
(NIP) PD arrays; these arrays are set up as PD arrays with an infinite pole planted relatively
close to or on the survey lines itself. A COMSOL finite element model were modelled with
varying infinite pole distances to identify the distance required for the retention of the depth
ii
of investigation and resolution of the PD array.
Modelling from GEA shows the NIP distances do not affect the 1D inversion accuracy of
the resistivity nor the layer thickness. The field data indicates that it is necessary to have
an infinite pole, but the distance to the infinite pole can be substantially less than 10 times
the array length, which is the usual rule of thumb for infinite pole placement. With 3 dimen-
sional (3D) COMSOL modelling results, it indicates a minimum pole distance to be 2.5-array
length to retain the depth of investigation and precision of the inverted sections of the PD
array.
iii
Acknowledgements
I would like to thank Cameco for funding this thesis and Dr. Jim Merriam for the opportu-
nity to work with him as his student.
I also would like to thank the following people:
Dr. Sam Butler (University of Saskatchewan)
Jochen Kursawe (Oxford University)
Lloyd Litwin (University of Saskatchewan)
Susan Nehzati (University of Saskatchewan)
Robert Zschuppe
Michael Zhang (University of Saskatchewan)
Lastly, I would like to thank all of my family and friends for their love and support.
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Contents
Permission to Use i
Abstract ii
Acknowledgements iv
List of Figures vii
List of Tables xx
1 Introduction 1
1.1 Thesis and research objective . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Electrical properties of materials . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.3 Basics of Resistivity arrays . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.4 Depth of investigation and sensitivity using the Frechet derivative . . . . . . 13
1.5 Cross-correlation analytics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
1.6 Uranium exploration using PD resistivity array . . . . . . . . . . . . . . . . 17
2 Methods 22
2.1 GEA 1D modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.2 Field data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.3 Geoelectric COMSOL modelling . . . . . . . . . . . . . . . . . . . . . . . . . 26
v
3 Data Analysis and Results 30
3.1 GEA 1D inversions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
3.2 Field data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
3.2.1 Field data inversions . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
3.2.2 Field data sensitivity sections . . . . . . . . . . . . . . . . . . . . . . 41
3.2.3 Field data cross-correlation sections . . . . . . . . . . . . . . . . . . . 46
3.3 Geoeletric COMSOL modelling . . . . . . . . . . . . . . . . . . . . . . . . . 50
3.3.1 COMSOL model inversions . . . . . . . . . . . . . . . . . . . . . . . 50
3.3.2 COMSOL model sensitivity sections . . . . . . . . . . . . . . . . . . . 55
3.3.3 COMSOL model cross-correlation sections . . . . . . . . . . . . . . . 61
4 Conclusion and Discussions 71
4.1 GEA 1D data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
4.2 Field Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
4.3 Geoelectric COMSOL modelling . . . . . . . . . . . . . . . . . . . . . . . . . 72
4.4 Discussions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
References 74
Appendix A Electrode positioning for 1D GEA inversions 79
Appendix B Inverted sections from Field Data 84
Appendix C Sensitivity sections from Field Data 92
Appendix D Cross-correlation sections from Field Data 99
Appendix E Inverted sections from COMSOL modelling 107
Appendix F Sensitivity sections from COMSOL modelling 117
vi
List of Figures
1.3.1 An illustration of a multi-node Wenner-α array system. C1 and C2 are in-
jection electrodes. P1 and P2 are potential electrodes. The red dot is where
the apparent resistivity measured is plotted using the pseudo-depth of unit
electrode spacing depth. The pseudodepth will be discussed in more details
in section 1.4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.3.2 An explanation of apparent resistivity. . . . . . . . . . . . . . . . . . . . . . 7
1.3.3 A comparison of a pseudosection and inverted section. . . . . . . . . . . . . 8
1.3.4 Diagram is showing a current flowing through an object with a cross sectional
area of A and length of l. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.3.5 Diagram is showing a single injection point of current into a homogeneous
half space. The half circles are the equipotential surfaces and the current flow
intercepts the equipotential surfaces at 90 degrees. . . . . . . . . . . . . . . 10
1.3.6 Diagram containing four-electrode array with 2 potential electrodes P1 and
P2 and 2 injection electrodes C1 and C2. The potential measurement between
each injection electrodes and potential electrodes are seen as V1, V2, V3 and V4.
The corresponding distances between the potential and injection electrodes
are denoted by r1, r2, r3 and r4. . . . . . . . . . . . . . . . . . . . . . . . . 10
viii
1.3.7 Most commonly used arrays are outlined with its configuration. The arrays
include: Wenner-α, Wenner- β, pole-pole, pole-dipole, dipole-dipole equato-
rial, dipole-dipole axial. C1 and C2 are current injection electrodes. P1 and
P2 are potential electrodes. . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.4.1 DOI values calculated using Butler’s (2015) Zmean equation. The array length
is normalized to 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
1.6.1 Map of the Athabasca Basin in northern Saskatchewan. Athabasca Basin is
outlined with the solid orange line. Seen here are some of the unconformity-
type uranium depositions. This map is taken from Overview of Cameco Ex-
ploration Athabasca Basin 2011, presentation by John Halaburda. . . . . . 18
1.6.2 A map of the Dawn Lake area in northern Saskatchewan. The PD electric
resistivity survey lines can be seen on the map. Each survey line is 1400
metres long. The electrode spacing is 25 metres. The data used to invert
Figure 1.6.3 is from line L15E seen in red. This map is modified from the
map provided by Cameco. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
1.6.3 Inverted cross section of PD resistivity survey of uranium ore deposit in
northern Saskatchewan. Part of the ore deposit is seen at the bottom of the
section. A fault can be seen at the 1000 metres mark diagonally going down
to the 1300 metres mark. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.1.1 Diagram of infinite one-dimensional GEA forward model. The model con-
tains two layers; the first layer is one meter and two meters thick with the
conductive models having 100Ωm in the top layer and the bottom contain-
ing infinite thickness with a 10Ωm resistivity. In reverse, the resistive models
have a 100Ωm top layer and a 10Ωm bottom layer. C1 and C2 are injection
electrodes. C2 is the infinite pole. P1 and P2 are potential electrodes. . . . . 23
ix
2.1.2 An example of a GEA inversion process. This is a conductive model with
one-meter upper layer thickness. The top layer is 10Ωm and the lower layer
is 100Ωm. In this graph, the circular dots are the forward model with added
five percent noise. The hollow dots are points eliminated due to large error
for inversion. The equivalent models can be seen in the dotted lines in the box
on the bottom right corner. The equivalent models are calculated to estimate
inversion error. The solid blue line in the box is the final inversion prediction.
The blue line across the full graph is the final predicted curve. The thickness
prediction is at 1.1m, the top resistivity is predicted to be 9.9Ωm and the
bottom resistivity is predicted to be 129.5Ωm. The data error estimate is at
5.6 percent. The predicted thickness and resistivity are summarized for each
of GEA models in Figure 3.1.1 to Figure 3.1.8. . . . . . . . . . . . . . . . . 24
2.2.1 Illustration of buried galvanized steel sheet and open pit at the U of S seismic
station. Field data is collected adjacent to buried steel sheet and open pit. 25
2.3.1 An example of a COMSOL model used to simulate PD resistivity data. There
are a total of 30 electrodes in this multi-node system. The electrode spacing
is 30 metres. The infinite pole is located 8.7 kilometres away from the end
of the array. The NIPs are located at 4.350km, 3.480km, 2.610km, 2.175km,
1.740km, 1.305km, 870m and 30m away from the end of the array. A 100-
metre cubed conductor is at a depth of 100 metres below the centre of the
array. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
3.1.1 GEA inverted depths with varying infinite pole distances. This is a conductive
model with a top layer thickness of one metre. The top resistivity is 10Ωm
and bottom layer is 100Ωm. Five percent noise was added to the modelled
forward resistivity before inversion. . . . . . . . . . . . . . . . . . . . . . . . 32
x
3.1.2 GEA inverted resistivity of conductive model with top layer thickness of one
metre. The model contains a top layer resistivity of 10Ωm and bottom layer
of 100Ωm. Five percent noise was added to the modelled forward resistivity
before inversion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.1.3 GEA inverted depths with varying infinite pole distances. This is a conductive
model with a top layer thickness of two metres. The top resistivity is 10Ωm
and bottom layer is 100Ωm. Five percent noise was added to the modelled
forward resistivity before inversion. . . . . . . . . . . . . . . . . . . . . . . . 34
3.1.4 GEA inverted resistivity of conductive model with top layer thickness of two
metres. The model contains a top layer resistivity of 10Ωm and bottom layer
of 100Ωm. Five percent noise was added to the modelled forward resistivity
before inversion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.1.5 GEA inverted depths with varying infinite pole distances. This is a conductive
model with a top layer thickness of one metre. The top resistivity is 100Ωm
and bottom layer is 10Ωm. Five percent noise was added to the modelled
forward resistivity before inversion. . . . . . . . . . . . . . . . . . . . . . . . 36
3.1.6 GEA inverted resistivity of resistive model with top layer thickness of one
metre. The model contains a top layer resistivity of 100Ωm and bottom layer
of 10Ωm. Five percent noise was added to modelled forward resistivity before
inversion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.1.7 GEA inverted depths with varying infinite pole distances. This is a resistive
model with a top layer thickness of two metres. The top resistivity is 100Ωm
and bottom layer is 10Ωm. Five percent noise was added to the modelled
forward resistivity before inversion. . . . . . . . . . . . . . . . . . . . . . . . 38
xi
3.1.8 GEA inverted resistivity of resistive model with top layer thickness of two
metres. The model contains a top layer resistivity of 100Ωm and bottom layer
of 10Ωm. Five percent noise was added to the modelled forward resistivity
before inversion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.2.1.1 Traditional PD cross-section inverted using RES2DINV. The array is 28 me-
tres long with one metre electrode spacing. The infinite pole is located 300
metres away from the end of the array. . . . . . . . . . . . . . . . . . . . . . 42
3.2.1.2 NIP array with pole distance at 100 metres. The array is 28 metres long with
one metre electrode spacing. . . . . . . . . . . . . . . . . . . . . . . . . . . 43
3.2.1.3 NIP array with pole distance at 50 metres. The array is 28 metres long with
one metre electrode spacing. . . . . . . . . . . . . . . . . . . . . . . . . . . 44
3.2.1.4 NIP PD array at 1 metre. The array is 28 metres long with one metre elec-
trode spacing. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.2.2.1 The bottom sensitivity section is calculated from NIP PD inverted field data.
The NIP is located at 100 metres away from the end of the array. The top
sensitivity section is the traditional PD inverted field data. . . . . . . . . . 47
3.2.2.2 The bottom sensitivity section is calculated from NIP PD inverted field data.
The NIP is located at 50 metres away from the end of the array. The top
sensitivity section is from the traditional PD inverted field data. . . . . . . 48
3.2.2.3 The bottom sensitivity section is calculated from NIP PD inverted field data.
The NIP is located at one metre away from the end of the array. The top
sensitivity section is from the traditional PD inverted field data. . . . . . . 49
3.2.3.1 Cross-correlation control graph for the traditional PD array. The field data
collected using infinite pole at 300 metres was inverted twice on RES2DINV.
The two sets of inverted data are cross-correlated. . . . . . . . . . . . . . . 51
xii
3.2.3.2 Cross-correlation image of the traditional PD inverted data and NIP inverted
data with pole distance at 100 metres. . . . . . . . . . . . . . . . . . . . . 52
3.2.3.3 Cross-correlation image of the traditional PD inverted data and NIP inverted
data with pole distance at 50 metres. . . . . . . . . . . . . . . . . . . . . . 53
3.2.3.4 Cross-correlation image of the traditional PD inverted data and NIP inverted
data with pole distance at 1 metre. . . . . . . . . . . . . . . . . . . . . . . 54
3.3.1.1 Traditional PD inverted section from COMSOL model. The infinite pole is
located 10 times the array length or at 8.7 kilometres away from the survey
line. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
3.3.1.2 NIP inverted section from COMSOL model with pole distance at 4.350 kilo-
metres or 5 times the array length. . . . . . . . . . . . . . . . . . . . . . . . 57
3.3.1.3 NIP inverted section from COMSOL model with pole distance at 2.175 kilo-
metres or 2.5 times the array length. . . . . . . . . . . . . . . . . . . . . . 58
3.3.1.4 NIP inverted section from COMSOL model with pole distance at 1.305 kilo-
metres or 1.5 times the array length. . . . . . . . . . . . . . . . . . . . . . . 59
3.3.1.5 NIP inverted section from COMSOL model with pole distance at 30 metres
or one electrode spacing away from the end of survey line. . . . . . . . . . . 60
3.3.2.1 Sensitivity section calculated from COMSOL model inverted data. The top
sensitivity section is from the traditional PD. This is shown here for com-
parison. The bottom is a NIP PD with pole distance at 4.350 kilometres or
5 times the array length. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
3.3.2.2 Sensitivity sections calculated from COMSOL model inverted data. The top
sensitivity section is calculated from the traditional PD. It is shown here for
comparison. The bottom section is a NIP PD with pole distance at 2.175
kilometres or 2.5 times the array length. . . . . . . . . . . . . . . . . . . . . 63
xiii
3.3.2.3 Sensitivity sections calculated from COMSOL model inverted data. The top
sensitivity section is calculated from the traditional PD. It is shown here for
comparison. The bottom section is a NIP PD with pole distance at 1.305
kilometres or 1.5 times the array length. . . . . . . . . . . . . . . . . . . . . 64
3.3.2.4 Sensitivity sections calculated from COMSOL model inverted data. The top
sensitivity section is calculated from the traditional PD. It is shown here for
comparison. The bottom section is a NIP PD with pole distance at 30 metres
or one electrode spacing from the end of the array. . . . . . . . . . . . . . . 65
3.3.3.1 Cross-correlated section made from data inversion control. This section was
cross-correlated using the traditional PD data computed from COMSOL.
The traditional PD data set was inverted twice in RES2DINV to obtain two
sets of data. These two sets of data are then compared for similarities. The
similarities are in red or a scale of one and the differences can be seen in blue.
Due to the non-uniqueness of the inversion process, differences were noted in
the two inverted data sets, although they came from the same forward data. 66
3.3.3.2 Cross-correlated section between inverted traditional PD data and NIP in-
verted data with pole distance at 4.35 kilometres or 5 times the array length. 67
3.3.3.3 Cross-correlated section between inverted traditional PD data and NIP in-
verted data with pole distance at 2.175 kilometres or 2.5 times the array
length. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
3.3.3.4 Cross-correlated section between inverted traditional PD data and NIP in-
verted data with pole distance at 1.74 kilometres or 2 times the array length. 69
3.3.3.5 Cross-correlated section between inverted traditional PD data and NIP in-
verted data with pole distance 30 metres or one electrode spacing from the
end of the survey line. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
xiv
B.1 Traditional PD cross-section inverted using RES2DINV. The array is 28 me-
tres long with 1 metre electrode spacing. The infinite pole is located 300
metres away from the end of the array. . . . . . . . . . . . . . . . . . . . . . 85
B.2 NIP array with pole distance at 200 metres. The array is 28 metres long with
1 metre electrode spacing. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
B.3 NIP array with pole distance at 100 metres. The array is 28 metres long with
1 metre electrode spacing. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
B.4 NIP array with pole distance at 50 metres. The array is 28 metres long with
1 metre electrode spacing. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
B.5 NIP array with pole distance at 25 metres. The array is 28 metres long with
1 metre electrode spacing. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
B.6 NIP array with pole distance at 10 metres. The array is 28 metres long with
1 metre electrode spacing. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
B.7 PD array with cheater pole at 1 metre. The array is 28 metres long with 1
metre electrode spacing. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
C.1 The bottom sensitivity section is calculated from NIP PD inverted field data.
The NIP is located at 200 metres away from the end of array. The top sen-
sitivity section is the traditional PD inverted field data. . . . . . . . . . . . 93
C.2 The bottom sensitivity section is calculated from NIP PD inverted field data.
The NIP is located at 100 metres away from the end of array. The top sen-
sitivity section is the traditional PD inverted field data. . . . . . . . . . . . 94
C.3 The bottom sensitivity section is calculated from NIP PD inverted field data.
The NIP is located at 50 metres away from the end of array. The top sensi-
tivity section is from the traditional PD inverted field data. . . . . . . . . . 95
xv
C.4 The bottom sensitivity section is calculated from NIP PD inverted field data.
The NIP is located at 25 metres away from the end of array. The top sensi-
tivity section is from the traditional PD inverted field data. . . . . . . . . . 96
C.5 The bottom sensitivity section is calculated from NIP PD inverted field data.
The NIP is located at 10 metres away from the end of array. The top sensi-
tivity section is from the traditional PD inverted field data. . . . . . . . . . 97
C.6 The bottom sensitivity section is calculated from NIP PD inverted field data.
The NIP is located at 1 metre away from the end of array. The top sensitivity
section is from the traditional PD inverted field data. . . . . . . . . . . . . 98
D.1 Cross-correlation control graph for the traditional PD array. The field data
collected using infinite pole at 300 metres was inverted twice on RES2DINV.
The 2 sets of inverted data are cross-correlated. . . . . . . . . . . . . . . . 100
D.2 Cross-correlation image of the traditional PD inverted data and the NIP
inverted data with pole distance at 200 metres. . . . . . . . . . . . . . . . . 101
D.3 Cross-correlation image of the traditional PD inverted data and the NIP
inverted data with pole distance at 100 metres. . . . . . . . . . . . . . . . . 102
D.4 Cross-correlation image of the traditional PD inverted data and the NIP
inverted data with pole distance at 50 metres. . . . . . . . . . . . . . . . . 103
D.5 Cross-correlation image of the traditional PD inverted data and the NIP
inverted data with pole distance at 25 metres. . . . . . . . . . . . . . . . . 104
D.6 Cross-correlation image of the traditional PD inverted data and the NIP
inverted data with pole distance at 10 metres. . . . . . . . . . . . . . . . . 105
D.7 Cross-correlation image of the traditional PD inverted data and NIP inverted
data with pole distance at 1 metre. . . . . . . . . . . . . . . . . . . . . . . 106
xvi
E.1 Traditional PD inverted section from COMSOL model. The infinite pole is
located 10 times the array length or at 8.700 kilometres away from the survey
line. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
E.2 NIP inverted section from COMSOL model with pole distance at 4.350 kilo-
metres or 5 times the array length. . . . . . . . . . . . . . . . . . . . . . . . 109
E.3 NIP inverted section from COMSOL model with pole distance at 3.480 kilo-
metres or 4 times the array length. . . . . . . . . . . . . . . . . . . . . . . . 110
E.4 NIP inverted section from COMSOL model with pole distance at 2.610 kilo-
metres or 3 times the array length. . . . . . . . . . . . . . . . . . . . . . . . 111
E.5 NIP inverted section from COMSOL model with pole distance at 2.175 kilo-
metres or 2.5 times the array length. . . . . . . . . . . . . . . . . . . . . . 112
E.6 NIP inverted section from COMSOL model with pole distance at 1.740 kilo-
metres or 2 times the array length. . . . . . . . . . . . . . . . . . . . . . . . 113
E.7 NIP inverted section from COMSOL model with pole distance at 1.305 kilo-
metres or 1.5 times the array length. . . . . . . . . . . . . . . . . . . . . . . 114
E.8 NIP inverted section from COMSOL model with pole distance at 870 metres
or one array length. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
E.9 NIP inverted section from COMSOL model with pole distance at 30 metres
or one electrode spacing away from the end of survey line. . . . . . . . . . . 116
F.1 Sensitivity section calculated from COMSOL model inverted data. The top
sensitivity section is from the traditional PD. The bottom is a NIP PD with
pole distance at 4.350 kilometres or 5 times the array length. . . . . . . . . 118
F.2 Sensitivity section calculated from COMSOL model inverted data. The top
sensitivity section is from the traditional PD. The bottom is a NIP PD with
pole distance at 3.480 kilometres or 4 times the array length. . . . . . . . . 119
xvii
F.3 Sensitivity section calculated from COMSOL model inverted data. The top
sensitivity section is from the traditional PD. The bottom is a NIP PD with
pole distance at 2.610 kilometres or 3 times the array length. . . . . . . . . 120
F.4 Sensitivity sections calculated from COMSOL model inverted data. The top
sensitivity section is calculated from the traditional PD. The bottom section
is a NIP PD with pole distance at 2.175 kilometres or 2.5 times the array . 121
F.5 Sensitivity section calculated from COMSOL model inverted data. The top
sensitivity section is from the traditional PD. The bottom is a NIP PD with
pole distance at 1.740 kilometres or 2 times the array length. . . . . . . . . 122
F.6 Sensitivity sections calculated from COMSOL model inverted data. The top
sensitivity section is calculated from the traditional PD. The bottom section
is a NIP PD with pole distance at 1.305 metres or 1.5 times the array . . . 123
F.7 Sensitivity section calculated from COMSOL model inverted data. The top
sensitivity section is from the traditional PD. The bottom is a NIP PD with
pole distance at 870 metres or one array length. . . . . . . . . . . . . . . . 124
F.8 Sensitivity sections calculated from COMSOL model inverted data. The top
sensitivity section is calculated from the traditional PD. The bottom section
is a NIP PD with pole distance at 30 metres or one electrode spacing from
the end of the array. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
xviii
G.1 Cross-correlated section made from data inversion control. This section was
cross-correlated using the traditional PD data computed from COMSOL.
The traditional PD data set was inverted twice in RES2DINV to obtain 2
sets of data. These 2 sets of data are compared to themselves for similarities.
The similarities are in red or a scale of one and the differences can be seen
in blue. Although the 2 inverted data sets are from the same forward data,
but there are still differences in the CC. This is due to the non-uniqueness of
the inversion process. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
G.2 Cross-correlated section between inverted traditional PD data and NIP in-
verted data with pole distance at 4.350 kilometres or 5 times the array length.128
G.3 Cross-correlated section between inverted traditional PD data and NIP in-
verted data with pole distance at 3.480 kilometres or 4 times the array length.129
G.4 Cross-correlated section between inverted traditional PD data and NIP in-
verted data with pole distance at 2.610 kilometres or 3 times the array length.130
G.5 Cross-correlated section between inverted traditional PD data and NIP in-
verted data with pole distance at 2.175 kilometres or 2.5 times the array
length. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
G.6 Cross-correlated section between inverted traditional PD data and NIP in-
verted data with pole distance at 1.740 kilometres or 2 times the array length.132
G.7 Cross-correlated section between inverted traditional PD data and NIP in-
verted data with pole distance at 1.305 kilometres or 1.5 times the array
length. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
G.8 Cross-correlated section between inverted traditional PD data and NIP in-
verted data with pole distance at 870 metres or one array length. . . . . . . 134
xix
G.9 Cross-correlated section between inverted traditional PD data and NIP in-
verted data with pole distance at 30 metres or one electrode spacing from
the end of survey line. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
xx
List of Tables
1.4.1 Median depths for Pole-Dipole resistivity arrays . . . . . . . . . . . . . . . . 14
A.1 GEA electrodes positioning . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
xxi
Chapter 1
Introduction
Electrical resistivity has been used for prospecting since the early 1900’s. Conrad Schlum-
berger and other pioneers were the first to use the direct current methods. Frank Wenner
developed a method to quantify resistivity. Wenner is considered to be the father of quan-
titative resistivity methods (Van Nostrand and Cook, 1984). The naming of Schlumberger
array and Wenner array are after Conrad Schlumberger and Frank Wenner.
Modern day electric resistivity can be used in a multitude of situations. Some examples are
archaeological investigations (Candansayar and Basokur, 2001; Candansayar, 2008), geother-
mal exploration (Longuevergne et al., 2009), hydrological exploration (Maxwell et al., 2014)
and mineral exploration (Sharma et al., 2014).
1.1 Thesis and research objective
The main objective of this thesis is to examine the infinite pole of the PD array to determine
a practical limit for the pole distance. It is in the best interest of the workers to have prac-
1
tical limits instead of placing the pole at ten times the array length, as placing the infinite
pole is time consuming and costly, especially in rough terrains. It can also be a liability in
populated areas. In this thesis, the minimum distance of an infinite pole that is needed to
retain the depth of investigation and relative precision of the inverted sections compared to
the traditional PD inverted sections will be determined.
The effects of the infinite pole in the PD array is relatively small; that effect is proportional
to the maximum r1 (C1 to P1) distance over the r2 (C2 to P1) distance squared. At infinite
pole distance of 5 array lengths, the effects of neglecting the infinite pole is about 4 percent.
The exact error will depend on the position of the infinite pole and the subsurface resistivity
distribution (Loke, 2001; 2004). By reducing the infinite pole distance to half of the rule of
thumb distance the error remains below 5 percent; this already brings the pole to a more
reasonable distance, while keeping an acceptable error.
Early mathematical models for PD resistivity inversions assume the infinite pole distance to
be at good approximation to infinity, which greatly simplifies the inversions process. This
approximation presented the advantage of only having to considered 3 electrodes. With mod-
ern programs, like RES2D and GEA, which can explicitly account for all locations of the 4
electrodes in an array, there is no need to have an approximation to infinity to simplify the
inversion process.
Considering the capabilities of modern inversion programs, is a good approximation of infin-
ity for the infinite pole still necessary? Will the depth of investigation remain adequate with
a non-infinite, infinite pole? Will it retain the resolution of the cross-sections? To answer the
above questions, GEA models, field data and COMSOL models are used.
2
This thesis will have an emphasis on mineral exploration, specifically uranium exploration,
using electric resistivity methods. The COMSOL model is modelled after an ore deposit.
1.2 Electrical properties of materials
There are three primary ways of direct current (DC) conduction: electronic, electrolytic and
dielectric (Telford et al., 1990). The two main modes for Earth materials are electronic con-
duction and electrolytic conduction (Loke, 2001; 2004). Electronic conduction is a current
flow through free electrons in materials such as metals or graphites. Electrolytic conduction
is the flow of current through ions in pore water.
In general, rocks with less porosity and little to no water content are more resistive; hence
the electrical conductivity will be low. Thus, metamorphic and igneous rocks are more resis-
tive and sedimentary rocks are more conductive (Loke, 2001; 2004). Sedimentary rocks are
more porous which allows greater water content; these 2 factors can greatly influence the
conductivity of rocks. Other factors such as the amount of ions dissolved in water and the
shape of the pores can also alter the resistivity of rocks.
Archie’s empirical formula, ρr=aφ−mS−nρw, relates the bulk resistivity of the rock, ρr, to the
porosity, φ, and the resistivity of the water, ρw. Where S is the fraction of pores containing
water, a and m are constants (Archie, 1950).
3
1.3 Basics of Resistivity arrays
Resistivity is a geophysical exploration technique developed to measure the Earth’s resistiv-
ity. In this thesis the focus will be on the Direct Current (DC) electric resistivity techniques.
The DC electric technique employs a time varying direct current with frequencies low enough
that it behaves like a DC current (Zhdanov, 2009). As a result, the depth of investigation
(DOI) is controlled by the geometry of the array and not by the frequency. In section 1.4,
the DOI will be discussed in more detail.
The three common groups of DC electric methods are vertical electric sounding (VES), hori-
zontal profiling and electrical mapping. VES assumes one dimension for geoelectric structure
in the area of exploration, where resistivity only varies with depth. The VES surveys are
carried out by using a fixed mid-point with increasing electrode spacing to ‘sense’ deeper
into the ground. Horizontal profiling assumes the resistivity only changes laterally. The array
with fixed electrodes are moved along a profile line for data collection. Electrical mapping
is used to map the spatial behaviour of a DC field in a given area of interest; current is
introduced into the ground and the resulting potential in the area is mapped. The surveys
can be carried out along a profile line with varying electrode spacing.
The basic idea of using a resistivity survey is as follows: using 4 electrodes, two are injection
electrodes, C1 and C2, which inject current into the ground and two are potential electrodes,
P1 and P2, which measure the resulting potential.
Multi-node array systems are becoming more common as this system is a less labour in-
tensive way to collect data. In a multi-node system, electrodes do not need to be manually
moved to a new position after each injection. This can save time and reduce human error. A
4
Figure 1.3.1: An illustration of a multi-node Wenner-α array system. C1 and C2 are injectionelectrodes. P1 and P2 are potential electrodes. The red dot is where the apparent resistivitymeasured is plotted using the pseudo-depth of unit electrode spacing depth. The pseudodepthwill be discussed in more details in section 1.4.
multi-node system contains many electrodes and is usually co-linear in arrangements (Figure
1.3.1).Only four electrodes are active at each injection: two as injectors and two as receivers.
The measured potentials are collected and the apparent resistivity is calculated with the ge-
ometric factor k (equation 1.3.12). The general equation for calculating apparent resistivity
is ρa=k∆V /I. Where ∆V is the change in potential, I is the current, ρa is the apparent
resistivity and k is the geometric factor. The geometric factor k is a very important variable
that controls the DOI. As mentioned above, the array’s DOI is controlled by its geometry,
and this geometry (electrode spacing and position) is used to calculate the geometric factor
k (equation 1.3.12).
The apparent resistivity is numerically the resistivity of homogeneous isotropic ground that
will give the same value as a field measured resistivity with the same array configuration
(Figure 1.3.2). This is not a true resistivity, but rather an overall resistivity of a current’s
5
travel path. Most instruments will measure the resulting potential of the current’s travel
path and the apparent resistivity is calculated from the measured potential.
A graph of apparent resistivity with pseudodepth and horizontal position is called a pseu-
dosection; a blurred and distorted image of the actual resistivity in the subsurface. The
pseudosection can be used to eliminate bad data points and also as an overview of what the
inverted cross-section would represent. However, the pseudosection is difficult to interpret
because of the blurring and distortion. An example of a pseudosection and its related in-
verted section is shown in Figure 1.3.3. The top cross-section is the pseudosection and the
bottom is the inverted section. The two sections are very different. The conductor (galvanized
metal sheet) located between 9 to 15 meters, and the resistor (open pit) located between 23
to 25 meters are not apparent in the top apparent resistivity section. There are no visible
indications of the conductor or the resistor in the pseudosection.
6
Figure 1.3.2: An explanation of apparent resistivity.
To derive the apparent resistivity equation above, a current, I, flowing through a wire with
a resistance R:
R =V
I(1.3.1)
causes a potential drop V .
Consider a direct current with uniform current density flowing through an object with a
cross sectional area A, length l, and resistivity of ρ (Figure 1.3.4). The resistance is:
R =ρL
A(1.3.2)
7
Figure 1.3.4: Diagram is showing a current flowing through an object with a cross sectionalarea of A and length of l.
We will now derive the resistivity of a point current source through a homogeneous half space.
Figure 1.3.5 shows an injection point with propagation of current and equipotential through
a homogeneous half space. We will assume spherical symmetry of current propagation. The
incremental distance of current propagation is dr and the area is A=2πr2. The incremental
resistance is:
dR =ρdr
2πr2(1.3.3)
Equating equations 1.3.1 and 1.3.3:
dV
I=
ρdr
2πr2(1.3.4)
Integrating equation 1.3.4 from r to ∞ and solving for ρ:
ρ = 2πr∆V
I(1.3.5)
9
Figure 1.3.5 shows the situation for a single electrode with a geometric factor of k =2πr. This
equation can be expanded to incorporate any 4-electrode arrangements. In Figure 1.3.6 the
diagram shows one possible electrode configuration. It shows V1 as the measured potential
between C1 and P1. V2 is the measured potential between C2 and P1. V3 is the measured
potential between C1 and P2. V4 is the measured potential between C2 and P2. While r1 is
the distance between C1 and P1. r2 is the distance between C2 and P1. r3 is the distance
between C1 and P2. r4 is the distance between P2 and C2.
Figure 1.3.5: Diagram is showing a single injection point of current into a homogeneous halfspace. The half circles are the equipotential surfaces and the current flow intercepts theequipotential surfaces at 90 degrees.
Figure 1.3.6: Diagram containing four-electrode array with 2 potential electrodes P1 andP2 and 2 injection electrodes C1 and C2. The potential measurement between each injectionelectrodes and potential electrodes are seen as V1, V2, V3 and V4. The corresponding distancesbetween the potential and injection electrodes are denoted by r1, r2, r3 and r4.
10
The potentials can be written as:
V1 =Iρ
2πr1(1.3.6)
V2 =Iρ
2πr2(1.3.7)
V3 =Iρ
2πr3(1.3.8)
V4 =Iρ
2πr4(1.3.9)
The total potential in VP1 = V1 +V2 and the total potential for VP2 = V3 +V4. The potential
difference between P1 and P2 is:
∆V = (V1 + (−V2))− (V3 + (−V4)) =Iρ
2π[(
1
r1− 1
r2)− (
1
r3− 1
r4)] (1.3.10)
Solving for ρ:
ρ =∆V
I
2π(1r1− 1
r2
)−(
1r3− 1
r4
) (1.3.11)
The general geometric factor that fits any configurations of electrodes including non-colinear
arrays is:
11
k =2π(
1r1− 1
r2
)−(
1r3− 1
r4
) (1.3.12)
Many configurations of electrodes placements are possible and each configuration of elec-
trodes is known as an array. Some of the more common arrays (Figure 1.3.7) are Wenner-α,
Wenner-β, pole-pole, PD, dipole axial and dipole equatorial. A total of 102 geoelectric arrays
were published as of 2008 (Szalai et al., 2011).
Figure 1.3.7: Most commonly used arrays are outlined with its configuration. The arraysinclude: Wenner-α, Wenner- β, pole-pole, pole-dipole, dipole-dipole equatorial, dipole-dipoleaxial. C1 and C2 are current injection electrodes. P1 and P2 are potential electrodes.
12
1.4 Depth of investigation and sensitivity using the
Frechet derivative
The depth of investigation (DOI) is a parameter that is used in geoelectric arrays. As men-
tioned in section 1.3, the DOI is controlled by the geometry of an array. Therefore, electrode
spacing and positioning are the major factors in controlling an array’s DOI. Hence, each
array has a specific DOI that is unique to it. To determine an array’s DOI, the sensitivity
function or the Frechet derivative is used; this function is also known as the depth of inves-
tigation characteristics (DIC).
While Evjen (1938) was the first to attempt to use the spatial distribution of current at
depths to determine the DOI, Roy and Apparao in 1971 were the first to define the DIC
function. This is used to determine the maximum depth a given array can detect a signal
from a thin sheet conductor in homogeneous grounds. This DIC function is also applicable
to inhomogeneous grounds. Roy (1974) later extended his DIC function to layered models.
Edwards (1977) suggested the use of median depth instead of the maximum depth for the
DIC function. He found the use of the median depth was a better match for his field results.
Loke uses the median depth in his 2 dimensional (2D) and 3D inverse resistivity programs
(Loke, 2001; 2004). A unique set of depth coefficients can be empirically derived for any
array. The coefficient values for PD array with one electrode at infinity can be seen on Table
1.4.1.
The above table outlines some PD array pseudodepths using effective depth coefficients. Ze
is the array’s coefficient, a is the electrode spacing, n is the integer multiple of a. Here a is 1
metre. For example, if a is 25 metres and n is 1, the median depth for this point is at 12.975m.
13
Table 1.4.1: Median depths for Pole-Dipole resistivity arrays
n Ze/a
1 0.5192 0.9253 1.3184 1.7065 2.0936 2.4787 2.8638 3.2479 3.63210 4.015
As mentioned above, each array has a specific DOI that is unique. Different arrays can have
different horizontal and vertical resolution with its DOI.
Roy and Apparao (1971) determined the pole-pole array to have the largest depth of inves-
tigation and the Wenner array to have the smallest depth of investigation. It is noted that
the large depth of investigation obtained by the pole-pole array has low vertical resolution.
Oldenburg and Li (1999) used their cross-correlation algorithm to evaluate the DOI of spe-
cific arrays. With known levels of noise in generated forward models, they deployed the
pole-pole, dipole-dipole and PD (right and left infinite pole) array to compare its relative
DOI curves using cross-correlation with the inverted array sections. They found the pole-pole
array to have the largest DOI curve. The dipole-dipole array has the smallest DOI curve.
The pole-dipole array has a good intermediate depth. It should also be noted that the PD
array’s DOI curve is showing less sensitivity to near surface features.
Szalai et al., (2011) using 2D and 1D forward models found the PD array has one of the
best depths of detection compared to other common arrays. The depth of detection is the
14
depth of investigation where the signal strength is above the noise level present. In other
words, the depth of detection is dependent on an array’s geometry and the level of signal to
noise strength. All data above the depth of detection will be useful data; whereas the depth
of investigation does not consider the noise present. Therefore, an array can contain a large
DOI but have little to no useful data. To compare depth of detection values Szalai et al.,
(2011) carried out several in-line and broadside geoelectric arrays in resistive and conductive
models. There are three models in total and each model contains a conductive and resistive
mode for the target. The conductive 1D model is a thin sheet with resistivity of 50Ωm and
surrounding to be 100Ωm. The resistive 1D model contains a thin sheet with resistivity of
200Ωm and surrounding to be 100Ωm. Two 2D models were used; one model is a 2-metre
square prism and the other is a 2-metre wide dyke. The conductive model of the prism is
10Ωm and the resistive model of the prism is 1000Ωm. The conductive dyke is 50Ωm and
the resistive dyke is 200Ωm. Five and ten percent noise was added to each of the conduc-
tive and resistive models. Using these models Szalai et al., (2011) found the PD array to
have an overall best result for depth of detection for models with five and ten percent noise
added. The second best result is from the Dipole axial array and the worst results are from
a Wenner-α array. Apparao et al., (1992, 1997) have also done similar geoelectric modelling
as Szalai et al., (2011).
Until recently, the mean of the sensitivity function for pseudodepth plotting was never con-
sidered. Butler (2015, in press) found using the mean instead of the median of the sensitivity
function for plotting the pseudodepth maps the apparent resistivity at a more accurate
depth. This will give a better initial parameter for data inversion and can potentially reduce
inversion time and error.
In figure 1.4.1 the DOI for the PD array with varying infinite pole distances in homogeneous
15
half-space was calculated using Butler’s (2015) Zmean equation. The array length is normal-
ized and the electrode spacing of 0.10, 0.15, 0.20, 0.25, and 0.30 was used. The DOI has an
inverse relationship with the electrode spacing. At infinite pole distance of one array length,
the electrode configuration is a Schlumberger array. The DOI increases to the maximum
depth sensitivity when the infinite pole is at two array length. The DOI levels off around 20
array length.
These values are pseudodepths and can be useful to a labourer planning a field survey. If the
rough estimate of the depth for an area of interest is known, a labourer can use this infor-
mation to plan the electrode spacing and array length, as well as the infinite pole distance
if using the PD array.
Figure 1.4.1: DOI values calculated using Butler’s (2015) Zmean equation. The array lengthis normalized to 1.
16
1.5 Cross-correlation analytics
Cross-correlation (CC) is a signal processing method used to measure the similarities be-
tween two signals. This is also known as the sliding inner dot product. CC comparisons can
be carried out with inverted resistivity data to analyze the similarities between 2 inverted
resistivity sections.
CC =
∑n1 (TPD − TPD)(NIP −NIP )√∑n
1 (TPD − TPD)2∑n
1 (NIP −NIP )2(1.5.1)
CC = Cross-correlation
TPD = Inverted traditional PD array data
TPD = Average of the traditional PD array data
NIP = Non-inifinite pole PD array data
NIP = Average of non-infinite pole PD array data
In this thesis, a CC analytical comparison of the traditional PD and NIP PD arrays are
done to compare the similarities between the inverted sections. This is done to observe the
inversion process of RES2DINV if a NIP PD array is inverted as a traditional PD array. At
what distance is an infinite pole sufficiently large to show little-to-no significant change in
the inverted section. CC is used in the field data and the COMSOL model data. The CC
equation (Equation 1.5.1) presented is from Oldenburg and Li (1999).
1.6 Uranium exploration using PD resistivity array
The northern Saskatchewan Athabasca Basin contains mostly unconformity-type uranium
deposits (Figure 1.6.1). A few other types of uranium deposits exist in the Athabasca basin.
17
These exceptions include Charlebois Lake which is of ultrametamorphic-type and Beaver-
lodge district which is of classical vein-type (Nash et al., 1981).
Figure 1.6.1: Map of the Athabasca Basin in northern Saskatchewan. Athabasca Basin isoutlined with the solid orange line. Seen here are some of the unconformity-type uraniumdepositions. This map is taken from Overview of Cameco Exploration Athabasca Basin 2011,presentation by John Halaburda.
Unconformity-type deposits are favourable to form under these specific first-order attributes:
conglomeratic sandstones as host rock, age between late Paleo-Proterozoic to Meso-Proterozoic,
fluvial, flat laying, intra-continental, have not gone through metamorphism and have a dia-
genetic redbed sequence (Jefferson et al., 2007). Some, like Spirakis (1996), have suggested
organic matter may also play a role in sedimentary uranium deposits.
The unconformity-type uranium deposits in northern Saskatchewan are amongst the high-
grade uranium resources around the world (Nash et al., 1981; Hoeve and Thomas 1978;
18
Jefferson et al., 2007, Chi et al., 2013). According to the World Nuclear Association, the
northern Saskatchewan mines were producing one third of the world’s uranium in 2001. This
number decreased to 23 percent by 2007.
The Wollaston group, which includes Dawn Lake, Rabbit Lake, Key Lake and other uranium
deposits, contains a basal graphitic sedimentary layer (Hoeve and Sibbald, 1978; Nash et al.,
1981; Jefferson et al., 2007; Pascal, 2014). These graphitic layers are strong electromagnetic
conductors. To detect and locate these conductors, geophysical electromagnetic techniques
could be used. One of these techniques is the geoelectric resistivity survey.
The unconformity-type deposits can sometimes be hundreds of metres deep, with faulting
features. In these cases, a PD array is best used for the survey, as the PD array has more
vertical sensitivity and a larger depth of investigation (Loke, 2001; 2004; Szalai et al., 2011).
An example of uranium exploration using PD resistivity is shown below.
Figure 1.6.2 is a map of the Dawn Lake area in northern Saskatchewan. In this map, PD
resistivity survey lines can be seen. Line L15E’s data has been inverted and displayed in
Figure 1.6.3. The survey lines are 1400 metres long and 200 metres apart. The electrode
spacing is 25 metres. The geological faults are outlined in the orange dotted lines.
Figure 1.6.3 is an example of a survey done using a PD array over an unconformity-type
uranium deposit. In the range 1000 to 1100 metres, a fault can be seen; the fault has lower
resistivity than the surrounding area, and is seen in the dark blue colour. The region of
deposit is seen at the bottom of the cross section from 1000 metres to 1600 metres.
19
Figure 1.6.2: A map of the Dawn Lake area in northern Saskatchewan. The PD electricresistivity survey lines can be seen on the map. Each survey line is 1400 metres long. Theelectrode spacing is 25 metres. The data used to invert Figure 1.6.3 is from line L15E seenin red. This map is modified from the map provided by Cameco.
20
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21
Chapter 2
Methods
2.1 GEA 1D modelling
Generalized Electrode Array (GEA) is used to investigate 2-layer 1D models. It is a forward
and inverse modelling program using the simplex method for inversion with Monte Carlo
error estimates. Hendrick Holmes originally wrote this program for his Master’s thesis at the
University of Saskatchewan in 1996. Dr. Jim Merriam later wrote the user interface and the
algorithm for the voltage prediction.
Four forward models are made in GEA with two resistive models and two conductive models.
The resistive models have a resistive upper layer that is 100Ωm and a bottom layer of 10Ωm.
The conductive models contain a conductive upper layer of 10Ωm and a bottom layer of
100Ωm. Each of the conductive and resistive models has an upper layer of one meter and
two meters thick. (Figure 2.1.1).
GEA takes each model with specified electrode spacing and pseudo-depths along with the
resistivity, layer thickness, infinite pole distance and generates a forward resistive 1D curve.
22
Figure 2.1.1: Diagram of infinite one-dimensional GEA forward model. The model containstwo layers; the first layer is one meter and two meters thick with the conductive modelshaving 100Ωm in the top layer and the bottom containing infinite thickness with a 10Ωmresistivity. In reverse, the resistive models have a 100Ωm top layer and a 10Ωm bottom layer.C1 and C2 are injection electrodes. C2 is the infinite pole. P1 and P2 are potential electrodes.
Five percent noise is added to this forward data. The forward data generated are inverted
to obtain the thickness of the first layer and the resistivity of both layers. Each model is
subjected to infinite pole distances located at 9000, 8000, 5000, 2000, 1000, 500, 200, 100, 50,
10 and 1 meter(s) away from the array. This is done to observe how much the infinite pole
affects the inverted models when its distance is decreased. The electrode spacing increases
one meter at a time to ‘sense’ deeper into the model subsurface (see Appendix A for electrode
positions). An example of the forward and inverse model of GEA is shown in Figure 2.1.2.
23
Figure 2.1.2: An example of a GEA inversion process. This is a conductive model with one-meter upper layer thickness. The top layer is 10Ωm and the lower layer is 100Ωm. In thisgraph, the circular dots are the forward model with added five percent noise. The hollowdots are points eliminated due to large error for inversion. The equivalent models can beseen in the dotted lines in the box on the bottom right corner. The equivalent models arecalculated to estimate inversion error. The solid blue line in the box is the final inversionprediction. The blue line across the full graph is the final predicted curve. The thicknessprediction is at 1.1m, the top resistivity is predicted to be 9.9Ωm and the bottom resistivityis predicted to be 129.5Ωm. The data error estimate is at 5.6 percent. The predicted thicknessand resistivity are summarized for each of GEA models in Figure 3.1.1 to Figure 3.1.8.
24
2.2 Field data
Field data were collected near Saskatoon, Saskatchewan, at the U of S seismic station. The
data is collected over a buried galvanized metal sheet; this is the target conductor. The metal
sheet is buried 1.5 metres at one end and slopes down to a depth of 2.5 metres at the other
end. The metal sheet is about 7 metre long and 1.5 metre wide (Figure 2.2.1). An open pit,
which acts as a resistor, is to the right end of the survey line. The pit is about one metre
wide, three metre long and 1.5 metre deep. The survey line is adjacent to both the conductor
and the resistor.
Figure 2.2.1: Illustration of buried galvanized steel sheet and open pit at the U of S seismicstation. Field data is collected adjacent to buried steel sheet and open pit.
A multi-node system with 30 nodes was used. The survey line is 28 metres long with 29 elec-
trodes placed 1 metre apart and an infinite pole at a distance. The infinite pole is planted
300 metres away from the end of the survey line. NIPs are planted along the survey line at
distances of 200, 100, 50, 25, 10 and 1 metre(s). All the NIP data sets are compiled and
inverted as a traditional PD array.
The apparent resistivity of the field data is calculated using the general apparent resistiv-
ity equation (Equation 1.3.11). Instead of using the shortened apparent resistivity equation
25
specific for PD array, which assumes an infinite distance for the infinite pole, the general
apparent resistivity equation uses the exact infinite pole distance for the calculation of the ap-
parent resistivity. The calculated apparent resistivity data is inverted using the least squares
method in Loke’s RES2DINV program.
The sensitivity sections of the measured field data are displayed from Figure 3.2.2.1 to
Figure 3.2.2.3. The NIP sensitivity sections are compared to the traditional PD sensitivity
section. The sensitivity sections are calculated using the two dimensional sensitivity function
in RES2D. The field data were collected using the same array geometry, with an array length
of 28 metres and a spacing of one metre, but with a varying infinite pole length. The sensi-
tivity function is dependent on the array’s geometry (Loke, 2001; 2004). Therefore, keeping
the array geometry constant is very important in this study. If any changes occur in the NIP
sensitivity sections, hence changes to the DOI, these will be caused by the varying infinite
pole distances and not by the array geometry.
CC similarity comparisons of the traditional PD inverted data to the NIP inverted data sets
are done to show the sections over all resolution change; this is done by using the traditional
PD data to cross-correlate with each of the NIP inverted data. The results of the CC sections
are shown in Figure 3.2.3.1 to Figure 3.2.3.4.
2.3 Geoelectric COMSOL modelling
COMSOL 4.2 was used to construct the resistivity models. COMSOL software solves forward
models using finite element analysis. Many useful modules exist in COMSOL, but the direct
current (DC) conductive module is used to construct the resistivity model (Figure 2.3.1). A
26
brief description of the solution to this module is discussed below.
In general, the current density depends linearly on the electric field E:
J = σE (2.3.1)
where J is the current density, and σ is the conductivity of the material. The electric field
can be defined as the gradient of the potential, V :
E = −∇V. (2.3.2)
The divergence of the current density is zero everywhere except at the location of the source
and sink:
∇ · J = 0 (2.3.3)
Combining equations 2.3.1, 2.3.2 and 2.3.3:
∇ · (σ∇V ) = 0 (2.3.4)
The Comsol DC conductive module solves this equation numerically.
The model’s array contains 30 electrodes with spacing of 30 metres apart; the total array
length is 870 metres. A 100-metre cubed conductor is located in the middle of the array at a
depth of 100 metres. The traditional infinite pole is located at 8.7 kilometres away from the
end of the array. The NIPs are located at 4.350km, 3.480km, 2.610km, 2.175km, 1.740km,
27
1.305km, 870m and 30m away from the end of the array. The NIP distances are chosen to
be multiples of the array length, with respect to the NIP distances listed above, the per
array lengths are 5, 4, 3, 2.5, 2, 1.5, 1 and one electrode spacing. The inverted sections are
displayed in Figure 3.3.1.1 to Figure 3.3.1.5.
The apparent resistivity is calculated using the measured model voltage from COMSOL.
The general apparent resistivity equation (Equation 1.3.11) is used instead of the shortened
traditional PD specific equation. The apparent resistivity data is inverted using the blocky
inversion method in RES2DINV. This inversion process is selected due to sharp edges on
the conductor. This method will attempt to minimize the absolute changes in the resistivity
values (Loke, 2001; 2004; Geoelectrical Imaging 2D & 3D manual, 2007).
The sensitivity sections for each of the inverted PD data are displayed from Figure 3.3.2.1
to Figure 3.3.2.4 for comparison of the DOI. CC of the traditional PD inverted data and
NIP inverted data were also analyzed. Each cross-correlated section uses the traditional PD
inverted data to correlate with each of the NIP data sets. The results are shown on Figures
3.3.3.1 to 3.3.3.5.
28
Figure 2.3.1: An example of a COMSOL model used to simulate PD resistivity data. Thereare a total of 30 electrodes in this multi-node system. The electrode spacing is 30 metres.The infinite pole is located 8.7 kilometres away from the end of the array. The NIPs arelocated at 4.350km, 3.480km, 2.610km, 2.175km, 1.740km, 1.305km, 870m and 30m awayfrom the end of the array. A 100-metre cubed conductor is at a depth of 100 metres belowthe centre of the array.
29
Chapter 3
Data Analysis and Results
3.1 GEA 1D inversions
Despite the large differences in the infinite pole distances within each of the models, all 4
models inverted thickness and resistivity values fall within the error of the forward model
values. The infinite pole distances for each of the forward models are located at 9km, 8km,
5km, 2km, 1km, 500m, 200m, 100m, 50m, 10m and 1m. For more details on the one dimen-
sional GEA models refer to section 2.1.
The estimated depths in Figure 3.1.1 (one metre) and Figure 3.1.3 (two metres) are results
from the conductive model, while Figure 3.1.5 (one metre) and Figure 3.1.7 (two metres)
are results from the resistive model. These plots contain points randomly scattered around
the forward model thicknesses. The scattered points do not show any systematic relationship
with the position of the NIP position and could be caused by the GEA inversion process
due to the five percent noise added to the forward data. The 1D inverted thickness data
does not reflect an effect of the infinite pole distances. It is noted in Figure 3.1.4 the in-
verted resistivity of the bottom layer (100Ωm) have inverted resistivity with an infinite pole
30
distance at 1 metre that are all below 100Ωm. This is due to a combination of the model
thickness (2 metres) and the infinite pole distance. As the infinite pole is only 1 metre away
from the end of the array the current travel path does not sense deep enough into the
second layer to get a better estimate of the second layer’s resistivity. The specific geometry
of this array which controls the DOI is also a factor in the under-estimation of the resistivity.
The inverted resistivity values for the GEA models are shown in Figure 3.1.2 (conductive),
Figure 3.1.4 (conductive), Figure 3.1.6 (resistive) and Figure 3.1.8 (resistive). The inverted
resistivity data hovers below and above the forward model resistivities. The points scatter
evenly through all pole distances, no specific pattern in inverted resistivity due to the pole
distances is found.
31
Figure 3.1.1: GEA inverted depths with varying infinite pole distances. This is a conductivemodel with a top layer thickness of one metre. The top resistivity is 10Ωm and bottom layeris 100Ωm. Five percent noise was added to the modelled forward resistivity before inversion.
32
Figure 3.1.2: GEA inverted resistivity of conductive model with top layer thickness of onemetre. The model contains a top layer resistivity of 10Ωm and bottom layer of 100Ωm. Fivepercent noise was added to the modelled forward resistivity before inversion.
33
Figure 3.1.3: GEA inverted depths with varying infinite pole distances. This is a conductivemodel with a top layer thickness of two metres. The top resistivity is 10Ωm and bottom layeris 100Ωm. Five percent noise was added to the modelled forward resistivity before inversion.
34
Figure 3.1.4: GEA inverted resistivity of conductive model with top layer thickness of twometres. The model contains a top layer resistivity of 10Ωm and bottom layer of 100Ωm. Fivepercent noise was added to the modelled forward resistivity before inversion.
35
Figure 3.1.5: GEA inverted depths with varying infinite pole distances. This is a conductivemodel with a top layer thickness of one metre. The top resistivity is 100Ωm and bottom layeris 10Ωm. Five percent noise was added to the modelled forward resistivity before inversion.
36
Figure 3.1.6: GEA inverted resistivity of resistive model with top layer thickness of onemetre. The model contains a top layer resistivity of 100Ωm and bottom layer of 10Ωm. Fivepercent noise was added to modelled forward resistivity before inversion.
37
Figure 3.1.7: GEA inverted depths with varying infinite pole distances. This is a resistivemodel with a top layer thickness of two metres. The top resistivity is 100Ωm and bottom layeris 10Ωm. Five percent noise was added to the modelled forward resistivity before inversion.
38
Figure 3.1.8: GEA inverted resistivity of resistive model with top layer thickness of twometres. The model contains a top layer resistivity of 100Ωm and bottom layer of 10Ωm. Fivepercent noise was added to the modelled forward resistivity before inversion.
39
3.2 Field data
3.2.1 Field data inversions
The field data was inverted using RES2DINV. Each set of data with the NIPs at 200m, 100m,
50m, 25m, 10m and 1m were prepared and inverted the same way as a traditional PD array.
The traditional pole is located 300 metres away from the end of the array. The full set of
inverted sections from the field data can be found in Appendix B. The difference in inversion
error if the infinite poles were not close to infinity is observed in the NIP cross-sections.
Here, the traditional inverted field data (Figure 3.2.1.1) and field data with NIP at 100 me-
tres (Figure 3.2.1.2), 50 metres (Figure 3.2.1.3) and 1 metre(Figure 3.2.1.4) are examined. In
Figure 3.2.1.2 with NIP at 100 metres, the inverted section is comparable to the traditional
PD array. By comparison of the two images, they are similar, as the conductor is the same
size and at the same depth with comparable resistivity. The open pit with high resistivity
is also seen to the right of both sections at the same distance and dimensions. It should be
noted that the differences in the images could be due to the non-uniqueness of the inver-
sion process and not due to the infinite pole distances. The differences found near surface
are due to the sensitivity of the PD array to near surface features. Overall, the features of
the conductor and resistor deteriorate with NIP distance of 50 metres and below. With the
50 metres NIP (Figure 3.2.1.3) the detection of the gravel and sand layer at 3.5 metres is
distorted and the near surface features are amplified. For NIPs at 25 metres (Figure B 5)
and 10 metres (Figure B 6) detection of the gravel and sand layer is the same as the NIP at
50 metres (Figure 3.2.1.3). The cross-section of NIP at 50 metres is also more sensitive to
near surface features than cross-sections with NIPs at 200 and 100 metres. With the NIP at
1 metre (Figure 3.2.1.4), the resolution in the cross-section breaks down with distortion of
the size and resistivity of the conductor and resistor.
40
3.2.2 Field data sensitivity sections
A visual comparison of the traditional PD sensitivity section and the NIP sensitivity sections
can be seen from Figure 3.2.2.1 to Figure 3.2.2.3. The top section at each of the figures are
from the traditional PD array with infinite pole distance at 300 metres, the bottom sensitiv-
ity sections are the NIP arrays with pole distances at 100, 50, and 1 metre(s). The full set
of sensitivity sections for field data can be found in Appendix C.
The visual differences between the NIP array sensitivity sections with pole distances at 100
metres (Figure 3.2.2.1) to the traditional PD array sensitivity sections have minimal differ-
ences. Any changes are very minor and sensitivity levels remain the same. Hence the DOI
is similar to both of the traditional PD array and the NIP array with pole distance at 100
metres. At NIP distance of 50 metres (Figure 3.2.2.2), greater changes within the sensitivity
sections are observed, notably below the 10 and 23 metre mark. The sensitivity section with
NIP at 1 metre (Figure 3.2.2.3) is not comparable to the traditional PD sensitivity section.
The NIP section has shallow sensitivity contours, therefore, giving the DOI a smaller value.
41
Fig
ure
3.2.
1.1:
Tra
dit
ional
PD
cros
s-se
ctio
nin
vert
edusi
ng
RE
S2D
INV
.T
he
arra
yis
28m
etre
slo
ng
wit
hon
em
etre
elec
trode
spac
ing.
The
infinit
ep
ole
islo
cate
d30
0m
etre
saw
ayfr
omth
een
dof
the
arra
y.
42
Fig
ure
3.2.
1.2:
NIP
arra
yw
ith
pol
edis
tance
at10
0m
etre
s.T
he
arra
yis
28m
etre
slo
ng
wit
hon
em
etre
elec
trode
spac
ing.
43
Fig
ure
3.2.
1.3:
NIP
arra
yw
ith
pol
edis
tance
at50
met
res.
The
arra
yis
28m
etre
slo
ng
wit
hon
em
etre
elec
trode
spac
ing.
44
Fig
ure
3.2.
1.4:
NIP
PD
arra
yat
1m
etre
.T
he
arra
yis
28m
etre
slo
ng
wit
hon
em
etre
elec
trode
spac
ing.
45
3.2.3 Field data cross-correlation sections
A cross-correlation analytical comparison of the traditional and NIP PD arrays are done
to compare the similarities between the inverted sections. The full set of CC sections for
the field data is in Appendix D. A control graph (Figure 3.2.3.1) is made to show the non-
uniqueness in the least squares inversion. The traditional PD array with infinite pole distance
at ten times the array length was used. The field data was inverted twice within RES2DINV
and two different inverted files were made. The control graph shows some differences (blue
colour) in the two inverted data sets, even though it is from the same field data set. The
non-uniqueness in the inversion process can create slight variations of inverted data. With
this in mind, other NIP cross-correlated sections may show differences, but these differences
may not be from the effects of the infinite pole. For more information on CC, please refer to
section 1.5.
The comparison of the traditional PD cross-section with NIP at 100 metres (Figure 3.2.3.2)
are negligible, the image mainly have a scale of one (red). This implies that the traditional
PD cross-section is very similar, if not the same as the cross-sections of the NIP at 100
metres. The cross-correlation of the traditional PD cross-section and the NIP at 50 metres
(Figure 3.2.3.3) shows variations in its resolution; this is seen as the blue on the image.
The cross-correlation of the traditional PD and NIP at one metre (Figure 3.2.3.4) shows
the biggest difference in the cross-sections. Figure 3.2.3.4 is the CC of the traditional PD
inverted data and the NIP inverted data at one metre. The blue curve in this CC section
is also known as the DOI curve. Once the differences in the two sections are large enough
a DOI curve will be present in the CC section, everything above the DOI curve is useful
interpretable data. This was developed by Oldenburg and Li (1999) to prevent over inter-
pretation of the inverted sections.
46
Fig
ure
3.2.
2.1:
The
bot
tom
sensi
tivit
yse
ctio
nis
calc
ula
ted
from
NIP
PD
inve
rted
fiel
ddat
a.T
he
NIP
islo
cate
dat
100
met
res
away
from
the
end
ofth
ear
ray.
The
top
sensi
tivit
yse
ctio
nis
the
trad
itio
nal
PD
inve
rted
fiel
ddat
a.
47
Fig
ure
3.2.
2.2:
The
bot
tom
sensi
tivit
yse
ctio
nis
calc
ula
ted
from
NIP
PD
inve
rted
fiel
ddat
a.T
he
NIP
islo
cate
dat
50m
etre
saw
ayfr
omth
een
dof
the
arra
y.T
he
top
sensi
tivit
yse
ctio
nis
from
the
trad
itio
nal
PD
inve
rted
fiel
ddat
a.
48
Fig
ure
3.2.
2.3:
The
bot
tom
sensi
tivit
yse
ctio
nis
calc
ula
ted
from
NIP
PD
inve
rted
fiel
ddat
a.T
he
NIP
islo
cate
dat
one
met
reaw
ayfr
omth
een
dof
the
arra
y.T
he
top
sensi
tivit
yse
ctio
nis
from
the
trad
itio
nal
PD
inve
rted
fiel
ddat
a.
49
3.3 Geoeletric COMSOL modelling
3.3.1 COMSOL model inversions
The computed potentials from the COMSOL model is exported to MATLAB for data sort-
ing and calculation of apparent resistivity. The calculated and sorted data is inverted using
RES2DINV. The data sets with the NIP are compiled and inverted as a traditional PD array.
This is done to analyze the effects of the pole distance on the inversion process. The full set
of inverted sections can be found in Appendix E.
Figures 3.3.1.1 to 3.3.1.5 displays the inverted PD cross-section data computed from the
COMSOL model. Figure 3.3.1.1 contains the traditional PD array with the infinite pole
located 8.7 kilometres away. Figures 3.3.1.2 to 3.3.1.5 contains the NIP cross-sections; the
NIP distances are located respectively at 4.35km, 2.175km, 1.305km and 30 metres. The NIP
distances are chosen according to the array length. The respective NIP distances are 5, 2.5,
1.5 times the array length and 30 metres or one electrode spacing.
The comparison of the inverted traditional PD section (Figure 3.3.1.1) to the NIP distance
at 4.35 kilometres (Figure 3.3.1.2) has no major changes. The conductor’s width, depth and
resistivity are all very similar. The very slight differences could be due to the non-uniqueness
of the inversion process and not the NIP distance. With the NIP distance at shorter than
1.305 kilometres (Figure 3.3.1.4), the resolution of the conductor is lost, the shape and size
of the conductor are not true to the COMSOL model. With the NIP at 30 metres (Figure
3.3.1.5), the inverted section is not recognizable to the COMSOL model. The size and depth
of the conductor are distorted.
50
Fig
ure
3.2.
3.1:
Cro
ss-c
orre
lati
onco
ntr
olgr
aph
for
the
trad
itio
nal
PD
arra
y.T
he
fiel
ddat
aco
llec
ted
usi
ng
infinit
ep
ole
at30
0m
etre
sw
asin
vert
edtw
ice
onR
ES2D
INV
.T
he
two
sets
ofin
vert
eddat
aar
ecr
oss-
corr
elat
ed.
51
Fig
ure
3.2.
3.2:
Cro
ss-c
orre
lati
onim
age
ofth
etr
adit
ional
PD
inve
rted
dat
aan
dN
IPin
vert
eddat
aw
ith
pol
edis
tance
at10
0m
etre
s.
52
Fig
ure
3.2.
3.3:
Cro
ss-c
orre
lati
onim
age
ofth
etr
adit
ional
PD
inve
rted
dat
aan
dN
IPin
vert
eddat
aw
ith
pol
edis
tance
at50
met
res.
53
Fig
ure
3.2.
3.4:
Cro
ss-c
orre
lati
onim
age
ofth
etr
adit
ional
PD
inve
rted
dat
aan
dN
IPin
vert
eddat
aw
ith
pol
edis
tance
at1
met
re.
54
3.3.2 COMSOL model sensitivity sections
The sensitivity sections of the inverted COMSOL data with varying infinite pole distances
are displayed for comparison in Figures 3.3.2.1 to 3.3.2.4. All figures contain two sensitivity
sections: the top section displays the traditional PD array and the bottom section displays
the NIP arrays. The set of sensitivity sections for all the NIP PD array can be found in
Appendix F.
The NIP array sensitivity sections with pole distance at 4.35 kilometres or five times the
array length (Figure 3.3.2.1) and pole distance at 2.175 kilometres or 2.5 times the array
length (Figure 3.3.2.2) both show a slight change in the contours under the 500-metre mark
along the profiles where the deepest sensitivity of the array lies. The change is very slight as
the sensitivity contours narrows at depths from 87 metres to 200 metres. The most sensitive
region shifts to the left as the infinite pole distance decreases, this is evident in Figure 3.3.2.3
with NIP distance at 1.305 kilometres or 1.5 times the array length. In Figure 3.3.2.4 the
most sensitive region is shifted toward the centre, as the infinite pole is one electrode spacing
or 30 metres from the end of the survey line. This electrode arrangement is a dipole-dipole
array; therefore the array is symmetrical, unlike the asymmetrical PD array.
55
Fig
ure
3.3.
1.1:
Tra
dit
ional
PD
inve
rted
sect
ion
from
CO
MSO
Lm
odel
.T
he
infinit
ep
ole
islo
cate
d10
tim
esth
ear
ray
lengt
hor
at8.
7kilom
etre
saw
ayfr
omth
esu
rvey
line.
56
Fig
ure
3.3.
1.2:
NIP
inve
rted
sect
ion
from
CO
MSO
Lm
odel
wit
hp
ole
dis
tance
at4.
350
kilom
etre
sor
5ti
mes
the
arra
yle
ngt
h.
57
Fig
ure
3.3.
1.3:
NIP
inve
rted
sect
ion
from
CO
MSO
Lm
odel
wit
hp
ole
dis
tance
at2.
175
kilom
etre
sor
2.5
tim
esth
ear
ray
lengt
h.
58
Fig
ure
3.3.
1.4:
NIP
inve
rted
sect
ion
from
CO
MSO
Lm
odel
wit
hp
ole
dis
tance
at1.
305
kilom
etre
sor
1.5
tim
esth
ear
ray
lengt
h.
59
Fig
ure
3.3.
1.5:
NIP
inve
rted
sect
ion
from
CO
MSO
Lm
odel
wit
hp
ole
dis
tance
at30
met
res
oron
eel
ectr
ode
spac
ing
away
from
the
end
ofsu
rvey
line.
60
3.3.3 COMSOL model cross-correlation sections
The CC of the traditional PD and the NIP arrays are calculated to compare its similarities
with the inverted data sets. This method is adopted from Oldenburg and Li (1999). For
more information on CC, please refer to section 1.5. The full set of CC sections from the
COMSOL model can be found in Appendix G.
Figures 3.3.3.2 to 3.3.3.5 are the results of cross-correlation between the traditional PD in-
verted section and the NIP inverted sections. Figure 3.3.3.1 is an inversion non-uniqueness
control graph for the cross-correlation sections using inverted data from RES2DINV. This
cross-correlation plot is made using the traditional PD array data generated from the COM-
SOL model with infinite pole at 8.7 kilometres. The forward data is inverted twice using
RES2DINV and two different sets of inverted data are generated; these two sets of inverted
data are cross-correlated to obtain Figure 3.3.3.1. Although the two inverted data sets come
from the same forward data notable differences are found. If the data sets were the same, the
CC section would contain the value of one everywhere. This shows the differences calculated
in CC could be due to the non-uniqueness in the inversion process and not to the infinite
pole distances. The control graph (Figure 3.3.3.1) is comparable to NIP of 4.35 kilometres or
5 times the array length (Figure 3.3.3.2) and NIP of 2.175 kilometres or 2.5 times the array
length (Figure 3.3.3.3). The change in resolution of the CC can be seen on Figure 3.3.3.4.
This figure is cross-correlated between the traditional PD inverted data and NIP inverted
data with pole distance at 1.75 kilometres. The 1.75 kilometres is twice the distance of the
array length. It is critical to highlight the change in resolution in the CC sections occurring
between 2.5 and 2 times the array length. Figure 3.3.3.5, which indicates the traditional PD
inverted data and the inverted data with NIP at 30 metres is non-coherent.
61
Fig
ure
3.3.
2.1:
Sen
siti
vit
yse
ctio
nca
lcula
ted
from
CO
MSO
Lm
odel
inve
rted
dat
a.T
he
top
sensi
tivit
yse
ctio
nis
from
the
trad
itio
nal
PD
.T
his
issh
own
her
efo
rco
mpar
ison
.T
he
bot
tom
isa
NIP
PD
wit
hp
ole
dis
tance
at4.
350
kilom
etre
sor
5ti
mes
the
arra
yle
ngt
h.
62
Fig
ure
3.3.
2.2:
Sen
siti
vit
yse
ctio
ns
calc
ula
ted
from
CO
MSO
Lm
odel
inve
rted
dat
a.T
he
top
sensi
tivit
yse
ctio
nis
calc
ula
ted
from
the
trad
itio
nal
PD
.It
issh
own
her
efo
rco
mpar
ison
.T
he
bot
tom
sect
ion
isa
NIP
PD
wit
hp
ole
dis
tance
at2.
175
kilom
etre
sor
2.5
tim
esth
ear
ray
lengt
h.
63
Fig
ure
3.3.
2.3:
Sen
siti
vit
yse
ctio
ns
calc
ula
ted
from
CO
MSO
Lm
odel
inve
rted
dat
a.T
he
top
sensi
tivit
yse
ctio
nis
calc
ula
ted
from
the
trad
itio
nal
PD
.It
issh
own
her
efo
rco
mpar
ison
.T
he
bot
tom
sect
ion
isa
NIP
PD
wit
hp
ole
dis
tance
at1.
305
kilom
etre
sor
1.5
tim
esth
ear
ray
lengt
h.
64
Fig
ure
3.3.
2.4:
Sen
siti
vit
yse
ctio
ns
calc
ula
ted
from
CO
MSO
Lm
odel
inve
rted
dat
a.T
he
top
sensi
tivit
yse
ctio
nis
calc
ula
ted
from
the
trad
itio
nal
PD
.It
issh
own
her
efo
rco
mpar
ison
.T
he
bot
tom
sect
ion
isa
NIP
PD
wit
hp
ole
dis
tance
at30
met
res
oron
eel
ectr
ode
spac
ing
from
the
end
ofth
ear
ray.
65
Fig
ure
3.3.
3.1:
Cro
ss-c
orre
late
dse
ctio
nm
ade
from
dat
ain
vers
ion
contr
ol.
This
sect
ion
was
cros
s-co
rrel
ated
usi
ng
the
trad
itio
nal
PD
dat
aco
mpute
dfr
omC
OM
SO
L.
The
trad
itio
nal
PD
dat
ase
tw
asin
vert
edtw
ice
inR
ES2D
INV
toob
tain
two
sets
ofdat
a.T
hes
etw
ose
tsof
dat
aar
eth
enco
mpar
edfo
rsi
milar
itie
s.T
he
sim
ilar
itie
sar
ein
red
ora
scal
eof
one
and
the
diff
eren
ces
can
be
seen
inblu
e.D
ue
toth
enon
-uniq
uen
ess
ofth
ein
vers
ion
pro
cess
,diff
eren
ces
wer
enot
edin
the
two
inve
rted
dat
ase
ts,
alth
ough
they
cam
efr
omth
esa
me
forw
ard
dat
a.
66
Fig
ure
3.3.
3.2:
Cro
ss-c
orre
late
dse
ctio
nb
etw
een
inve
rted
trad
itio
nal
PD
dat
aan
dN
IPin
vert
eddat
aw
ith
pol
edis
tance
at4.
35kilom
etre
sor
5ti
mes
the
arra
yle
ngt
h.
67
Fig
ure
3.3.
3.3:
Cro
ss-c
orre
late
dse
ctio
nb
etw
een
inve
rted
trad
itio
nal
PD
dat
aan
dN
IPin
vert
eddat
aw
ith
pol
edis
tance
at2.
175
kilom
etre
sor
2.5
tim
esth
ear
ray
lengt
h.
68
Fig
ure
3.3.
3.4:
Cro
ss-c
orre
late
dse
ctio
nb
etw
een
inve
rted
trad
itio
nal
PD
dat
aan
dN
IPin
vert
eddat
aw
ith
pol
edis
tance
at1.
74kilom
etre
sor
2ti
mes
the
arra
yle
ngt
h.
69
Fig
ure
3.3.
3.5:
Cro
ss-c
orre
late
dse
ctio
nb
etw
een
inve
rted
trad
itio
nal
PD
dat
aan
dN
IPin
vert
eddat
aw
ith
pol
edis
tance
30m
etre
sor
one
elec
trode
spac
ing
from
the
end
ofth
esu
rvey
line.
70
Chapter 4
Conclusion and Discussions
4.1 GEA 1D data
The GEA 1D inversion process using the Simplex method is not affected by the forward
models’ decreasing infinite pole distances. The pole distances ranging from 9 kilometres to
1 metre did not have an affect on the inverted thickness nor the inverted resistivity of the
models. We can conclude here using GEA 1D data that planting an infinite pole close to
infinity is not necessary. By considering all four electrode positions, we have inversion results
that are seemingly unaffected by the infinite pole distance. It is important to still have an
infinite pole at some distance as resistivity arrays are dependent on its geometry for DOI.
It should be noted that the inverted resistivity values for the conductive model (Figure 3.1.2
and Figure 3.1.4) with thicknesses of one and two metres have high error values; these high
error values could be due to VES insensitivity to lateral changes in resistivity.
71
4.2 Field Data
Inverted field data indicates the resolution deteriorates somewhere between the NIP distance
of 100 to 50 metres. One hundred metres is about 3.5 times the array length and 50 metres is
1.8 times the array length. This can be seen in the resolution change in the inverted sections
on Figure 3.2.1.2 (NIP at 100 metres) and Figure 3.2.1.3 (NIP at 50 metres). Furthermore,
the sensitivity section on Figure 3.2.2.1 (NIP at 100 metres) is comparable to the traditional
PD array with infinite pole at 300 metres. However, the results displayed in Figure 3.2.2.2
(NIP at 50 metres) are not comparable to the traditional PD array.
These results indicate that a distance of ten times the array length for an infinite pole is not
necessary. but a distance greater than 1.8 times the array length should be used. Further
studies should be carried out to test the generality of this recommendation.
In figure 3.2.1.4 with the NIP located one metre from the end of the array results in a
dipole-dipole electrode arrangement. The geometry of the array has eliminated the infinite
pole making it a dipole-dipole array. The resolution and depth distortion in this figure indi-
cate that an infinite pole at a distance is still needed for the PD arrays.
4.3 Geoelectric COMSOL modelling
Using the cross-correlation analysis, the NIP distance needed to retain resolution and depth
of investigation is determined to be at a minimum of 2.5 times of the array length. This
coincides with the field data results of change in resolution for infinite pole distances be-
tween 3.5 to 1.8 times the array length. The change in resolution in Figure 3.3.3.4 shows
that NIP at two times the array length is not comparable to the traditional PD inverted
72
data. Whereas Figure 3.3.3.3, with NIP at 2.5 times the array length is comparable to the
traditional PD inverted data. The changes in sensitivity sections in Figure 3.3.2.2 (2.5 times
the array length) and Figure 3.2.2.3 (1.5 times the array length) compared to the traditional
PD sensitivity section have a subtle shift to the centre. This change is subtle but still no-
ticeable. The change in resolution happens between 2.5 to 2 times the array length. For this
particular situation, we can conclude with the observation of the resolution changes in the
sensitivity and CC sections that 2.5 times the array length is the minimum distance for an
infinite pole to be planted.
4.4 Discussions
If this study proofs anything other than that 10 times the infinite pole distance is not needed,
one planning a field survey could still save a large amount of time and money by planting
an infinite pole that is only 3 or 4 times the array length.
Further field studies are needed to confirm the COMSOL modelling results. One could further
this study by modelling different scenarios to see what effects it may have on the infinite
pole and its distance.
73
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77
Appendix A
Electrode positioning for 1D GEA
inversions
Each of the models contains a set of varying infinite pole distances (9km, 8km, 5km, 2000m,
1km, 500m, 200m, 100m, 50m, 10m, 1m) with increasing P1, P2 electrode distances and a
fixed C1 position. The forward models takes into consideration of the first layer thickness
and resistivities of the first and second layer. The electrode positions is then applied to this
forward model to generate a forward resistivity curve. The forward data is added with five
percent noise and inverted to obtain the estimated model thickness and resistivity.
Table A.1: GEA electrodes positioning
C2 position (m) C1 position (m) P1 position (m) P2 position (m) Pseudodepth (m)
9000 0 1 2 0.75
9000 0 2 3 1.25
9000 0 3 4 1.75
9000 0 4 5 2.25
9000 0 5 6 2.75
79
9000 0 6 7 3.25
9000 0 7 8 3.75
9000 0 8 9 4.25
9000 0 9 10 4.75
8000 0 1 2 0.75
8000 0 2 3 1.25
8000 0 3 4 1.75
8000 0 4 5 2.25
8000 0 5 6 2.75
8000 0 6 7 3.25
8000 0 7 8 3.75
8000 0 8 9 4.25
8000 0 9 10 4.75
5000 0 1 2 0.75
5000 0 2 3 1.25
5000 0 3 4 1.75
5000 0 4 5 2.25
5000 0 5 6 2.75
5000 0 6 7 3.25
5000 0 7 8 3.75
5000 0 8 9 4.25
5000 0 9 10 4.75
2000 0 1 2 0.75
2000 0 2 3 1.25
2000 0 3 4 1.75
2000 0 4 5 2.25
80
2000 0 5 6 2.75
2000 0 6 7 3.25
2000 0 7 8 3.75
2000 0 8 9 4.25
2000 0 9 10 4.75
1000 0 1 2 0.75
1000 0 2 3 1.25
1000 0 3 4 1.75
1000 0 4 5 2.25
1000 0 5 6 2.75
1000 0 6 7 3.25
1000 0 7 8 3.75
1000 0 8 9 4.25
1000 0 9 10 4.75
500 0 1 2 0.75
500 0 2 3 1.25
500 0 3 4 1.75
500 0 4 5 2.25
500 0 5 6 2.75
500 0 6 7 3.25
500 0 7 8 3.75
500 0 8 9 4.25
500 0 9 10 4.75
200 0 1 2 0.75
200 0 2 3 1.25
200 0 3 4 1.75
81
200 0 4 5 2.25
200 0 5 6 2.75
200 0 6 7 3.25
200 0 7 8 3.75
200 0 8 9 4.25
200 0 9 10 4.75
100 0 1 2 0.75
100 0 2 3 1.25
100 0 3 4 1.75
100 0 4 5 2.25
100 0 5 6 2.75
100 0 6 7 3.25
100 0 7 8 3.75
100 0 8 9 4.25
100 0 9 10 4.75
50 0 1 2 0.75
50 0 2 3 1.25
50 0 3 4 1.75
50 0 4 5 2.25
50 0 5 6 2.75
50 0 6 7 3.25
50 0 7 8 3.75
50 0 8 9 4.25
50 0 9 10 4.75
10 0 1 2 0.75
10 0 2 3 1.25
82
10 0 3 4 1.75
10 0 4 5 2.25
10 0 5 6 2.75
10 0 6 7 3.25
10 0 7 8 3.75
10 0 8 9 4.25
10 0 9 10 4.75
1 0 1 2 0.75
1 0 2 3 1.25
1 0 3 4 1.75
1 0 4 5 2.25
1 0 5 6 2.75
1 0 6 7 3.25
1 0 7 8 3.75
1 0 8 9 4.25
1 0 9 10 4.75
83
Fig
ure
B.1
:T
radit
ional
PD
cros
s-se
ctio
nin
vert
edusi
ng
RE
S2D
INV
.T
he
arra
yis
28m
etre
slo
ng
wit
h1
met
reel
ectr
ode
spac
ing.
The
infinit
ep
ole
islo
cate
d30
0m
etre
saw
ayfr
omth
een
dof
the
arra
y.
85
Fig
ure
B.2
:N
IPar
ray
wit
hp
ole
dis
tance
at20
0m
etre
s.T
he
arra
yis
28m
etre
slo
ng
wit
h1
met
reel
ectr
ode
spac
ing.
86
Fig
ure
B.3
:N
IPar
ray
wit
hp
ole
dis
tance
at10
0m
etre
s.T
he
arra
yis
28m
etre
slo
ng
wit
h1
met
reel
ectr
ode
spac
ing.
87
Fig
ure
B.4
:N
IPar
ray
wit
hp
ole
dis
tance
at50
met
res.
The
arra
yis
28m
etre
slo
ng
wit
h1
met
reel
ectr
ode
spac
ing.
88
Fig
ure
B.5
:N
IPar
ray
wit
hp
ole
dis
tance
at25
met
res.
The
arra
yis
28m
etre
slo
ng
wit
h1
met
reel
ectr
ode
spac
ing.
89
Fig
ure
B.6
:N
IPar
ray
wit
hp
ole
dis
tance
at10
met
res.
The
arra
yis
28m
etre
slo
ng
wit
h1
met
reel
ectr
ode
spac
ing.
90
Fig
ure
B.7
:P
Dar
ray
wit
hch
eate
rp
ole
at1
met
re.
The
arra
yis
28m
etre
slo
ng
wit
h1
met
reel
ectr
ode
spac
ing.
91
Fig
ure
C.1
:T
he
bot
tom
sensi
tivit
yse
ctio
nis
calc
ula
ted
from
NIP
PD
inve
rted
fiel
ddat
a.T
he
NIP
islo
cate
dat
200
met
res
away
from
the
end
ofar
ray.
The
top
sensi
tivit
yse
ctio
nis
the
trad
itio
nal
PD
inve
rted
fiel
ddat
a.
93
Fig
ure
C.2
:T
he
bot
tom
sensi
tivit
yse
ctio
nis
calc
ula
ted
from
NIP
PD
inve
rted
fiel
ddat
a.T
he
NIP
islo
cate
dat
100
met
res
away
from
the
end
ofar
ray.
The
top
sensi
tivit
yse
ctio
nis
the
trad
itio
nal
PD
inve
rted
fiel
ddat
a.
94
Fig
ure
C.3
:T
he
bot
tom
sensi
tivit
yse
ctio
nis
calc
ula
ted
from
NIP
PD
inve
rted
fiel
ddat
a.T
he
NIP
islo
cate
dat
50m
etre
saw
ayfr
omth
een
dof
arra
y.T
he
top
sensi
tivit
yse
ctio
nis
from
the
trad
itio
nal
PD
inve
rted
fiel
ddat
a.
95
Fig
ure
C.4
:T
he
bot
tom
sensi
tivit
yse
ctio
nis
calc
ula
ted
from
NIP
PD
inve
rted
fiel
ddat
a.T
he
NIP
islo
cate
dat
25m
etre
saw
ayfr
omth
een
dof
arra
y.T
he
top
sensi
tivit
yse
ctio
nis
from
the
trad
itio
nal
PD
inve
rted
fiel
ddat
a.
96
Fig
ure
C.5
:T
he
bot
tom
sensi
tivit
yse
ctio
nis
calc
ula
ted
from
NIP
PD
inve
rted
fiel
ddat
a.T
he
NIP
islo
cate
dat
10m
etre
saw
ayfr
omth
een
dof
arra
y.T
he
top
sensi
tivit
yse
ctio
nis
from
the
trad
itio
nal
PD
inve
rted
fiel
ddat
a.
97
Fig
ure
C.6
:T
he
bot
tom
sensi
tivit
yse
ctio
nis
calc
ula
ted
from
NIP
PD
inve
rted
fiel
ddat
a.T
he
NIP
islo
cate
dat
1m
etre
away
from
the
end
ofar
ray.
The
top
sensi
tivit
yse
ctio
nis
from
the
trad
itio
nal
PD
inve
rted
fiel
ddat
a.
98
Fig
ure
D.1
:C
ross
-cor
rela
tion
contr
olgr
aph
for
the
trad
itio
nal
PD
arra
y.T
he
fiel
ddat
aco
llec
ted
usi
ng
infinit
ep
ole
at30
0m
etre
sw
asin
vert
edtw
ice
onR
ES2D
INV
.T
he
2se
tsof
inve
rted
dat
aar
ecr
oss-
corr
elat
ed.
100
Fig
ure
D.2
:C
ross
-cor
rela
tion
imag
eof
the
trad
itio
nal
PD
inve
rted
dat
aan
dth
eN
IPin
vert
eddat
aw
ith
pol
edis
tance
at20
0m
etre
s.
101
Fig
ure
D.3
:C
ross
-cor
rela
tion
imag
eof
the
trad
itio
nal
PD
inve
rted
dat
aan
dth
eN
IPin
vert
eddat
aw
ith
pol
edis
tance
at10
0m
etre
s.
102
Fig
ure
D.4
:C
ross
-cor
rela
tion
imag
eof
the
trad
itio
nal
PD
inve
rted
dat
aan
dth
eN
IPin
vert
eddat
aw
ith
pol
edis
tance
at50
met
res.
103
Fig
ure
D.5
:C
ross
-cor
rela
tion
imag
eof
the
trad
itio
nal
PD
inve
rted
dat
aan
dth
eN
IPin
vert
eddat
aw
ith
pol
edis
tance
at25
met
res.
104
Fig
ure
D.6
:C
ross
-cor
rela
tion
imag
eof
the
trad
itio
nal
PD
inve
rted
dat
aan
dth
eN
IPin
vert
eddat
aw
ith
pol
edis
tance
at10
met
res.
105
Fig
ure
D.7
:C
ross
-cor
rela
tion
imag
eof
the
trad
itio
nal
PD
inve
rted
dat
aan
dN
IPin
vert
eddat
aw
ith
pol
edis
tance
at1
met
re.
106
Fig
ure
E.1
:T
radit
ional
PD
inve
rted
sect
ion
from
CO
MSO
Lm
odel
.T
he
infinit
ep
ole
islo
cate
d10
tim
esth
ear
ray
lengt
hor
at8.
700
kilom
etre
saw
ayfr
omth
esu
rvey
line.
108
Fig
ure
E.2
:N
IPin
vert
edse
ctio
nfr
omC
OM
SO
Lm
odel
wit
hp
ole
dis
tance
at4.
350
kilom
etre
sor
5ti
mes
the
arra
yle
ngt
h.
109
Fig
ure
E.3
:N
IPin
vert
edse
ctio
nfr
omC
OM
SO
Lm
odel
wit
hp
ole
dis
tance
at3.
480
kilom
etre
sor
4ti
mes
the
arra
yle
ngt
h.
110
Fig
ure
E.4
:N
IPin
vert
edse
ctio
nfr
omC
OM
SO
Lm
odel
wit
hp
ole
dis
tance
at2.
610
kilom
etre
sor
3ti
mes
the
arra
yle
ngt
h.
111
Fig
ure
E.5
:N
IPin
vert
edse
ctio
nfr
omC
OM
SO
Lm
odel
wit
hp
ole
dis
tance
at2.
175
kilom
etre
sor
2.5
tim
esth
ear
ray
lengt
h.
112
Fig
ure
E.6
:N
IPin
vert
edse
ctio
nfr
omC
OM
SO
Lm
odel
wit
hp
ole
dis
tance
at1.
740
kilom
etre
sor
2ti
mes
the
arra
yle
ngt
h.
113
Fig
ure
E.7
:N
IPin
vert
edse
ctio
nfr
omC
OM
SO
Lm
odel
wit
hp
ole
dis
tance
at1.
305
kilom
etre
sor
1.5
tim
esth
ear
ray
lengt
h.
114
Fig
ure
E.8
:N
IPin
vert
edse
ctio
nfr
omC
OM
SO
Lm
odel
wit
hp
ole
dis
tance
at87
0m
etre
sor
one
arra
yle
ngt
h.
115
Fig
ure
E.9
:N
IPin
vert
edse
ctio
nfr
omC
OM
SO
Lm
odel
wit
hp
ole
dis
tance
at30
met
res
oron
eel
ectr
ode
spac
ing
away
from
the
end
ofsu
rvey
line.
116
Fig
ure
F.1
:Sen
siti
vit
yse
ctio
nca
lcula
ted
from
CO
MSO
Lm
odel
inve
rted
dat
a.T
he
top
sensi
tivit
yse
ctio
nis
from
the
trad
itio
nal
PD
.T
he
bot
tom
isa
NIP
PD
wit
hp
ole
dis
tance
at4.
350
kilom
etre
sor
5ti
mes
the
arra
yle
ngt
h.
118
Fig
ure
F.2
:Sen
siti
vit
yse
ctio
nca
lcula
ted
from
CO
MSO
Lm
odel
inve
rted
dat
a.T
he
top
sensi
tivit
yse
ctio
nis
from
the
trad
itio
nal
PD
.T
he
bot
tom
isa
NIP
PD
wit
hp
ole
dis
tance
at3.
480
kilom
etre
sor
4ti
mes
the
arra
yle
ngt
h.
119
Fig
ure
F.3
:Sen
siti
vit
yse
ctio
nca
lcula
ted
from
CO
MSO
Lm
odel
inve
rted
dat
a.T
he
top
sensi
tivit
yse
ctio
nis
from
the
trad
itio
nal
PD
.T
he
bot
tom
isa
NIP
PD
wit
hp
ole
dis
tance
at2.
610
kilom
etre
sor
3ti
mes
the
arra
yle
ngt
h.
120
Fig
ure
F.4
:Sen
siti
vit
yse
ctio
ns
calc
ula
ted
from
CO
MSO
Lm
odel
inve
rted
dat
a.T
he
top
sensi
tivit
yse
ctio
nis
calc
ula
ted
from
the
trad
itio
nal
PD
.T
he
bot
tom
sect
ion
isa
NIP
PD
wit
hp
ole
dis
tance
at2.
175
kilom
etre
sor
2.5
tim
esth
ear
ray
121
Fig
ure
F.5
:Sen
siti
vit
yse
ctio
nca
lcula
ted
from
CO
MSO
Lm
odel
inve
rted
dat
a.T
he
top
sensi
tivit
yse
ctio
nis
from
the
trad
itio
nal
PD
.T
he
bot
tom
isa
NIP
PD
wit
hp
ole
dis
tance
at1.
740
kilom
etre
sor
2ti
mes
the
arra
yle
ngt
h.
122
Fig
ure
F.6
:Sen
siti
vit
yse
ctio
ns
calc
ula
ted
from
CO
MSO
Lm
odel
inve
rted
dat
a.T
he
top
sensi
tivit
yse
ctio
nis
calc
ula
ted
from
the
trad
itio
nal
PD
.T
he
bot
tom
sect
ion
isa
NIP
PD
wit
hp
ole
dis
tance
at1.
305
met
res
or1.
5ti
mes
the
arra
y
123
Fig
ure
F.7
:Sen
siti
vit
yse
ctio
nca
lcula
ted
from
CO
MSO
Lm
odel
inve
rted
dat
a.T
he
top
sensi
tivit
yse
ctio
nis
from
the
trad
itio
nal
PD
.T
he
bot
tom
isa
NIP
PD
wit
hp
ole
dis
tance
at87
0m
etre
sor
one
arra
yle
ngt
h.
124
Fig
ure
F.8
:Sen
siti
vit
yse
ctio
ns
calc
ula
ted
from
CO
MSO
Lm
odel
inve
rted
dat
a.T
he
top
sensi
tivit
yse
ctio
nis
calc
ula
ted
from
the
trad
itio
nal
PD
.T
he
bot
tom
sect
ion
isa
NIP
PD
wit
hp
ole
dis
tance
at30
met
res
oron
eel
ectr
ode
spac
ing
from
the
end
ofth
ear
ray.
125
Fig
ure
G.1
:C
ross
-cor
rela
ted
sect
ion
mad
efr
omdat
ain
vers
ion
contr
ol.
This
sect
ion
was
cros
s-co
rrel
ated
usi
ng
the
trad
i-ti
onal
PD
dat
aco
mpute
dfr
omC
OM
SO
L.
The
trad
itio
nal
PD
dat
ase
tw
asin
vert
edtw
ice
inR
ES2D
INV
toob
tain
2se
tsof
dat
a.T
hes
e2
sets
ofdat
aar
eco
mpar
edto
them
selv
esfo
rsi
milar
itie
s.T
he
sim
ilar
itie
sar
ein
red
ora
scal
eof
one
and
the
diff
eren
ces
can
be
seen
inblu
e.A
lthou
ghth
e2
inve
rted
dat
ase
tsar
efr
omth
esa
me
forw
ard
dat
a,but
ther
ear
est
ill
diff
eren
ces
inth
eC
C.
This
isdue
toth
enon
-uniq
uen
ess
ofth
ein
vers
ion
pro
cess
.
127
Fig
ure
G.2
:C
ross
-cor
rela
ted
sect
ion
bet
wee
nin
vert
edtr
adit
ional
PD
dat
aan
dN
IPin
vert
eddat
aw
ith
pol
edis
tance
at4.
350
kilom
etre
sor
5ti
mes
the
arra
yle
ngt
h.
128
Fig
ure
G.3
:C
ross
-cor
rela
ted
sect
ion
bet
wee
nin
vert
edtr
adit
ional
PD
dat
aan
dN
IPin
vert
eddat
aw
ith
pol
edis
tance
at3.
480
kilom
etre
sor
4ti
mes
the
arra
yle
ngt
h.
129
Fig
ure
G.4
:C
ross
-cor
rela
ted
sect
ion
bet
wee
nin
vert
edtr
adit
ional
PD
dat
aan
dN
IPin
vert
eddat
aw
ith
pol
edis
tance
at2.
610
kilom
etre
sor
3ti
mes
the
arra
yle
ngt
h.
130
Fig
ure
G.5
:C
ross
-cor
rela
ted
sect
ion
bet
wee
nin
vert
edtr
adit
ional
PD
dat
aan
dN
IPin
vert
eddat
aw
ith
pol
edis
tance
at2.
175
kilom
etre
sor
2.5
tim
esth
ear
ray
lengt
h.
131
Fig
ure
G.6
:C
ross
-cor
rela
ted
sect
ion
bet
wee
nin
vert
edtr
adit
ional
PD
dat
aan
dN
IPin
vert
eddat
aw
ith
pol
edis
tance
at1.
740
kilom
etre
sor
2ti
mes
the
arra
yle
ngt
h.
132
Fig
ure
G.7
:C
ross
-cor
rela
ted
sect
ion
bet
wee
nin
vert
edtr
adit
ional
PD
dat
aan
dN
IPin
vert
eddat
aw
ith
pol
edis
tance
at1.
305
kilom
etre
sor
1.5
tim
esth
ear
ray
lengt
h.
133
Fig
ure
G.8
:C
ross
-cor
rela
ted
sect
ion
bet
wee
nin
vert
edtr
adit
ional
PD
dat
aan
dN
IPin
vert
eddat
aw
ith
pol
edis
tance
at87
0m
etre
sor
one
arra
yle
ngt
h.
134