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AUTOMATIC INTERPRETATION OF
MAGNETIC DATA USING EULER
DECONVOLUTION WITH MODIFIED
ALGORITHM
NURADDEEN USMAN
UNIVERSITI SAINS MALAYSIA
2018
AUTOMATIC INTERPRETATION OF
MAGNETIC DATA USING EULER
DECONVOLUTION WITH MODIFIED
ALGORITHM
by
NURADDEEN USMAN
Thesis submitted in fulfilment of the requirements
for the degree of
Doctor of Philosophy
April 2018
ii
ACKNOWLEDGEMENT
All praises are due to Allah (S.A.W) Who spare my life up to this moment
and destined me to pursue this PhD program. May the peace and blessing of Allah be
upon Prophet Mohammad (S.A.W), his companions and those that followed them up
to the day of resurrection. I’m in short of words to appreciate the solid upbringing,
support and guidance I received from my parent, may Allah reward them abundantly.
Special thanks to my supervisor Professor (Dr) Khiruddin Abdullah, my
mentor who makes my dream to become true. His vast knowledge, tolerance,
humility, hardworking and decades of experience in academia have helps me to
achieve my aim. May Allah (S. A. W) keeps him (and his family members) healthy,
accept his prayers and forgive all his shortcomings. I also appreciate the support I
received from my co supervisor in person of Professor (Dr) Mohd Nawawi. I
appreciate the effort I received from Dr. Amin Esmail Khalil for his enomous effort
to improve my manuscripts. The efforts of Arisona Arisona, Fathi Al-hamadi and
Talha Anees are highly appreciated. I thank all the staff and students of Geophysics
unit, Universiti Sains Malaysia who help in one way or the other in the course of
carrying out this research.
I acknowledge the moral, financial and spiritual support I received from the
entire members of Alhaji Usman Koguna’s family. To my parent, Alhaji Usman
Koguna and Hajiya Tamagaji Usman Koguna, may Allah for give them and reward
them with Jannatul Firdaus, without your support I would have never achieved this
dream. The assistance I received from my brother, Alhaji Aminu Usman (Kogunan
Katsina II), Halima Usman and Hadiza Usman during the trial moment of my life will
never be forgotten. I thank you very much for your support. I also remain indebted to
iii
my wife and children for their patients and understanding to stay away from them
during my academic pursuit, I highly appreciate it. I acknowledge the support I
received from my step mothers, right from my primary school to date.
For Umaru Musa Yar’adua University Katsina, I appreciate the opportunity
given to me to serve my people and pursue higher degrees. I remained resolute, loyal
and steadfast to render service as enshrined in the condition of service of the
university. I thank the management of this great University for given me full
sponsorship to pursue PhD exploration geophysics program. Special thanks to Dr.
Bashir Muhammad Gide, Dr. Sama’ila Batagarawa, Dr. Nura Liman Chiromawa and
Dr. Abdullahi Abdulrahaman who introduced me to Universiti Sains Malaysia where
I fulfilled my destiny.
iv
TABLE OF CONTENTS
Acknowledgement
ii
Table of Contents
iv
List of Tables
x
List of Figures
xii
List of Symbols
xxi
List of Abbreviations
xxiv
Abstrak
xxvi
Abstract
xxviii
CHAPTER 1 – INTRODUCTION
1.1 Background
1
1.2 Problem statements
7
1.3 Research question
11
1.4 Research objectives
11
1.5 Novelty of the study
12
1.6 Scope and limitation of the study
12
1.7 Thesis outline
13
CHAPTER 2 - LITERATURE REVIEW
2.1 Introduction
15
2.2 Magnetic susceptibility
15
2.2.1 Magnetic elements
17
2.3 Reduction of magnetic observations 17
v
2.3.1 Diurnal variation correction
18
2.3.2 Geomagnteic correction
18
2.3.3 Reduction to the pole and equator
19
2.3.4 Applications of magnetic method
20
2.3.5 Limitations of magnetic method
20
2.4 Euler homogeneity
21
2.4.1 Previus work
23
2.4.2 2D Euler deconvolution
25
2.4.3 3D Euler deconvolution
27
2.4.4 Differential similarity transform
28
2.4.5 Euler deconvolution using gravity tensor gradient
31
2.4.6 Extended Euler deconvolution
32
2.4.7 Euler deconvolution of higher derivatives
33
2.4.8 Analytical Euler (AN-EUL)
34
2.4.9 Euler deconvolution of analytic signal
35
2.4.10 Depth estimation using tilit derivatives
36
2.4.11 Related literitures
36
2.4.12 Drawbacks of the reviewed technique
41
2.5 Research Gap 41
2.6 Filtering of Euler deconvolution solution
42
2.7 Accuracy assessment
43
2.8 Chapter summary
47
vi
CHAPTER 3 - RESEARCH METHODOLOGY
3.1 Introduction
49
3.2 The present technique (AUTO-EUD)
51
3.2.1 Description of the technique
52
3.2.2 Data inversion
54
3.2.3 Filtering using AUTO-EUD
56
3.2.4 Plotting of Euler solution 57
3.3 Synthetic models
58
3.3.1 Enhancement filter assessment
68
3.3.1(a) Sphere and box model
70
3.3.2 Validation using synthetic models
72
3.3.2(a) Assessing the effect of inclination on AUTO-
EUD
72
3.3.2(b) Accuracy assessment of AUTO-EUD using
synthetic models
80
3.3.2(c) Effect of interfernce
84
3.3.2(d) Effect of noise
87
3.3.2(e) Effect of window size
88
3.3.2(f) Comparison between AUTO-EUD and CED
91
3.4 Validation using field model
92
3.4.1 Multiple sources field model, Leister (UK)
93
3.4.1(a) Comparison between total field and RTP results
94
3.4.2 Concrete wall field model, Leister (UK)
94
3.5 Application of AUTO-EUD to magnetic data of Nevada basin,
USA
96
3.5.1 Description of some part of Nevada basin, USA
96
3.5.2 Geology of Nevada, USA 96
vii
3.5.3 Data description 99
3.5.4 Data processing 100
3.6 Application of AUTO-EUD to Seberang Jaya, Malaysia 101
3.6.1 Description of some part of Sebarang Jaya, Malaysia 101
3.6.2 Geology of Sebarang Jaya, Malaysia
101
3.6.3 Data acquisition
102
3.6.4 Comparison between measured and computed vertical
derivatives of total field
104
3.6.5 Data inversion
105
3.7 Application of AUTO-EUD to Upper Benue Trough, Nigeria
105
3.7.1 Description of some part of Upper Benue Trough, Nigeria
105
3.7.2 Geology and tectonics of of Upper Benue Trough, Nigeria 106
3.7.3 Data acquisition
108
3.7.4 Data pre-processing
109
3.7.5 Reduction to the equator
110
3.7.6 Data inversion
110
3.7.7 Data filtering
111
3.8 Chapter summary 111
CHAPTER 4 - RESULTS AND DISCUSSIONS
4.1 Introduction
113
4.2 The present technique
113
4.3 Synthetic models
116
4.3.1 Enhancement filter assessment 116
4.3.1(a) Sphere and box model 117
viii
4.3.2 Validation using synthetic models 138
4.3.2(a) Assessing the effect of inclination on AUTO-
EUD
138
4.3.2(b) Accuracy assessment of AUTO-EUD 171
4.3.2(c) Effect of interference 177
4.3.2(d) Effect of noise 182
4.3.2(e) Effect of window size 184
4.3.2(f) Comparison between AUTO-EUD and CED 197
4.3.3 Structural index of synthetic models 200
4.4 Field model 201
4.4.1 Multiple source field model 201
4.4.1(a) Comparison between total field and RTP data 202
4.4.1(b) Inversion output result after filtering 206
4.4.2 Concrete wall field model 208
4.4.3 Structural index of field models 210
4.5 Application of AUTO-EUD to magnetic data of Nevada basin,
USA
212
4.5.1 Comparison between estimated position and geology 217
4.6 Application of AUTO-EUD to magnetic data of Sebarang Jaya 221
4.6.1 Measured and computed vertical derivatives 224
4.6.2 Comparison between CED and AUTO-EUD 227
4.7 Application of AUTO-EUD to magnetic data Upper Benue Trough,
Nigeria
228
4.7.1 Reduction to the equator of total aeromagnetic intensity 229
4.7.2 Analytic signal of total aeromagnetic field intensity 230
4.7.3 Analytic signal of RTE of total aeromagnetic field
intensity
231
ix
4.7.4 Comparison between the AUTO-EUD outputs inverted
using total field and it’s RTE data
232
4.7.5 Comparison between CED and AUTO-EUD
238
4.8 Structural index of application site
240
4.9 Chapter summary
242
CHAPTER 5 - CONCLUSION AND RECOMMENDATIONS
5.0 Introduction
243
5.1 Conclusion 243
5.2 Limitation and sources of error
245
5.3 Recommendations
245
REFERENCES
246
APPENDICES
LIST OF PUBLICATIONS
x
LIST OF TABLES
Page
Table 1.1 Geophysical methods 2
Table 2.1 Magnetic susceptibility of rocks and minerals 16
Table 2.2 Structural index of magnetic source 22
Table 2.3 Grading scale of synthetic models 46
Table 3.1 Dimension of the targets of multiple sphere model 84
Table 3.2 Dimension of the targets of complex model 2 86
Table 3.3 Description of the targets in the field model 94
Table 4.1 Summary of the result (Profile AA’) 118
Table 4.2 Summary of the result (Profile BB’) 118
Table 4.3 Summary of the result (Profile CC’) 119
Table 4.4 Inversion and filtering output of isolated box model 140
Table 4.5 Inversion and filtering output of isolated contact model 144
Table 4.6 Inversion and filtering output of isolated cylinder model 148
Table 4.7 Inversion and filtering output of isolated dike model 152
Table 4.8 Inversion and filtering output of isolated sphere model 157
Table 4.9 Inversion and filtering output of complex model 1 164
Table 4.10 Inversion and filtering output of slanted box model 167
Table 4.11 Inversion and filtering output of infinite box model 172
Table 4.12 Inversion and filtering output of multiple cylinder model 174
Table 4.13 Inversion and filtering output of multiple sphere model 176
Table 4.14 Solutions of complex sphere model after filtering 178
Table 4.15 Estimated parameters of concrete wall model
180
xi
Table 4.16 Estimated parameters of two cylinder model 184
Table 4.17 Solutions of finite box model using different windows 186
Table 4.18 Result of combined two sphere and cylinder model 195
Table 4.19 Average inversion result of two sphere model
198
Table 4.20 Total field inversion result of multiple source field model
204
Table 4.21 Reduction to the pole of field model inversion result
204
Table 4.22 Total field inversion result of multiple source field model
after filtering using 4 parameters
207
Table 4.23 Inversion result of concrete wall model after filtering
209
Table 4.24 Statistics of solutions obtained from the inversion of total
field derivatives
216
Table 4.25 Statistics of solutions obtained from the inversion of RTP
derivatives
216
Table 4.26 Comparison between CED and AUTO-EUD
216
Table 4.27 Correlation coefficient (R) of the profiles analysed
227
Table 4.28 Inversion result of Sebarang Jaya site using CED and
AUTO-EUD
227
Table 4.29 AUTO-EUD solution (SI deviation = 0.2) summary of the
depth to the magnetic basement source indicating depth
interval and percentage obtained from the number of
solution from the RTE of total field
237
Table 4.30 AUTO-EUD solution (SI deviation = 0.2) summary of the
depth to the magnetic basement source indicating depth
interval and percentage obtained from the number of
solution from the RTE of total field
237
Table 4.31 Statistics of the inversion result using AUTO-EUD and
conventional Euler (Program) deconvolution technique
239
xii
LIST OF FIGURES
Page
Figure 3.1 Flow chart of the methodology for the application site and
aeromagnetic grid
50
Figure 3.2 Figure 3.2: Description of data inversion and filtering
procedure
53
Figure 3.3 A plane of box model indicating directions of some
magnetic element
60
Figure 3.4 A section of box model indicating the coordinate system
60
Figure 3.5 A section of contact model
62
Figure 3.6 Plane of horizontal cylinder model indicating directions of
the magnetic element
64
Figure 3.7 A section of horizontal cylinder model
64
Figure 3.8 Plane of dike model indicating directions of the magnetic
element
66
Figure 3.9 Section of a dike model
66
Figure 3.10 Plane of sphere model indicating directions of the magnetic
element
67
Figure 3.11 Section of sphere model indicating directions of the
magnetic element
68
Figure 3.12 Sphere and box model
71
Figure 3.13 Three Profiles (AA’, BB’ and CC’) on Sphere and box
model
71
Figure 3.14 Base map of isolated box model
73
Figure 3.15 Base map of isolated contact model
74
Figure 3.16 Base map of isolated cylinder model
75
Figure 3.17 Base map of isolated dike model
76
Figure 3.18 Base map of isolated sphere model
77
xiii
Figure 3.19 Base map of complex model 1
79
Figure 3.20 Slanted finite box model
80
Figure 3.21 Base map infinite box model
82
Figure 3.22 Base map of multiple cylinder model
83
Figure 3.23 Multiple sphere model
84
Figure 3.24 Base map of complex model 2
85
Figure 3.25 Base map of concrete wall model
87
Figure 3.26 Base map of two cylinder model
88
Figure 3.27 Base map of finite box model showing the true position of
the target
89
Figure 3.28 Base map of finite box model showing the true position of
the target
91
Figure 3.29 Base map of two sphere model indicating true position of
the target
92
Figure 3.30 Base map of multiple source model
93
Figure 3.31 Base map of concrete wall model
95
Figure 3.32 Geological map of Nevada
97
Figure 3.33 Survey stations super imposed on base map of geotechnical
site
103
Figure 3.34 Layout of GEM gradiometer system
104
Figure 3.35 Map of Nigeria showing the location of Upper Benue
Trough
106
Figure 3.36 Geological map of the study area
108
Figure 4.1 Anomalous magnetic field (nT) of sphere and box model
117
Figure 4.2 Anomalous magnetic field (nT) of a box and sphere model
with Profile AA’, BB’ and CC’
118
Figure 4.3 Vertical derivative of the field
120
Figure 4.4 Vertical derivative (dz) of Profile AA’
120
xiv
Figure 4.5 Vertical derivative (dz) of Profile BB’
121
Figure 4.6 Vertical derivative (dz) of Profile CC’ 121
Figure 4.7 Horizontal derivative in x-direction (nT/km) of the field
122
Figure 4.8 Horizontal derivative in y-direction (nT/km) of the field
123
Figure 4.9 Total horizontal derivatives (nT/km) of the field
123
Figure 4.10 Total horizontal derivatives (nT/km) of Profile AA’
124
Figure 4.11 Total horizontal derivatives (THDR) of Profile BB’
124
Figure 4.12 Total horizontal derivatives (THDR) of Profile CC’
125
Figure 4.13 Analytic signal (nT/km) of the field
126
Figure 4.14 Analytic signal (AS) of Profile (AA’)
126
Figure 4.15 Analytic signal (AS) of Profile (BB’)
127
Figure 4.16 Analytic signal (AS) of Profile (CC’)
127
Figure 4.17 Tilt derivative (nT/km) of the field
128
Figure 4.18 Tilt derivative (nT/km) of Profile AA’
129
Figure 4.19 Tilt derivative (nT/km) of Profile BB’
129
Figure 4.20 Tilt derivative (nT/km) of Profile CC’
130
Figure 4.21 Tilt angle
130
Figure 4.22 Total horizontal derivative of tilt derivative (nT/km) of the
field
131
Figure 4.23 Total horizontal derivative of tilt derivative (nT/km) of
Profile AA’
132
Figure 4.24 Total horizontal derivative of tilt derivative (nT/km) of
Profile BB’
132
Figure 4.25 Total horizontal derivative of tilt derivative (nT/km) of
Profile CC’
133
Figure 4.26 Laplacian of the field (nT2/km
2)
134
Figure 4.27 Laplacian Profile AA’
134
xv
Figure 4.28 Laplacian Profile BB’
135
Figure 4.29 Laplacian Profile CC’
135
Figure 4.30 Enhancement filters along Profile AA’
136
Figure 4.31 Enhancement filters along Profile BB’
137
Figure 4.32 Enhancement filters along Profile CC’
137
Figure 4.33 Anomalous magnetic field intensity (nT) of a box model
(a) I = 0(b) I = 15° (c) I = 30° (d) I = 45° (e) I = 60°
(f) I = 75° and (g) I = 90°
141
Figure 4.34 Depth solutions super-imposed on AS (nT/km) of a box
model
(a) I = 0° (b) I = 15° (c) I = 30° (d) I = 45° (e) I = 60°
(f) I = 75° and (g) I = 90°
142
Figure 4.35 Anomalous magnetic field intensity of a contact model
(a) I = 0° (b) I = 15° (c) I = 30° (d) I = 45° (e) I = 60°
(f) I = 75° and (g) I = 90°
145
Figure 4.36 Depth solutions super-imposed on AS of contact model
(a) I = 0° (b) I = 15° (c) I = 30° (d) I = 45° (e) I = 60°
(f) I = 75° and (g) I = 90°
146
Figure 4.37 Anomalous magnetic field intensity of cylinder model
(a) I = 0° (b) I = 15° (c) I = 30° (d) I = 45° (e) I = 60°
(f) I = 75° and (g) I = 90°
149
Figure 4.38 Depth solution of a cylinder model super-imposed on AS
for:
(a) I = 0° (b) I = 15° (c) I = 30° (d) I = 45°(e) I = 60°
(f) I = 75° and (g) I = 90°
150
Figure 4.39 Anomalous magnetic field intensity of dike model
(a) I = 0° (b) I = 15° (c) I = 30° (d) I = 45° (e) I = 60°
(f) I = 75° and (g) I = 90°
153
Figure 4.40 Depth solutions super imposed on AS of dike model
(a) I = 0° (b) I = 15° (c) I = 30° (d) I = 45° (e) I = 60°
(f) I = 75° and (g) I = 90°
154
Figure 4.41 Anomalous magnetic field intensity of sphere model
(a) I = 0° (b) I = 15° (c) I = 30° (d) I = 45° (e) I = 60°
(f) I =75° and (g) I = 90°
158
xvi
Figure 4.42 Depth solutions super imposed on AS of anomalous
magnetic field intensity of sphere model
(a) I = 0°(b) I = 15° (c) I = 30° (d) I = 45° (e) I = 60°
(f) I = 75° and (g) I = 90°
159
Figure 4.43 Anomalous magnetic field intensity of complex model
(a) I = 0° (b) I = 15° (c) I = 30° (d) I = 45° (e) I = 60°
(f) I = 75° and (g) I = 90°
162
Figure 4.44 Depth solutions super imposed on AS of anomalous
magnetic field intensity of complex model
(a) I = 0°(b) I = 15° (c) I = 30° (d) I = 45° (e) I = 60°
(f) I = 75° and (g) I = 90°
163
Figure 4.45 Anomalous magnetic field intensity of slanted model
(a) I = 0° (b) I = 15° (c) I = 30° (d) I = 45° (e) I = 60°
(f) I = 75° and (g) I = 90°
168
Figure 4.46 Depth solutions super imposed on AS of anomalous
magnetic field intensity of slanted box model
(a) I = 0° (b) I = 15° (c) I = 30° (d) I = 45° (e) I = 60°
(f) I = 75° and (g) I = 90°
169
Figure 4.47 Anomalous field (nT) of infinite box model
172
Figure 4.48 Depth solutions of infinite box model super imposed on
AS (nT/km)
173
Figure 4.49 Anomalous field (nT) of multiple cylinder model
174
Figure 4.50 Depth solutions of multiple cylinder model super imposed
on AS (nT/km)
175
Figure 4.51 Anomalous magnetic field (nT/km)
176
Figure 4.52 Depth solutions super imposed on AS (nT/km)
177
Figure 4.53 Anomalous magnetic field (nT) of complex model 2
179
Figure 4.54 Depth solutions super imposed on AS (nT/km)
179
Figure 4.55 Anomalous magnetic field (nT)
181
Figure 4.56 Depth solutions super imposed on AS (nT/km)
181
Figure 4.57 Anomalous magnetic field of two cylinder model
183
Figure 4.58 Depth solutions super imposed on AS of anomalous
magnetic field with10% noise with Gaussian distribution
183
xvii
Figure 4.59 Depth solutions of finite box model using the window size
of 5×5 grid points super-imposed on AS
187
Figure 4.60 Depth solutions of finite box model using the window size
of 7×7 grid points and convolution window of 7 units
super-imposed on AS
187
Figure 4.61 Depth solutions of finite box model using the window size
of 7×7 grid points and convolution window of 11 units (all
SI)super-imposed on AS
188
Figure 4.62 Depth solutions of finite box model using the window size
of 7×7 grid points and convolution window of 11 units
super-imposed on AS
188
Figure 4.63 Depth solutions of finite box model using the window size
of 7×7 grid points and convolution window of 11 units
super-imposed on AS
189
Figure 4.64 Depth solutions of finite box model using the window size
of 9×9 grid points and convolution window of 11 units
super-imposed on AS (all values of N)
190
Figure 4.65 Depth solutions of finite box model using the window size
of 9×9 grid points and convolution window of 11 units
super-imposed on AS (-0.44≤N≤0.44)
191
Figure 4.66 Depth solutions of finite box model using the window size
of 9×9 grid points and convolution window of 11 units
super-imposed on AS (0.45≤N≤1.44)
191
Figure 4.67 Depth solutions of finite box model using the window size
of 11×11 grid points and convolution window of 11 units
super-imposed on AS (all values of N)
192
Figure 4.68 Depth solutions of finite box model using the window size
of 11×11 grid points and convolution window of 11 units
super-imposed on AS (-0.44≤N≤0.44)
192
Figure 4.69 Depth solutions of finite box model using the window size
of 11×11 grid points and convolution window of 11 units
super-imposed on AS (0.45≤N≤1.44)
193
Figure 4.70 Anomalous magnetic field (nT) of two sphere and cylinder
model
195
Figure 4.71 Depth solutions super-imposed on AS of anomalous
magnetic field (nT) of two sphere model inversed using
window size 9×9 grid points (cw = 9 units; SI dev = 0.2;
error = 1% and threshold of AS)
196
xviii
Figure 4.72 Depth solutions super-imposed on AS of anomalous
magnetic field (nT) of two sphere and cylinder model
inversed using window size 13×13 grid points (cw = 13; SI
dev = 0.2 error = 1 and threshold of AS = 2)
196
Figure 4.73 Depth solutions super-imposed on AS of anomalous
magnetic field (nT) of two sphere model inversed using
window size 21×21 grid points (ws = 21; SI dev = 0.2;
error = 1 and threshold of AS = 2)
197
Figure 4.74 Anomalous field of two sphere model
199
Figure 4.75 Estimated position (obtained from inversion using a
commercial software, CED) super imposed on AS
199
Figure 4.76 Estimated (obtained from inversion using AUTO-EUD)
super imposed on AS
200
Figure 4.77 Total field (nT) of multiple source field model
203
Figure 4.78 RTP of multiple source field model
203
Figure 4.79 Depth solutions (obtained from inversion of the total field)
super imposed on AS (nT/m) of total field of field model
205
Figure 4.80 Depth solutions (obtained from inversion of RTP of the
total field) super imposed on AS (nT/m) of Reduction to
the pole of field model
205
Figure 4.81 AS (nT/m) of total field of field model (after filtering)
207
Figure 4.82 Total magnetic field intensity of the field model
209
Figure 4.83 Depth solutions of buried walls model super imposed on
AS
209
Figure 4.84 Total magnetic field (nT) of some part of Nevada
213
Figure 4.85 RTP of total magnetic field (nT) of some part of Nevada
214
Figure 4.86 Analytic signal (nT/km) map of the total magnetic field
214
Figure 4.87 Analytic signal (nT/km) of RTP
215
Figure 4.88 Depth solutions (from the inversion of total field
derivatives using AUTO-EUD) super imposed on analytic
signal (nT/km) map of the total magnetic field
215
xix
Figure 4.89 Depth solutions (from the inversion of RTP data
derivatives using AUTO-EUD) super imposed on analytic
signal (nT/m) of RTP data
216
Figure 4.90 Depth solutions (from the inversion of RTP data
derivatives) super imposed on analytic signal (nT/m) of
RTP data
218
Figure 4.91 Range of valued of the estimated depth solutions (obtained
from the inversion of total field data)
218
Figure 4.92 Depth solutions (obtained from the inversion of total field
data) super imposed on geological map of the study area
219
Figure 4.93 Range of valued of the estimated structural index (obtained
from the inversion of total field data)
220
Figure 4.94 Total magnetic field intensity map of application site
222
Figure 4.95 Depth solutions (application site) super imposed on AS
(nT/m)
223
Figure 4.96 Solutions of structural index super imposed on AS (nT/m)
of Sebarang Jaya, Malaysia
223
Figure 4.97 Measured vertical derivatives (gradiometer reading) of
application site
224
Figure 4.98 Computed vertical derivatives of application site
225
Figure 4.99 Computed vertical derivative (contour lines) of application
site superimposed on measured vertical derivative with
three selected profiles (AA’, BB’ and CC’)
225
Figure 4.100 Profile AA’: Measured (Meas) and computed (Comp)
vertical derivatives of application site
226
Figure 4.101 Profile BB’: Measured (meas) and computed (comp)
vertical derivatives of application site
226
Figure 4.102 Profile CC’: Measured (Meas) and computed (Comp)
vertical derivatives of application site
226
Figure 4.103 Total aeromagnetic magnetic field intensity (nT) of some
part of Upper Benue Trough
228
Figure 4.104 Reduction to the equator of total aeromagnetic magnetic
field intensity (nT) of some part of Upper Benue Trough
230
xx
Figure 4.105 Analytic signal (nT/km) of total aeromagnetic magnetic
field intensity of some part of Upper Benue Trough
231
Figure 4.106 Analytic signal (nT/km) of RTE of total aeromagnetic
magnetic field intensity of some part of Upper Benue
Trough
232
Figure 4.107 AUTO-EUD depth solutions super-imposed on AS of total
field map, SI deviation = 0.2, the maximum acceptable
regression error was10%, the threshold value of AS was
0.005 nT/m, the centre of convolution window was 13 and
negative SI values are rejected
234
Figure 4.108 AUTO-EUD depth solutions (obtained from RTE data)
super-imposed on AS of RTE map, SI dev = 0.2, the
maximum acceptable regression error was10%, the
threshold value of AS was 0.005 nT/m, the centre of
convolution window was 13 and negative SI values are
rejected
235
Figure 4.109 AUTO-EUD depth solutions super-imposed on AS map, SI
deviation = 0, the maximum acceptable regression error
was10%, the threshold value of AS was 0.005 nT/m, the
centre of convolution window was 13 and negative N
values are rejected
236
Figure 4.110 Lineament obtained from AS super-imposed on AS (nT/m)
map
240
xxi
LIST OF SYMBOLS
A Angle between positive x-axis and magnetic north (0° to 90°) of dike
model
a order of differentiation
B Background field/ base level of the total magnetic field
c Magnetic constant
Partial differentiation
δ: 90 - I
d dip
d1 Depth to top of the structure of contact model
d2 Depth to the bottom of the structure of contact model
E True model parameter
Estimated model parameter
f field
F Force
f(x) Observed field at x
General function
H magnetizing field
h depth of the element (box model) below the level of observation
I inclination of the geomagnetic field
i Positive and negative in the Northern and Southern Hemisphere
respectively
Ip Polarization
effective inclination of cylinder model
real inclination of cylinder model
xxii
k Magnetic susceptibility
K Dipole moment
L1 direction cosine characterizing polarization vector of the volume
element (box model)
L2 direction sine characterizing polarization vector of the volume element
(box model)
M Magnetization
m magnetic induction
N Structural index
n direction cosine vector of the earth’s field (box model)
n Degree of homogeneity
O’ Origin
∫
magnetic flux
Adjacent distance from the observation point to the depth to the top of
the dike model
Ro’p vector originate at O’ towards P
r Radius between the poles
Vector directed from the source point to observation point
S the cross-sectional area of cylinder model
operator of the differential similarity transform
: effective total intensity of cylinder model
T Tesla
t Number of observations
xxiii
u Inclination
v Declination
ui Set of variables
w Width of the sheet
x observation point along x
x0 Position of source in x direction
Xi Independent variable in MLR
y0 Position of source in y direction
Yi Dependent variable in MLR
y observation point along y
z observation point along z
z0 Position of source in z direction/depth
Coordinate of the volume element d
ρ Density of magnetic flux
β0 intercept of dependent variable
βi Slope of regression/ coefficient of variable
εi Regression error
∆T(x) Total magnetic intensity at x
Permeability of free space
µ 1+K
τ threshold value of Euler depth solutions
θ the azimuth angle of box model
strike angle of the structure measured clockwise from north of contact
model
∆n Absolute error
xxiv
LIST OF ABBREVIATIONS
AN-EUL Analytical Euler
AS Analytic Signal
AAS Amplitude of Analytic Signal
AS_tot Analytic signal of total field
AS_rtp Analytic signal of RTP data
AUTO-EUD A technique of magnetic data interpretation for the fully
automation (AUTO) of Euler deconvolution (EUD) relation
CED Conventional Euler Deconvolution
cw distance from the centre of convolution window
dz Vertical derivative
DST Differential Similarity Transform
ED Euler deconvolution
fd declination of the ambient field
ft Feet
IAF Integrated and Automated Filter
IGRF International Geomagnetic Reference Field
Lap Laplacian
md declination of magnetization
MLR Multiple Linear Regression
Ng Geographic north
nT Nano Tesla
Oe Reduction to the equator
Op Reduction to the pole
THDR Total Horizontal Derivative
xxv
RTP Reduction to the pole
RTE Reduction to The Equator
S.I International System of Units
SI dev Deviation of structural index from the integer value
SD Standard Deviation
TA Tilt Angle
TD Tilt derivative
THD_TD Total horizontal derivative of tilt derivative
TDR Tilt Derivative
TMI Total Magnetic Field Intencity
USGS United States Geological Survey
ws Window size
2D Two dimensional
3D Three dimensional
pmic Micro scale clustering
pmac Macro scale clustering
xxvi
PENAFSIRAN AUTOMATIK DATA MAGNETIK MENGGUNAKAN
KAEDAH DEKONVOLUSI EULER DENGAN ALGORITMA TERUBAH
SUAI
ABSTRAK
Dekonvolusi Euler konvensional mempunyai lima parameter yang tidak
diketahui untuk diselesaikan iaitu lokasi sumber (x0, y0 and z0), medan latar belakang
(B) dan indeks struktur (N). Antara lima perkara yang tidak diketahui ini, indeks
struktur dipilih secara manual oleh pengguna. Input manual indeks struktur ke dalam
persamaan Euler menjadikan teknik ini semi-automatik dan tafsiran menjadi
subjektif. Untuk menangani masalah ini, penyelidikan ini bertujuan untuk
mengautomasikan teknik dekonvolusi Euler dan memperkenalkan teknik penurasan
untuk membezakan penyelesaian yang boleh dipercayai daripada output
penyongsangan. Ia juga merupakan sebahagian daripada objektif kajian ini, untuk
menilai kesan kecondongan teknik baru dan menyiasat ketepatan algoritma yang
diubahsuai. Regresi linear berganda digunakan untuk menyelesaikan lima parameter
hubungan dekonvolusi Euler yang tidak diketahui untuk data magnetik bergrid.
Untuk menyediakan penurasan yang berkesan, enam penuras dianalisis untuk
memilih yang terbaik yang akan digunakan sebagai bantuan untuk penuras
penyelesaian Euler. Kriteria lain yang digunakan untuk penurasan output
penyongsangan ialah jarak dari pusat tetingkap konvolusi, sisihan indeks struktur dan
ralat regresi. Kriteria ini disepadukan, automatik dan digunakan untuk memilih
penyelesaian yang boleh dipercayai dari output penyongsangan. Kesan kecondongan
pada teknik ini dinilai menggunakan kajian model sintetik (mudah dan gabungan) dan
model lapangan. Setiap model disimulasikan menggunakan kecondongan yang
xxvii
belainan (0°, 15°, 30°, 45°, 60°, 75° dan 90°) dengan parameter lain yang tetap.
Terbitan bagi setiap set data dikira dan disongsangkan. Penyelesaian yang boleh
dipercayai dipilih dan hasilnya dibandingkan. Untuk data sebenar, keputusan
tersongsang dan terturas dari jumlah medan dan data yang dikurangkan kepada kutub
juga dibandingkan. Kajian model sintetik dan lapangan atas sumber magnet
digunakan untuk menunjukkan keupayaan algoritma yang diubah suai untuk
menyelesaikan lokasi sumber dan sifat sasaran. Hasil songsangan (fail) terdiri
daripada 5 parameter yang tidak diketahui yang terdapat dalam persamaan
dekonvolusi Euler. Isyarat analitik didapati mempunyai banyak kelebihan terhadap
penuras yang dianalisis dan ia dipilih sebagai salah satu kriteria (sebagai tambahan
kepada tiga kriteria yang disebutkan) untuk penapisan. Hasil model sintetik
menggunakan kecondongan yang berlainan adalah sama. Hasil yang diperolehi dari
penyongsangan jumlah medan dan data yang dikurangkan kepada kutub dari model
medan pelbagai sumber juga adalah sama. Anggaran kedalaman min yang diperoleh
dari penyongsangan jumlah medan dan data yang dikurangkan kepada kutub bagi
data aeromagnet Nevada adalah 801 dan 787 m masing-masing. Keputusan yang
diperolehi daripada analisis data Nevada telah memperkuatkan hasil yang diperolehi
daripada pemodelan sintetik. Dalam kebanyakan ujian dijalankan, algorithma yang
diperkenalkan menentukan kedudukan sasaran dengan kejituan yang baik dan teknik
ini tidak bergantung pada kecondongan. Teknik ini adalah mod cepat tafsiran data
magnetik dan mudah dilaksanakan kerana ia melibatkan terbitan tertib pertama
medan tersebut.
xxviii
AUTOMATIC INTERPRETATION OF MAGNETIC DATA USING EULER
DECONVOLUTION WITH MODIFIED ALGORITHM
ABTRACT
The conventional Euler deconvolution has five unknown parameters to be
solve which are the location of source (x0, y0 and z0), the background field (B) and the
structural index (N). Among these 5 unknowns, the structural index is to be manually
selected by the user. The manual input of structural index into the Euler equation
makes the technique to be subjective and semi-automated. The objectives of this
research are, to automate Euler deconvolution equation and introduce a filter for
discriminating reliable solution from the inversion output. It is also part of the
objectives of this research, to assess the effect of inclination on the new technique
and investigate the accuracy of the introduced algorithm. Multiple linear regression
was used to solve the five unknown parameters of Euler deconvolution relation for
gridded magnetic data. To provide an effective filtering, six filters were analysed in
order to select a best one that would be used as an aid for filtering Euler solutions.
Other criteria used for filtering of the inversion output are distance from the centre of
convolution window, deviation of structural index and regression error. These
criterions are integrated, automated and used for selecting more reliable solutions
from the inversion output. The effect of inclination on this technique is assessed
using synthetic (simple and combined) and field model’s studies. Each model is
simulated using different inclinations (0°, 15°, 30°, 45°, 60°, 75° and 90°) with other
parameters kept constant. The derivatives of each data set were computed, inverted,
more reliable solutions are selected and the results were compared. For real data, the
inverted and filtered results from the total field and it’s reduced to the pole data were
xxix
also compared. The synthetic and field models studies over magnetic sources were
used to demonstrate the ability of the modified technique to solve for the source
location and nature of the target. The inversion (file) result comprises of 5 unknown
parameters contained in Euler deconvolution equation. The inversion can be achieved
by prescribing the window size which is the only choice a user has to make. Analytic
signal is found to have so many advantages over the filters analysed and it is chosen
as one of the criteria (in addition to the three mentioned criteria) for filtering. The
results of synthetic models using different inclinations are about the same. The result
obtained from the inversion of total field and it’s reduced to the pole data of multiple
source field model are about the same. The mean depth estimates obtained from the
inversion of total field and reduced to the pole data of aeromagnetic data from
Nevada are 801 and 787 m respectively. The results obtained from the analysis of
Nevada data have further corroborated the result obtained from the synthetic
modeling. In most of the tests carried out, the introduced algorithm located the
position of the target with good precision and the technique does not depend on
inclination. The technique is fast mode of magnetic data interpretation and easy to
implement as it involves first order derivatives of the field.
1
CHAPTER 1
INTRODUCTION
1.1 Background
Now a days geophysical methods are widely applied to investigate subsurface
of the Earth in order to explore geological structures of economic interest (in most
cases) in areas of hydrology, solid minerals (Arisona et al. 2016), hydrocarbons,
engineering, archaeological, geothermal studies (Khalil et al. 2017), geo-hazard
assessment, geochemical (Yang et al. 2015) and environmental studies (Loke et al.
2013; Yang et al. 2015). The choice of geophysical methods over other techniques is
partly due their nondestructive nature and cost effective in large area investigation.
Geophysical survey can be carried out on land, from the air or over water because of
the improved sensitivity of the measuring instruments. The speed of operation from
air geophysical survey is another feature that attracted many Earth scientists to these
techniques. The use of geophysical methods permits geophysicist to investigate the
conceal features beneath the Earth’s surface. These features appear in the form of
anomaly due to different physical properties in the subsurface that need to be
interpreted in to its geological relevance. The methods are Seismic, Electrical,
Ground penetration radar, Transient electromagnetic (TEM), Gravity and Magnetic
method among others.
Geophysical methods are classified as those that make use of the natural field
of the Earth e.g. gravity and magnetic methods, and methods that require the input of
artificially generated energy, e.g. seismic reflection and electrical methods. The
geophysical surveying methods, measured parameters together with their respective
operative physical properties are shown in the Table 1.1. It is the operative physical
2
property that determines the specific use of any method. Thus for example, seismic
or electrical method are suitable for locating water table because saturated rock may
be distinguished from dry rock by its higher seismic velocity and higher electrical
conductivity. Nevertheless, other considerations also determine the type of methods
employed in a geophysical exploration program. For example, reconnaissance
surveys are often carried out from air because of the high speed of operation. In such
cases the electrical or seismic methods are not applicable since these require physical
contact with the ground for the direct input of energy.
Table 1.1: Geophysical methods (Kearey et al., 2002)
Method Measured Parameters Operative Physical
Properties
Seismic Travel times of reflected/ refracted
seismic waves
Density and elastic moduli,
which determine the
propagation velocity of
seismic waves
Gravity Spatial variations in the strength of the
gravitational field of the Earth
Density
Magnetics
Spatial variations in the strength of the
geomagnetic field of the Earth
Magnetic susceptibility
Electrical
-Resistivity
-Induced
polarization
-Earth resistance
-Polarization voltages or frequency-
dependent ground resistance
-Electrical conductivity
-Electrical capacitance
Self-potential Electrical potential
Electrical conductivity
Electromagnetic Response to electromagnetic radiation
Electrical conductivity and
inductance
Radar Travel times of reflected radar pulses
Dielectric constant
Measurement of geomagnetic field can be used to determine the structure of
the Earth since many rocks have magnetization. Magnetic method can be used as a
tool for detecting shallow structure of local, regional and global scales. With the aid
3
of techniques used for the inversion of magnetic data (Gerovska, and Bravo 2003;
Gerovska, et al. 2010;Cooper, 2014; Cooper and Whitehead 2016; Salem, 2007), it is
possible to determine the horizontal and vertical position of concealed metallic
objects in the near vicinity of the earth’s surface in addition to the delineation of
deep-seated structures. The advantages of magnetic method include its ability to
detect near surface weak magnetic signal produced by the buried objects and its
relative ease of operation.
The choice of a geophysical method to locate a particular geological structure
depends on its mineral content. Some of the reasons for choosing the magnetic
method are:
i. This method is widely used in mineral and petroleum explorations,
engineering, environmental, geothermal and global applications.
Magnetic method is the most versatile of geophysical prospecting
techniques.
ii. Magnetic measurements are made more easily and cheaply than most
geophysical measurements (Telford et al., 1990).
iii. In order to understand this field (geophysics) very well, magnetic
method needs to be studied since the study of the Earth’s magnetism is
the oldest branch of geophysics.
iv. Magnetic method is one of the methods that use the natural field of the
Earth, unlike some other methods that requires the artificially generated
field. It is therefore provide room to understand the variation of a
certain phenomenon on the Earth.
v. Aeromagnetic maps of most of the areas around the globe are available
for free or at nominal amount.
4
Aeromagnetic survey is carried out in order to detect rocks or minerals that
have abnormal magnetic properties which can be identified by causing anomalies in
the geomagnetic field. It is fast, cost effective and accessible technique used for
regional geological mapping, mineral and petroleum exploration (Chinwuka, 2012).
Euler deconvolution can assist in the interpretation of aeromagnetic data by
indicating the nature of the basement topography (undulating), depth and the
direction of steepness. Overburden thickness of the sedimentary sediment is very
essential in hydrocarbon exploration.
Generally, potential field data interpretation can be categorized into three
sections; forward modeling, inverse modeling, data enhancement and display
(Blakely, 1996). Modeling is an essential aspect of geophysics because it can be used
to predict a particular geological structure based on known model parameters. It can
also be used to determine feasible subsurface distribution of physical properties of
the target. The former and latter processes are known as forward and inversion
modeling. Mathematical modeling can be divided into three main groups which are
analytic, empirical and numerical models. Analytical modeling applies to simple
cases only and it provides error free solution. Analytical modeling is a vital tool used
in potential field data inversion. In general, modeling of geophysical data is
addressed in terms of depth to simple magnetic or gravity sources. Modeling leads to
a distinct inversion techniques as a result of non-uniqueness nature of the causative
sources.
The difficulties attached on seeking an inverse solution are:
i. Scientifically, error is present in all the measurements collected due to
instrumental and systematic error.
5
ii. The presence of sub-surface features that are not properly addressed by
a model
iii. Superposition of close features. However, these effects can be
constrained by using geological map and borehole information.
There are so many depth estimation techniques that assist geophysicist in
potential field data interpretation such as Werner deconvolution, source parameter
imaging (SPI), source location using total field homogeneity, depth from extreme
points, tilt depth and so on. In addition to the mentioned manual or automatic/semi-
automatic depth estimation techniques, Euler deconvolution is powerful technique
designed to analyze large amount of potential field data. It has been applied
extensively in delineating geologic boundaries (Hsu et al., 1996; Ugalde and Morris,
2010; Barbosa and Silva, 2011), and locating geothermal sources or hot springs
(Nouraliee et al., 2015); and is combined with other geophysical methods to ensure
enhanced interpretation of subsurface geology. It is one of the techniques that can be
used to provide fast means of data interpretation. Euler deconvolution technique uses
field and its derivatives in the system of linear equation in relation to the source
coordinate to estimate depth and location of anomalous source. This technique can
assist geoscientist by indicating portion of interest which can then further be
analyzed in detail. Some of the justifications on the need of depth estimation
technique are:
i. Large amount of potential field data sets (especially magnetic and
gravity methods) have been collected using aeroplane, ships and
satellites in regional/global scale. These data sets need to be process
and interpreted in to its geological relevance.
6
ii. The thickness of the sedimentary section and depth of ore bodies (that
contains magnetic minerals) are highly interested in hydrocarbon and
mineral exploration respectively.
iii. Euler deconvolution technique (Thompson, 1982) has been popularly
used by Chevron Oil Companies and within the Gulf, EULDPH is also
applied by Durrheim (1983), Corner and Wisher (1989) to determine
magnetic markers in search for gold in Witwatersrand Basin.
In addition, Euler deconvolution technique does not assume any geological
model, but it requires (prior knowledge of the rate of decay of the field of a particular
source) structural index which gives the nature of the geological structure. The
anomaly source is considered as singular point that consists of elementary potential
field distribution such as point poles or dipoles. An anomaly is considered as the
field caused by local variation in the geomagnetic field given rise by a singular point
of source. With the aid of Euler homogeneity relation, magnetic method can be used
to delineate the presence of metallic structures in the subsurface. Therefore, some of
the advantages of this technique are speed of operation, ability to interpret large data
sets and its implementation is less tedious.
The conventional Euler deconvolution (Thompson 1982; Reid et al 1990;
Ugalde and Morris 2010; Barbosa and Silva 2011; Oruç and Selim 2011; Chen et al.
2014) has 5 unknown parameters which are the location of source in x, y and z-
directions (x0, y0 and z0), the background field and the structural index (N). Among
these 5 unknowns, N is to be manually selected by the interpreter/user. An interpreter
has to solve the equation using different N and finally select the best set of solution.
The interpreter is left with the decision that has the highest impact on the depth
7
solutions: which N should be chosen? Much of the interpreter’s efforts will be
exhausted on choosing the solution produced by the appropriate structural index.
1.2 Problem statement
i. The manual input of structural index into the Euler equation makes the
technique to be semi-automated. This makes the procedure too
subjective (Interpreters can make different decisions), tedious and time
consuming. Moreover, the geology of the earth is comprises of different
structures (it is very complex) which may not be fitted by a fixed N.
Hsu (2002) stated that the use of wrong N can cause bias on depth
estimate and scattered solution on target’s locations. Therefore, the used
of fixed structural index may not estimate the parameters of different
sources in the real geology with desired accuracy.
One of the disadvantages of conventional Euler deconvolution is
that the interpreter/user has to select N manually. This property is a
setback to one of the most important attribute of the technique which is
fast means of interpreting large volume of data. However, attempts
made to address this problem using Differential Similarity Transform
(DST) (Stavrev, 1997; Gerovska and Arouzo-Bravo 2003; Grerovska et
al. 2010) and other related techniques that does not require the use of
structural index (Mushayandebvu et al., 1999, 2001; Nabighian and
Hansen, 2001; Salem and Ravat, 2003; FitzGeral et al, 2004; Keating
and Pilkington, 2004; Salem et al., 2007) surfer some drawbacks. DST
is less implemented because of the complexity involved in operation
(Reid and Thurston, 2014). According to Florio et al., 2006, the
8
estimation of N using AS (Salem and Ravat, 2003) could lead to error.
Tilt depth (Salem et al., 2007) technique uses higher order derivatives
(Reid and Thurston, 2014).
A procedure for solving five unknown parameters (including the
structural index) of magnetic anomaly using Euler deconvolution that
can be implemented without the use of complex mathematics, the use of
analytic signal and higher order derivatives is missing in the literature.
Multiple linear regression (MLR) methodology can be used to
solve positions (x0, y0 and z0) of magnetic source, background (B) and
structural index (N) simultaneously. The use of multiple linear
regression to solve the unknown parameters of Euler deconvolution
technique of magnetic anomaly is not available in the geophysical
literature. Unlike the past works, this technique allows the use of first
order derivatives, the inversion is independent of analytic signal (AS)
and it does not involve complex mathematical operations. It is simple to
apply and the derivatives are computed directly from the total field grid.
ii. Euler deconvolution treats the potential field sources as consisting of
elementary points with different parameters (such as N) as such large
number of solutions is usually obtained and it needs effective filtering
technique. Because of the complicated nature of the Earth subsurface,
some of these solutions are spurious/artifacts caused by interference of
other sources. Many studies have been carried out to address this issue
and they come out with different procedures in determining the correct
solution. Reid and Thurston (2014) has advocated that when depth and
9
N of the source are to be estimated simultaneously, rigorous means of
data filtering is required to choose the valid solutions.
The use of standard deviation of estimated depth and clustered
solutions had been the preferred means of selecting valid solutions
(Thompson, 1982; Reid et al., 1990, Grerovska et al., 2010). Other
researchers used various traditional filtering techniques (FitzGerald et
al., 2004) to discriminate the most accurate solutions. However,
rigorous filtering technique still remained one of the challenges of using
Euler deconvolution technique. Euler deconvolution method is built
based on the potential field and its derivatives; so, the accuracy of Euler
deconvolution method relies largely on the derivatives. Thus, Euler
deconvolution solutions should be filtered based on the area of the data
to be convolve, rather than focusing on the sprays of solutions. It is
crucial to study how potential derivatives based filters can be used as an
aid of choosing the correct range of depth solutions. The coupling
problem that exists between depth and structural index can be avoided
by identifying and using the locations immediately above the source
body’s critical points (Reid and Thurston, 2014).
iii. The pattern of magnetic anomaly depends on its position on the earth
surface. The same structure placed at different geographical locations
would give different anomaly’s shape because of the variation in
magnetic latitude. The dipolar nature of the magnetic field causes
distortion in the anomaly’s shape and as a result of this effect, error will
be introduced to the data and there by affecting the estimate of the
anomaly’s location (Araffa et al, 2012). While the use of RTP is
10
recommended to be applied on the data prior to the application of Euler
deconvolution (Thompson, 1982), other researchers (Reid et al., 1990)
are of the opinion that it should not be applied.
However, this problem remain unsolved, no attempts have been made
to investigate this effect on Euler deconvolution related techniques.
Also, no inclination’s assessment was carried out on the present
technique. The use of synthetic models and real data is very essential in
understanding the effect of inclination of the introduced technique.
Because, the introduced technique is not available in the literature,
evaluating the effect of inclination will definitely be added or otherwise
to the strength of the technique.
iv. The limit of the accuracy of depth estimation technique from magnetic
data is well established in the literature (Breiner, 1973; 1999). The
accuracy of conventional Euler deconvolution (Thompson, 1982; Reid
et al., 1990) and other related techniques have been evaluated. The
traditional approach for evaluating the performance of Euler
deconvolution technique has been the use of deviation from a certain
referenced value (mean).
However, in this research where a new approach is introduced, its
accuracy remains unknown. Moreover, the accuracy of interpretation
techniques determines its applicability in various geophysical
applications. Therefore, there is need to assess the present technique in
order to know its accuracy. Synthetic modeling using different models
such as box, contact, cylinder, dike and sphere can be used to assess the
accuracy of the introduced technique. In this case, the theoretical basis
11
of the technique can be established. The assessment can also be carried
out using field model data and the site where the detail geological
information of the area is known. In addition to the deviation of the
parameters, percentage of minimum/maximum error permits easier
assessment of the output parameters.
1.3 Research question
i. Which approach shall be adopted to automate Euler deconvolution
technique?
ii. What are the criteria for choosing valid Euler solutions?
iii. What is the effect of inclination on AUTO-EUD?
iv. How accurate is AUTO-EUD?
1.4 Research objective
An algorithm/procedure based on Euler’s homogeneity relation for fully
automation (hence the acronym, AUTO-EUD) of magnetic data interpretation is
presented in this research. Some of the objectives of this research are:
i. to automate Euler deconvolution equation in order to estimate the
horizontal coordinates (x0 and y0), depth, background (B) and structural
index (N) of a magnetic source,
ii. to propose a filter for discriminating reliable solution from the inversion
output of Euler homogeneity equation,
iii. to assess the effect of inclination on the introduced algorithm (AUTO-
EUD) and
12
iv. to investigate the accuracy of AUTO-EUD’s solutions using synthetic
and real magnetic data.
1.5 Novelty of the study
i. The introduced algorithm for solving the unknown parameters
of magnetic anomaly using multiple linear regression is not available
in the geophysical literature.
ii. The integrated and automated filter introduced in this study is
unique in the geophysical literature and therefore, it is a novel.
iii. This study has empirically deduced the structural index of a
box which is also not available in the literature.
iv. An application of the technique in engineering and
environmental site has been demonstrated. This application is rarely
found in the literature and it is therefore a new contribution to the
knowledge.
1.6 Scope and limitation of the study
The scope of this study is limited to forward modeling and inversion of 3D
magnetic field only. The accuracy of the introduced technique is determined using
synthetic models and field model data. The test of this inversion program using
synthetic model (in this research) is also limited to certain type of structures, namely
box, contact, dike, horizontal cylinder and sphere. These structures are designed with
the intention to simulate field with simple geologic structures. For synthetic and field
model data, the solutions provided by the introduced technique are compared to true
13
parameters of the models, whereas for real magnetic data, the depth solutions are
compared to thickness of rocks available in the literature.
1.7 Thesis outline
This thesis consists of 5 chapters. The first chapter introduces geophysical
prospecting methods and data interpretation with emphasize on Euler deconvolution.
The introductory chapter also presents problem statements, research questions,
objective of this study, novelty of the study, the scope and limitation of the study.
Other component of this chapter, although not the least, is significance of findings,
organization of the thesis.
The second chapter provides fundamentals of magnetic field and some
background of Euler’s homogeneity concept, which is the basis of Euler
deconvolution methodology. This chapter also includes the previous works to give
overview of how Euler deconvolution has evolved and modified through the past
decades, and also to sort out the research gap in Euler deconvolution methodology.
The third chapter presents the methodology used in this study and it consists
of (i) the introduction of the new technique using Multiple Linear Regression
methodology, (ii) the accuracy assessment of AUTO-EUD using synthetic modeling
and (iii) the accuracy assessment of AUTO-EUD using real magnetic data. This
chapter also explains how the solutions are filtered based on analytic signal and the
comparison between Conventional Euler Deconvolution (CED) and the present
technique (AUTO-EUD).
The fourth chapter presents the results of the forward modeling and inversion
of synthetic models as well as the inversion of real magnetic data of field models
application site. This chapter also discusses the accuracy of AUTO-EUD based on
14
the comparison between solutions from AUTO-EUD and the true models, and
between solutions and geological map. Besides these, the discussion also includes the
limitations of AUTO-EUD based on the results obtained.
The last chapter concludes the study by relating the findings to the objectives
of this study, emphasizing the significance of AUTO-EUD in locating the source of
magnetic field. This chapter includes some recommendation for future study as well.
15
CHAPTER 2
LITERITURE REVIEW
2.1 Introduction
This chapter introduces magnetic susceptibility and Euler homogeneity
relation as a depth estimation technique. An overview on the development of
Conventional Euler Deconvolution (CED) methodology through the past decades is
included in order to provide theoretical basis of the present algorithm. Introduction
on filtering and accuracy assessment of Euler deconvolution are presented.
2.2 Magnetic susceptibility
The quantity of magnetic moment per unit volume is called magnetization
(also called magnetization intensity, dipole per unit volume or magnetic polarization)
and it is denoted by a symbol M. It is the vector field that expresses the density of
permanent dipole moments contained in a magnetic material. The arrangement/line-
up of internal dipoles gives rise to a field M which is added to the magnetizing field
H. The S.I unit for magnetization is ampere per meter. For low magnetic fields
(Equation 2.1)
M α H
M = kH (2.1)
The constant in Equation 2.1 is called magnetic susceptibility (k), it is determined the
degree to which a body is magnetized. The total field including the effect of
magnetization is called magnetic induction (m) and it is given by (Equation 2.2)
m (H+M) (1+k) H
H (2.2)
16
The S.I and electromagnetic units for m is the tesla (T) and gauss (10-4
T)
respectively. Gamma ) is the unit of magnetic induction that is generally used for
geophysical work. The magnetic flux ( ) is given by (Equation 2.3)
= m.A (2.3)
Where A is a vector area. Thus
|m|
A and B are parallel, that is, m is the density of magnetic flux. The S.I unit for
magnetic flux is the Weber.
Magnetic susceptibility is the significant variable in magnetics. Although
instruments are available for measuring susceptibilities in the field, they can only be
used on outcrops or on rock samples and such measurement do not give the bulk
susceptibility of the formation. Table 2.1 lists magnetic susceptibilities for a variety
of rocks. Sedimentary and basic igneous rocks have the lowest and the highest
average values of magnetic susceptibility respectively:
Table 2.1: Magnetic susceptibility of rocks and minerals (source: Telford et al.,
2001)
Rock/mineral type Susceptibility (S.I Unit)
Range Average
Metamorphic
Schist
Gneiss
Slate
0.3-3
0.1-25
0-35
1.4
-
6
Igneous
Granite
Porphyry
Peridotite
Diabase
Pyroxenite
Diorite
0-50
0.3-200
90-200
1-160
-
0.6-120
25
60
150
55
125
85
17
Minerals
Magnetite
Pyrrhotite
Ilmenite
Clays
Graphite
Casiterite
Limonite
Pyrite
1200-19200
1-6000
300-3500
-
-
-
-
0.05-5
6000
1500
1800
0.2
0.1
0.9
2.5
1.5
Sedimentary
Dolomite
Sandstones
Limestone
0-0.9
0-20
0-3
0.1
0.4
0.3
2.2.1 Magnetic Elements
i. Inclination of the geomagnetic field: It is the angle between magnetic
north and the direction of the Earth field (Telford et al., 1990)
ii. Declination of the geomagnetic field: This is the angle between
geographic north and magnetic north (Telford et al., 1990).
iii. The angle of dip at a place: Is the angle between the direction of
earth’s magnetic field and the horizontal component of the earth’s
magnetic field in the magnetic meridian at that place
iv. Strike angle of the cylinder: Is the angle between the cylinder axis and
magnetic north
v. Azimuth angle: Is the angle between geographic north and horizontal
of a plane of box model.
2.3 Reduction of magnetic observations
The magnetic field readings measure from survey stations varies with time.
Diurnal effect and magnetic storms are the most significant causes of the changes in
magnetic field. This effect must be corrected from the data using appropriate
techniques. The short term, spikes and erratic changes in magnetic field are known
18
as micro-pulsations. These effects range from few to 100 nT in terms of intensity.
When these changes become large (amplitude and period), they are called magnetic
storm. It is a short term disturbances in the intensity of magnetic field associated
with sun spot activity and charged particles from sun. To achieve a successful
interpretation, the reduction has to be carried out to enhance and isolate the
contribution of the field due to concealed structure (Ismail, 2015). The correction or
reduction of magnetic data is necessary to remove all causes of magnetic variation
and noises from the observations other than those arising from the magnetic
anomalies in the subsurface.
2.3.1 Diurnal variation correction
Diurnal variation correction accounts for the temporal variation caused by the
electromagnetic radiation of the sun, which disrupts the geomagnetic field and thus
our survey, it can be checked by using two magnetometers with one acting as a base
station and recording the magnetic field every 1 minute or so, while the other would
be the primary machine to survey the area, however this method and magnetic survey
in general must not be used during magnetic storms (Sharma, 1997). After the
measurements, the primary machine readings would be time-synced with the base
station and then subtracting the results would give us the corrected readings.
2.3.2 Geomagnetic correction
The earth’s magnetic field strength varies from 25000 nT at the magnetic
equator to 69000 nT at the magnetic pole. This correction is carried out by
subtracting the theoretical field from International Geomagnetic Reference Field
(IGRF). This theoretical value changes with time (Sharma, 1997)
19
2.3.3 Reduction to the pole and equator
Reduction to the pole (RTP) is an operation use to transform the magnetic
anomaly into an anomaly that would have been obtained if the measurements were
taken at the magnetic pole, the area where the magnetic inclination is vertical, with
the assumption that the source is magnetized by induction (Silva, 1986). This effect
causes asymmetry and lateral shift of the anomaly of measured total magnetic field
(Aina, 1986). It has some advantages when applied to magnetic data which include
simplification of the interpretation of anomaly, it removes dipolar nature of magnetic
anomaly and it changes the asymmetric shape of magnetic anomaly to its symmetric
form. The dipolar nature of the magnetic field causes distortion in the anomaly’s
shape and as a result of this effect, error will be introduced to the data and there by
affecting the estimate of the anomaly’s location (Araffa et al., 2012). For reduction to
the pole technique to be applied on magnetic data, the information about the
remanent magnetization is required. However this information is not available in
most areas. An alternative method to RTP is the reduction to the equator (RTE)
which transforms the magnetic measurement as the one that would be observed if the
anomaly were located at the equator.
The direction cosines of geomagnetic field vector are , m and n. the
geomagnetic field vector is assume to be parallel to the polarization vector, in this
case, the remanent magnetization is zero. u and v are the Cartesian spatial
coordinates of angular frequency given the reduction to the pole (Op) and equator
(Oe) operators as Equation 2.4 and 2.5 respectively:
) ){ ) ) } )
⁄ )
[ ) )]
(2.4)
20
) ) *{ ) ) } )
⁄ )+
[ ) )]
(2.5)
In Equation 2.4 and 2.5 above, j = -1. Both Op and Oe can be used as:
i. Find the Fourier transform of the measured field, apply Op/Oe on the
transformed field and obtain the inverse Fourier transform.
ii. Evaluate Op/Oe for various values of u and v for the inclination and
declination. Compute the inverse transform of various inclinations and
declination of the magnetic field. The output is the space domain
operator which can be convolved with the measured magnetic field.
The first approach is more accurate, however, it can only be applied to
relatively small areas. On the other hand, the second approach is less
accurate, the operator requires to be truncated to manageable
dimensions but it can be applied to maps of any size (Jain, 1988).
2.3.4 Applications of magnetic method
As for the applications of the magnetic survey, it can be used to map dikes
blocking groundwater flow in the subsurface, structural trends and basement
features. It can also be used for the detection of archaeological objects, buried metal
drums and investigations over landfills.
2.3.5 Limitations of magnetic method
The magnetic survey has some limitations, it can only detect ferrous materials
including volcanic rocks, its image resolution deteriorates quickly with the increase of
depth, and it becomes almost useless near buildings, vehicles, or areas with reinforced
concrete and where the ferrous materials or volcanic rocks are underlain by strongly
magnetic rocks (Zohdy et al., 1974).
21
2.4 Euler homogeneity
Euler Deconvolution method evolves from the Euler’s homogeneity
relationship (John, 1965); it is first initiated to solve 2D magnetic field by Thompson
(1982). In the homogeneity concept (Equation 2.6), a function f(u) is considered
homogenous of degree n if
) ) (2.6)
where u = (u1, u2, . . ., ui) is the set of variables with respect to the homogeneity of
the field f, t is a real number and n is the degree of homogeneity (Reid and Thurston,
2014). This function also satisfies the Euler’s differential equation given in Equation
2.7. as
) ) (2.7)
Equation 2.7 can be written in three Cartesian coordinates x, y and z form, such that
) ) (2.8)
and the partial differential equation then would be (Equation 2.9)
(2.9)
Based on this, the N is defined as (Equation 2.10)
(2.10)
According to the power law, the field’s fall off function for both f (u) and f (x, y, and
z) can be expressed in the form of (Equation 2.11)
f = C/rN (2.11)
22
where C is a constant which includes any factor that affects the field, r is the distance
between the source and the observation point. The degree of homogeneity explains
the rate at which the potential field strength reduces over increasing radius, and often
referred to as structural index (N) in the literature.
One of the advantages of Euler deconvolution is that no geological
assumption is required prior to inversion. However, some understanding of the study
area is necessary since N represents type of source that produces the field anomaly
(Thompson, 1982). Ideas about the geometry of the structure will elevate the
confidence in choosing the correct N. Besides this, the application of Euler equation
should be limited only to single-point sources, which give off potential field
conforming to Equation 2.10. Sources with homogenous magnetic field are listed in
Table 2.2.
Table 2.2: Structural index of magnetic source
Source Smellier model N
Sphere Dipole 3
Vertical/horizontal cylinder Line of dipoles 2
Dike/thin sheet Line of poles 1
Infinite contact - 0
In order to find the source location (x0, y0, z0), Equation 2.9 is then further
redefined as (Equation 2.12)
)
)
)
) (2.12)
where x, y, z are the observation point coordinates; x0, y0, and z0 are the source
locations;
are the potential derivatives; N is the structural index; B is the
background of field f (Thompson, 1982). Equation (2.12) forms the basis for the
methodology used in this research.
23
2.4.1 Previous work
Information that can be obtained from magnetic method can be used to
estimate the position of a concealed body if it has enough magnetization. Among the
parameters of the body that can be estimated is depth. Depth estimation techniques
are divided into manual and automated techniques. Some of the manual depth
estimation techniques include maximum slope, Peter half slope, half width, Sokolov
distance and Hannel distance (Samuel, 2017). The computerized techniques can be
further divided into 2; semi-automated and automated techniques. Euler
deconvolution belongs to a group of semi-automated techniques, in its operation, it
allows user to make choice of structural index. Numerous attempts have been made
to make it fully automated.
Examples of other automated techniques include spectral depth technique
Techniques for the estimation of depth to the bottom of magnetic sources has been
presented (Bansal et al., 2011; Nwankwo, 2015). The techniques used for detailed
analysis operates on limited amount of data and these are characteristics curve,
iterative or inverse curve matching.
There were so many computer-assisted interpretation techniques in the
literature that are belong to computerized class of depth estimation methods.
Automatic interpretation techniques have advantages of operating directly on the
recorded digital field data in addition to providing rapid means of analyzing large
amount of data. The technique for locating vertices of polygon model has been
presented (O’Brein, 1971). A technique based on vertical prism and thin plate
models was described by Koulomzine et al. (1970). Werner (1953) simplified
equation for the interpretation of two-dimensional thin dike. The position of the dike
could be obtained/ devised by choosing appropriate point along a profile. This
24
technique has been successfully applied in oil and mining companies as a result of
the advancement in Aero Service and further researches based on Hartman et al.
(1971). The analysis of magnetic discontinuities using derivatives (vertical and
horizontal) of the total magnetic field intensity was incorporated into Werner’s
equation (Hartman et al., 1971).
A depth estimation technique using statistical approach that makes use of
slope of the power spectral density has been presented (Vacqier et al., 1951). The
location of the boundary is obtained through the computation of the horizontal
gradient of the pseudo gravity that peaks over a vertical contact (Grauch and Cordell,
1987). For dipping contacts, the peak is somewhat offset (Thompson, 1982). Another
related technique is the use of analytic signal and total gradient with the peaks
directly over a contact model with arbitrary dip (Nabighian, 1972). Hansen et al.
(1987) has shown that the peak over a contact model using total gradient and analytic
signal were noisy estimator. Using this technique, the depth is obtained from the
breadth of the peak (Hansen et al., 1987). So many automatic processing techniques
that estimate both source location and depth have been presented. Naudy (1971)
introduced an approach similar to Werner deconvolution (Jain, 1976) that makes use
of prism and thin-plate models.
Euler deconvolution is a quick means of transforming field measurements
into location and depth estimates of the magnetic source. The technique operates on a
subset of data using a moving window in which the body coordinates are solved.
Intermediate bodies have non-integer structural index, the technique is only
approximate. The term Euler deconvolution was obtained by the analogy of Euler
equation with the established Werner deconvolution technique. In strong term, Euler
deconvolution is only valid for functions that are homogeneous. The field of most
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