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UBC Theses and Dissertations

Electromagnetic coupling in frequency domain induced polarisation data Routh, Partha Sarathi 1999

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E L E C T R O M A G N E T I C C O U P L I N G I N F R E Q U E N C Y D O M A I N I N D U C E D P O L A R I S A T I O N D A T A B y P a r t h a S a r a t h i R o u t h B . S c . ( E a r t h S c i e n c e s ) I n d i a n I n s t i t u t e o f T e c h n o l o g y , K h a r a g p u r 1 9 9 1 M . S c . ( E x p l o r a t i o n G e o p h y s i c s ) I n d i a n I n s t i t u t e o f T e c h n o l o g y , K h a r a g p u r , 1 9 9 3 A T H E S I S S U B M I T T E D I N P A R T I A L F U L F I L L M E N T O F T H E R E Q U I R E M E N T S F O R T H E D E G R E E O F D O C T O R O F P H I L O S O P H Y i n T H E F A C U L T Y O F G R A D U A T E S T U D I E S D E P A R T M E N T O F E A R T H A N D O C E A N S C I E N C E S ( G E O P H Y S I C S ) W e a c c e p t t h i s t h e s i s as c o n f o r m i n g t o t h e r e q u i r e d s t a n d a r d T H E U N I V E R S I T Y O F B R I T I S H C O L U M B I A N o v e m b e r 1 9 9 9 © P a r t h a S a r a t h i R o u t h , 1 9 9 9 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of Earth and Ocean Sciences (Geophysics) The University of British Columbia 129-2219 Main Mali Vancouver, Canada V6T 1Z4 Date: ft A b s t r a c t Frequency domain induced polarization (IP) surveys are commonly carried out to provide information about the chargeability structure of the earth. The goals might be as diverse as trying to delineate a mineralized and/or alteration zone for mineral exploration, or to find a region of contaminants for an environmental problem. Unfortunately, the measured responses can have contributions from inductive and galvanic effects of the ground. The inductive components are called E M coupling effects. They are considered to be "noise" and much of this thesis is devoted towards either removing these effects, or reformulating the inverse problem so that inductive effects are part of the "signal". If the forward modeling is based on galvanic responses only, then the inductive responses must first be removed from the data. The motivation for attacking the problem in this manner is that it is easier to solve D.C. resistivity equation than the full Maxwell's equation. The separation of the inductive response from the total response is derived by ex-pressing the total electric field as a product of an IP response function, and an electric field which depends on E M coupling response. This enables me to generate formulae to obtain IP amplitude (PFE) and phase response from the raw data. The data can then be inverted, using a galvanic forward modeling. I illustrate this with ID and 3D synthetic examples. To handle field data sets, I have developed an approximate method for estim-ating the E M coupling effects based upon the assumption that the earth is locally ID. The ID conductivity is obtained from a 2D inversion of the low frequency D C resistivity data. Application of this method to a field data set has shown encouraging results. I also examine the E M coupling problem in terms of complex conductivity. I show that if u the forward modeling is carried out with full Maxwell's equation, then there is no need to remove E M coupling. I illustrate this with I D synthetic example. In summary, I have investigated the E M coupling problem in IP and developed a practical removal methodology that can be applied to data sets from I D , 2 D and 3 D earth structures. i n Table of Contents Abstract ii List of Tables ix List of Figures x Acknowledgement xvii Dedication xviii 1 Introduction 1 1.1 Background 1 1.2 The E M coupling problem in IP 4 1.3 Organization of the thesis 9 2 I D C S A M T inversion 17 2.1 Introduction 17 2.2 Forward Problem 19 2.3 Inverse Problem 25 2.4 Computation of Sensitivity 27 2.4.1 Adjoint Green's function method 28 2.4.2 Differentiation of Propagator Matrices 33 2.5 Synthetic Example 39 2.6 M T inversion of Near-Field Corrected Data 43 iv 2.7 Inversion of electric field data acquired in C S A M T surveys 45 2.8 Field Examples 49 2.8.1 Example from a resistive environment 49 2.8.2 Example from conductive environment 55 2.9 Conclusions 58 3 IP and complex resistivity Method 62 3.1 Introduction 62 3.2 Time Domain IP 62 3.3 Frequency Domain IP 65 3.4 Complex Resistivity Measurements 66 4 E M Coupling and a method for its approximate removal 74 4.1 Introduction 74 4.2 Effects of E M Coupling on IP data 78 4.2.1 DC Effect: EDC . . 80 4.2.2 DCIP Effect: EDCIP 80 4.2.3 EMIP Effect: EEMIP 81 4.2.4 E M Effect: EEM 81 4.3 The Removal of E M Coupling 83 4.3.1 Amplitude Response 88 4.3.2 Phase Response 88 4.3.3 The P F E response and its relation to IPAMP 89 4.3.4 Removal of E M coupling from the P F E responses 91 4.4 Removal methodology for inhomogeneous earth 92 4.5 Removal methodology in ID: Synthetic Examples 93 4.5.1 Synthetic test for P F E data 94 v 4.5.2 Synthetic test for phase data 94 4.5.3 Effect of conductivity on E M coupling removal 96 4.5.4 Synthetic test for the pole-dipole and pole-pole data 98 4.6 E M Coupling removal in 3D 98 4.6.1 Example-1: Conductive block in half-space 100 4.6.2 Example-2: Conductive block in a layered medium 104 4.7 E M coupling in topographic terrains 106 4.8 Removal methodology in 2D: An approximate method 112 4.9 Field Examples 118 4.9.1 Data from Acropolis 118 4.9.2 Data from Elura Ore Deposit 127 4.10 Conclusions 131 5 I D complex conductivity inversion without E M coupling removal 138 5.1 Forward Problem 140 5.2 Inversion Methodology 142 5.2.1 Sensitivity 144 5.2.2 Synthetic Example 145 5.3 Conclusions 148 6 I D Complex conductivity inversion after E M Coupling removal 156 6.1 Introduction 156 6.2 Forward Problem 158 6.3 Removal of E M coupling from complex apparent resistivity data 162 6.4 Inversion Methodology 163 6.5 Synthetic Examples 164 6.5.1 Inversion of true complex resistivity data 165 v i 6.5.2 Inversion of complex resistivity data contaminated by E M coupling 165 6.5.3 Inversion of complex resistivity data after E M coupling removal . 166 6.6 Conclusions 168 7 2D Inversion for Cole-Cole Parameters 179 7.1 Introduction 179 7.2 Forward Problem 183 7.3 Sensitivity 185 7.4 Inverse Problem 186 7.5 Synthetic Example 188 7.6 Field Example 193 7.7 Conclusions 203 8 Conclusions and Future scope 208 Appendices 215 A Derivation for the vector potentials A and F for I D earth 215 A.0.1 Solution for Az potential 217 A.0.2 Solution for Fz potential 221 B Derivation for the model norm matrices 225 C Validity of the approximation kEMIP « kEM 227 D Linear inversion with complex quantities 232 D . l Introduction 232 D.2 Linear Inverse Problem 232 D.2.1 Method-1 233 vii D.2 .2 M e t h o d - 2 236 D . 3 S y n t h e t i c E x a m p l e 237 D . 4 Conc lus ions 240 R e f e r e n c e s 2 4 3 v i i i L i s t o f T a b l e s 5.1 M o d e l used to generate the da t a 146 6.1 M o d e l used to generate the d a t a 160 ix L i s t o f F i g u r e s 1.1 S i m p l i s t i c m o d e l t o e x p l a i n t h e I P p h e n o m e n a . C u r r e n t f l o w t h r o u g h ( a ) u n b l o c k e d p o r e ( b ) b l o c k e d p o r e s p a c e 3 1.2 T h e I P e x p e r i m e n t i n f r e q u e n c y d o m a i n 5 1.3 O r g a n i z a t i o n o f t h e t h e s i s 10 1.4 T h r e e d i f f e r e n t f o r m u l a t i o n s t o o b t a i n c h a r g e a b i l i t y i n f o r m a t i o n f r o m E M c o u p l i n g c o n t a m i n a t e d f i e l d d a t a 12 2.1 C o n d u c t i v i t y s t r u c t u r e f o r h o r i z o n t a l l y l a y e r e d e a r t h m o d e l 2 0 2.2 S y n t h e t i c m o d e l t o c o m p a r e I D C S A M T a n d M T r e s p o n s e s 2 3 2.3 I D C S A M T i n v e r s i o n o f a p p a r e n t r e s i s t i v i t y a n d p h a s e d a t a : s y n t h e t i c e x a m p l e 41 2.4 I D C S A M T i n v e r s i o n o f o n l y a p p a r e n t r e s i s t i v i t y a n d o n l y p h a s e d a t a : s y n t h e t i c e x a m p l e 42 2.5 C o m p a r i s o n o f C S A M T i n v e r s i o n w i t h M T i n v e r s i o n o f n e a r - f i e l d c o r r e c t e d d a t a : I D s y n t h e t i c e x a m p l e 4 6 2.6 C S A M T i n v e r s i o n o f e l e c t r i c f i e l d d a t a : I D s y n t h e t i c e x a m p l e 4 8 2.7 O b s e r v e d f i e l d d a t a f r o m t h e r e s i s t i v e e n v i r o n m e n t a n d p r e d i c t e d d a t a f r o m t h e C S A M T i n v e r s i o n 51 2.8 C o m p a r i s o n o f t h e r e c o v e r e d m o d e l f r o m C S A M T i n v e r s i o n a n d i n v e r s i o n o f n e a r - f i e l d c o r r e c t e d d a t a : e x a m p l e f r o m r e s i s t i v e e n v i r o n m e n t 53 2.9 R e c o v e r e d m o d e l f r o m r e s i s t i v i t y a n d p h a s e o n l y d a t a i n v e r s i o n 5 4 x 2.10 Observed field data from the conductive environment and predicted data from the C S A M T inversion 56 2.11 Comparison of the recovered model from C S A M T inversion and inversion of near-field corrected data: example from conductive environment. . . . 60 2.12 Drillhole information from geologic logs 61 3.1 Time domain IP signals 64 3.2 Frequency domain IP signals for P F E measurements 66 3.3 Frequency domain IP signals for phase measurement 67 3.4 Data representation for multiple frequency measurements in a complex resistivity survey 68 3.5 Cole-Cole model for complex resistivity 73 4.1 Different formulation to recover chargeability from the data 75 4.2 X-directed horizontal electric dipole on a homogeneous complex conductive half-space 78 4.3 The characterization of various effects based on an electric field generated by a point horizontal electric dipole 80 4.4 Factors affecting E M coupling responses for P F E data 84 4.5 Factors affecting E M coupling responses for phase data 85 4.6 Plot for the complex function tp(iuj) 86 4.7 ID synthetic example to test E M coupling removal 95 4.8 DC resistivity inversion of (a) amplitude data at 0.1 Hz. (b) The recovered model from the inversion 97 4.9 E M coupling corrections using different conductivity models 97 4.10 E M coupling corrections for the pole-dipole and pole-pole survey 99 4.11 3D conductive and polarizable block in a non-polarizable half-space. . . . 100 x i 4.12 EM coupling removal for phase data generated from the model in Figure 4.11. The EM coupling responses are computed using the true conductivity (Too of the medium 102 4.13 EM coupling removal for phase data generated from the model in Figure 4.11. The EM coupling responses are computed using the DC conductivity <r0 = <M1 - 7 7 ) 103 4.14 EM coupling removal for phase data generated from the model in Fig-ure 4.11. The EM coupling responses are computed using the half-space background conductivity 104 4.15 EM coupling removal for PFE data generated from the model in Figure 4.11. The PFEEMIP is generated by computing the amplitude of the im-pedance at two frequencies (30 Hz and 3 Hz) for a dipole-dipole geometry with dipole spacing of 100m 105 4.16 3D conductive block in a layered earth 106 4.17 3D EM coupling removal example for a prism i\ layered earth: correction using (Too 107 4.18 3D EM coupling removal example for a prism in a layered earth: correction using (To 108 4.19 3D EM coupling removal example for a prism in a layered earth: correction using ID layered background 109 4.20 EM coupling phases for topographic terrains I l l 4.21 The model used to test the EM coupling removal method for phase data on a topographic terrain 112 4.22 E M coupling removal in topographic terrain 113 4.23 EM coupling removal in topographic terrain 114 4.24 ID local average of 2D conductivity model 116 xii 4.25 M a p o f A u s t r a l i a 118 4.26 ( a ) T h e o b s e r v e d a p p a r e n t c o n d u c t i v i t y p s e u d o s e c t i o n a t 0.125 H z f o r t h e d i p o l e - d i p o l e s u r v e y , ( b ) T h e r e c o v e r e d c o n d u c t i v i t y m o d e l a f t e r i n v e r s i o n 120 4.27 ( a ) T h e o b s e r v e d p s e u d o s e c t i o n f o r t h e u n d e c o u p l e d p h a s e f o r t h e d i p o l e -d i p o l e a r r a y a t 0.125 H z . ( b ) T h e r e c o v e r e d 2 D c h a r g e a b i l i t y m o d e l a f t e r i n v e r s i o n 121 4.28 T h e c o m p a r i s o n o f t h e u n d e c o u p l e d p h a s e s h o w n i n ( a ) a n d E M c o u p l i n g p h a s e c o m p u t e d u s i n g o u r m e t h o d ( s h o w n i n ( b ) ) . T h e l i n e a r t r e n d o f t h e i n c r e a s i n g p h a s e w i t h N - s p a c i n g i s a c h a r a c t e r i s t i c o f E M c o u p l i n g c o n t a m i n a t i o n 122 4.29 ( a ) T h e c o r r e c t e d p s e u d o s e c t i o n o b t a i n e d b y s u b t r a c t i n g t h e E M c o u p l i n g p h a s e f r o m t h e u n d e c o u p l e d p h a s e , ( b ) T h e r e c o v e r e d c h a r g e a b i l i t y m o d e l o b t a i n e d a f t e r i n v e r s i o n o f c o r r e c t e d d a t a 123 4.30 ( a ) T h e c o r r e c t e d p h a s e p s e u d o s e c t i o n o b t a i n e d b y a t h r e e - p o i n t d e c o u p -l i n g m e t h o d , ( b ) T h e r e c o v e r e d c h a r g e a b i l i t y m o d e l o b t a i n e d a f t e r i n v e r -s i o n o f t h r e e - p o i n t d e c o u p l e d d a t a 124 4.31 ( a ) T h e c o r r e c t e d p h a s e p s e u d o s e c t i o n o b t a i n e d b y m a n u a l d e c o u p l i n g m e t h o d , ( b ) T h e r e c o v e r e d c h a r g e a b i l i t y m o d e l o b t a i n e d a f t e r i n v e r s i o n o f m a n u a l d e c o u p l e d d a t a 125 4.32 E M c o u p l i n g r e m o v a l u s i n g a h o m o g e n e o u s h a l f - s p a c e m o d e l 1 2 7 4.33 ( a ) T h e p s e u d o s e c t i o n o f t h e u n d e c o u p l e d p h a s e s a t 1 H z . ( b ) T h e r e -c o v e r e d m o d e l a f t e r i n v e r s i o n o f u n d e c o u p l e d d a t a a t 1 H z 128 4.34 ( a ) T h e p s e u d o s e c t i o n o f t h e d e c o u p l e d p h a s e s a t 1 H z u s i n g o u r m e t h o d , ( b ) T h e r e c o v e r e d c h a r g e a b i l i t y m o d e l o b t a i n e d a f t e r i n v e r s i o n o f d e c o u p l e d d a t a a t 1 H z 129 xin 4.35 (a) The pseudosection of the apparent conductivity (b) The recovered conductivity model obtained after inversion 130 4.36 The pseudosection of the undecoupled phases at (a) 0.125 Hz, (b) 0.25 Hz, (c) 0.5 Hz and (d) 1 Hz. 134 4.37 The recovered chargeability model obtained after inversion of undecoupled data at (a) 0.125 Hz, (b) 0.25 Hz, (c) 0.5 Hz and (d) 1 Hz 135 4.38 The pseudosection of the decoupled phases at (a) 0.125 Hz, (b) 0.25 Hz, (c) 0.5 Hz and (d) 1 Hz 136 4.39 The recovered chargeability model obtained after inversion of decoupled data at (a) 0.125 Hz, (b) 0.25 Hz, (c) 0.5 Hz and (d) 1 Hz 137 5.1 Different formulations to recover chargeability from the data 139 5.2 Physical situation of the complex conductivity problem in ID 141 5.3 Inversion of complex apparent resistivity (pa(ioj)) data generated at 1Hz. 150 5.4 Inversion of complex apparent resistivity (pa(iu))) data generated at 0.001Hz.l51 5.5 Inversion of complex apparent resistivity (pa(iu>)) data generated at 0.01Hz.l52 5.6 Inversion of complex apparent resistivity (pa(iu>)) data generated at 0.1Hz. 153 5.7 Inversion of complex apparent resistivity (pa(ico)) data generated at 10Hz. 154 5.8 Recovered complex conductivity model as a function of frequency 155 6.1 Different formulations to recover chargeability from the data 157 6.2 Synthetic data generated using a three-layer model at / = 1Hz 161 6.3 Inversion of true IP data generated using a three-layer model at / = 1Hz. 169 6.4 Inversion of EM coupling contaminated data using the forward mapping based on the complex DC resistivity equation 170 6.5 Inversion of data that have been corrected for EM coupling at / = 1Hz. . 171 6.6 DC resistivity inversion of amplitude data 0.1 Hz 172 xiv 6.7 Inversion of data that have been corrected for EM coupling at / = 1Hz. Here the correction is obtained by using the conductivity structure ob-tained from DC resistivity inversion 173 6.8 Inversion of data that have been corrected for EM coupling at / = 0.001Hz.l74 6.9 Inversion of data that have been corrected for EM coupling at / = 0.01Hz. 175 6.10 Inversion of data that have been corrected for EM coupling at / = 0.1Hz. 176 6.11 Inversion of data that have been corrected for EM coupling at / = 10Hz. 177 6.12 Recovered complex conductivity model as a function of frequency 178 7.1 Synthetic model used to generate the data, (a) Chargeability ( 7 7 ) (b) time constant (r) and (c) relaxation constant (c) 189 7.2 Real part of the observed data in percentage at (a) 0.125 Hz (c) 1.0 Hz and (e) 10 Hz. The predicted data from inversion at (b) 0.125 Hz (d) 1.0 Hz and (f) 10 Hz 191 7.3 Imaginary part of the observed data in percentage at (a) 0.1 Hz (c) 1.0 Hz and (e) 10 Hz. The predicted data from inversion at (b) 0.125 Hz (d) 1.0 Hz and (f) 10 Hz 192 7.4 Recovered model from inversion: synthetic example 194 7.5 Recovered model from inversion in region of high chargeability: synthetic example 195 7.6 (a) Observed apparent resistivity data obtained at the lowest frequency. (b) Predicted apparent resistivity data from the inversion 197 7.7 2D Conductivity model obtained from the inversion 197 7.8 Real part of the data 198 7.9 Real part of the data 200 7.10 Imaginary part of the data 201 xv 7.11 Imaginary part of the data 202 7.12 Recovered model obtained from inversion of field data from the North Silver Bell deposit 204 7.13 Geological information overlayed on the inversion model 205 8.1 Different formulation to recover chargeability from the data 209 C . 1 The validity of the approximation kEMIP kEM for varying conductivity and intrinsic chargeabilities 229 C.2 The validity of the approximation kEMIP m kEM for varying time constant and relaxation constant 230 C . 3 The validity of the approximation kEMIP f=a kEM for varying dipole length and dipole spacings 231 D. l Plot of kernel as function of depth x. (a) Real part (b) Imaginary part . 237 D.2 Inversion results with method-1 239 D.3 Fit to the data with method-1 240 D.4 Fit to the data with method-2 241 D.5 Inversion results with method-2 242 x v i A c k n o w l e d g e m e n t I w o u l d l i k e t o t a k e t h i s o p p o r t u n i t y t o e x p r e s s m y d e e p e s t g r a t i t u d e t o m y s u p e r v i s o r , D r . D o u g O l d e n b u r g , f o r h i s m o t i v a t i o n , e n c o u r a g e m e n t a n d s u p p o r t o v e r t h e s e y e a r s t o c a r r y o u t t h i s r e s e a r c h . I e s p e c i a l l y t h a n k h i m f o r h i s m o r a l u p l i f t m e n t d u r i n g t h e d a r k e r d a y s . I a m g r a t e f u l t o m e m b e r s o f m y c o m m i t t e e , D r s . B r u c e B u f f e t t , R o s e m a r y K n i g h t , D o n R u s s e l l a n d M a t t Y e d l i n , f o r t h e i r a d v i c e a n d e n c o u r a g e m e n t . I w o u l d l i k e t o t h a n k P e t e r K o w a l c z y k o f P l a c e r D o m e f o r s u g g e s t i n g t h e n e e d t o s o l v e t h e E M c o u p l i n g p r o b l e m . I w o u l d l i k e t o t h a n k D r s . E l d a d H a b e r , C o l i n F a r q u a r s o n a n d Y a o g u o L i f o r m a n y u s e f u l d i s c u s s i o n s o n i n v e r s e t h e o r y , e l e c t r o m a g n e t i c s , c o m p u t a t i o n a l i s s u e s a n d R o m a n S h e k t m a n , f o r h i s h e l p w i t h c o m p u t e r s a n d s o f t w a r e s . I a m g r a t e f u l t o , D r s . D a v i d A l u m b a u g h a n d G r e g N e w m a n o f S a n d i a L a b s f o r p r o v i d i n g u s t h e 3 D E M m o d e l i n g c o d e t h a t i s u s e d f o r 3 D E M c o u p l i n g p r o b l e m , D r . K e n Z o n g e o f Z o n g e E n g i n e e r i n g R e s e a r c h a n d D i c k W e s t o f B H P f o r p r o v i d i n g t h e f i e l d d a t a f o r t h i s r e s e a r c h . I a m g r a t e f u l t o , t h e J A C I c o n s o r t i u m m e m b e r s f o r t h e i r f i n a n c i a l s u p p o r t a n d m a n y u s e f u l d i s c u s s i o n s d u r i n g t h e a n n u a l m e e t i n g s . I w o u l d l i k e t o t h a n k m y p a s t a n d p r e s e n t c o l l e a g u e s , Y u v a l , D a v i d M c M i l l a n , C h r i s W i j n s , R o b L u z i t a n o , G w e n n F l o w e r s , C h r i s t o p h e H y d e , L e n P a s i o n a n d o t h e r s f o r t h e i r f r i e n d s h i p a n d m a n y h e l p . T h e f r i e n d l y a t m o s p h e r e t h a t i s u n i q u e o f t h i s d e p a r t m e n t m a d e m y s t a y v e r y e n j o y a b l e . I t h a n k P a r t h a , D e b a s i s , T h a v a , K o u s h i k , A n i r b a n a n d R a n j a n f o r m a n y d i s c u s s i o n s o n s c i e n c e a n d l i f e , a w a y f r o m h o m e . N o n e o f t h i s w o r k w o u l d h a v e b e e n p o s s i b l e w i t h o u t i m m e n s e s u p p o r t t h a t I r e c e i v e d , f r o m m y p a r e n t s a n d s i s t e r . I r e s e r v e m y b i g g e s t t h a n k s t o m y l o v i n g w i f e M a n t u , w i t h o u t h e r p a t i e n c e a n d i n s p i r a t i o n t h i s w o r k w o u l d n o t h a v e b e e n c o m p l e t e . x v n D e d i c a t i o n In the loving memory of my father xviii Chapter 1 . Introduction 1.1 Background Earth materials can effectively act like small capacitors when an electric current is ap-plied. This results in a charge build-up of the capacitors, that alters the electric potential measured on the surface or in boreholes. This is referred to as the induced polarization (IP) effect. The word "induced polarization" derives its name from the fact that the technique has the ability to detect electrical surface polarization of metallic minerals that are induced by the currents in the ground. An excellent account of the IP method is available in Sumner (1976) and Bertin and Loeb (1976). The IP method is a tool that is primarily used in conjunction with the DC resistivity method for base metal exploration. IP was first discovered by the Schlumberger brothers in 1920. Although it was first rec-ognized in 1920's, the major development came after the 1950's when field tests were carried out and a macroscopic theory for IP was established. Two important seminal papers on the theoretical exposition of IP are by Seigel (1959) and Wait (1959a,b). The first major extensive work of using IP as a geophysical tool was carried out by Newmont Exploration Ltd in the early- to mid-fifties. Since then IP has been routinely used for exploration of mining targets in various parts of the world. Although the physio-chemical mechanism to explain the microscopic theory of IP is still not well understood, the macroscopic basis for IP as a distinct phenomenon has been established. Many laboratory and field investigations have been carried out to understand 1 Chapter 1. Introduction 2 the m e c h a n i s m caus ing I P effects ( W a i t , 1959b; W o n g , 1979; M a d d e n a n d C a n t w e l l , 1967; F i n k , 1980; K l e i n et a l . , 1984). In a s impl i s t i c sense, the macroscop ic I P effect can be e x p l a i n e d i n the fo l lowing manner ( M a d d e n and C a n t w e l l , 1967; B e r t i n a n d L o e b , 1976): T h e currents i n the g round are the consequence of ion ic flow t h r o u g h the pore spaces i n the rocks . W h e n a po t en t i a l difference is created between two po in t s i n the m e d i u m , the ions flow i n the d i r ec t ion of the po t en t i a l gradient shown i n F i g u r e 1.1(a) r e su l t ing i n a cur rent flow. However , w h e n these pore spaces are b locked b y m i n e r a l gra ins , the ions accumula t e at the interface of these b l o c k i n g par t ic les resu l t ing i n a charge b u i l d - u p as shown i n F i g u r e 1.1(b). T h i s results i n a net r educ t ion i n the current dens i ty g iven by J — rjJ where n is a unit less n u m b e r rang ing f rom 0 to 1. T h u s , to m a i n t a i n the same cur ren t , some a d d i t i o n a l vol tage is r equ i red to d r ive the current across th is bar r ie r . T h i s a d d i t i o n a l vol tage is sometimes refered to as "overvol tage effect" ( W a i t , 1959a). W h e n the e x t e r n a l current source is r emoved , the charges r e t u r n to the i r e q u i l i b r i u m state re su l t ing i n vol tage discharge. T h e decay ing po t en t i a l measured after the current source is r e m o v e d is the i n d u c e d p o l a r i z a t i o n response. In the t i m e d o m a i n , the voltage decay is measured after the e x t e r n a l cur rent is re-m o v e d . T h e ra t io of the measured vol tage jus t before a n d after the cessat ion of the current is an i n d i c a t o r of chargeable mater ia l s i n the g round . T h e I P effect can also be measured i n the f requency d o m a i n since there is a f ini te t i m e i n v o l v e d i n b u i l d i n g up these charges. In the frequency d o m a i n , voltages are measured as a func t i on of frequency. T o o b t a i n i n f o r m a t i o n about chargeabi l i ty there are two approaches i n f r equency-domain surveys: (a) o b t a i n a ra t io of the po t en t i a l at two different frequencies a n d (b) measure the phase difference between the observed po t en t i a l and the t r a n s m i t t e d cur ren t . A l t h o u g h I P has been successful as an exp lo ra t i on too l , there are some l i m i t a t i o n s encoun te red w i t h the m e t h o d . O n e of the i m p o r t a n t l i m i t a t i o n s is tha t the I P d a t a are often c o n t a m i n a t e d by i n d u c t i v e effects of the g round ( c o m m o n l y k n o w n as E M coupl ing) Chapter 1. Introduction 3 (a) "—+~ F i g u r e 1.1: S i m p l i s t i c m o d e l to e x p l a i n the I P phenomena . C u r r e n t flow t h r o u g h (a) u n b l o c k e d pore (b) b locked pore space. a n d this makes i t difficult to recognize the I P anomaly . B o t h I P and E M coup l ing effects occur together a n d i t is difficult to d i s t ingu i sh their i n d i v i d u a l con t r ibu t ions . Because of th is , m a n y d a t a sets acqui red i n the f ield are not in te rpre tab le ei ther by rou t ine process ing or by invers ion . T h i s was the s ta r t ing poin t of this research, a n d the i n i t i a l i m p e t u s was generated at the U B C - G e o p h y s i c a l Invers ion F a c i l i t y c o n s o r t i u m mee t ing (1994) to revis i t the E M coup l ing p r o b l e m i n IP . Chapter 1. Introduction 4 1.2 T h e E M c o u p l i n g p r o b l e m i n I P In an IP survey a harmonic current I(iu>) is transmitted into the ground using a finite length grounded source, and complex potentials V(icu) are measured on the surface as shown in Figure 1.2 (where I(ILO) = Joe""'). The two basic Maxwell's equations for electric and magnetic field in the frequency domain are given by V x E = -iu>fiH (1.1) V x H = (<T + iuje) E + Js where to is the angular frequency, a is the conductivity, e is the dielectric permittivity and Js is the source current density. In eqn (1.1) the current density (<r + iu>e) E has contributions from both the conduction and displacement currents in the medium. For many geophysical applications, the conductivity (cr) is adequately assumed to be inde-pendent of the frequency of the applied electric field. In the IP method, low frequency currents (0.1 — 1000iJ,z) are most commonly used in the surveys, and at these low fre-quencies the displacement current due to dielectric (e) material is negligible. However at low frequencies, due to the IP effect, the voltage measured (V(zu>)) is not in gen-eral in-phase with the input current (J(zu>)) and the measured apparent conductivity increases with the frequency (Ward and Fraser, 1967; Sumner, 1976). This implies that the conductivity is frequency dependent and is a complex quantity. In other words the conventional Ohm's law J(iu>) = crE(ioj), which assumes the conductivity to be frequency independent, is represented by J(ito) = a(iuj)E{iu>) with a complex conductivity given by <r(i(jj) — CTT(UJ) + i<Ti(u>), where ar and <7; are the real and imaginary part of conductivity respectively. Thus the second equation in (1.1) can be written as V x H = + iwe) E + J3. (1.2) The imaginary part of the conductivity can also be interpreted in terms of a dielectric Chapter 1. Introduction 5 Figure 1.2: The IP experiment in frequency domain constant by writing V x H = (<rr(u) + i<Ti{u) + iue) E + Ja (1.3) = (<rr(u) + iu>(e + <ri(u>)/w))E + Js where the effective dielectric permittivity is represented by ee^(u>) = e + ai(w)/u> and the effective dielectric constant {Ke^ = ee^(u>)/eo) is given by J T « " M = - + ^  (1.4) e 0 e0uj For general earth materials exhibiting IP effects at the low frequencies, the effective dielectric constant yields abnormally high values of dielectric constant. Equation (1.4) shows that at low frequencies, <7;(u;) need not be very large in order to make the measured Ke*f(uj) much larger than true dielectric constant (e/eo) (Fuller and Ward, 1970). For example, Evjen (1948) reports a dielectric constant of 105 for some rocks, and Lesmes and Morgan (1999) report variable dielectric constant of 106 — 104 for a Berea sandstone measured in the frequency range of 0.1 — 100Hz. These values are abnormally high Chapter 1. Introduction 6 c o m p a r e d t o d i e l e c t r i c c o n s t a n t (ewater/eo = 81.1) o f w a t e r . I t i s i m p o r t a n t t o n o t e t h a t t h e e f f e c t i v e d i e l e c t r i c c o n s t a n t i n e q n (1.4) i s f r e q u e n c y d e p e n d e n t a n d t h e a b n o r m a l d i e l e c t r i c v a l u e s o c c u r a t l o w f r e q u e n c i e s . E x p e r i m e n t a l d a t a p r e s e n t e d b y L e s m e s a n d M o r g a n (1999) s h o w t h a t t h e e f f e c t i v e d i e l e c t r i c c o n s t a n t i n c r e a s e s w i t h t h e d e c r e a s e i n f r e q u e n c y . T h e c o n f u s i o n a b o u t t h e a b n o r m a l d i e l e c t r i c v a l u e s a r i s e s b e c a u s e t h e i m a g i n a r y p a r t o f t h e c o n d u c t i v i t y <7;(u>) is i n c l u d e d w i t h t h e i m a g i n a r y t e r m i n v o l v i n g e i n e q n (1.3). T h i s <7j(u>) c o m p o n e n t d o e s n o t h a v e a f u n d a m e n t a l w a v e s o l u t i o n c a r r i e d w i t h i t , r a t h e r t h i s w a v e s o l u t i o n is m a n u f a c t u r e d b y m u l t i p l y i n g a n d d i v i d i n g b y u> s h o w n i n e q n (1.3). H a v i n g d o n e t h i s t h e n t h e e f f e c t i v e d i e l e c t r i c c o n s t a n t ee^(u>) c a n b e c o m e l a r g e a t l o w f r e q u e n c i e s . H o w e v e r , w h e t h e r w e h a v e a w a v e s o l u t i o n d e p e n d s o n w h e t h e r t h e q u a n t i t y |u>(e + &i(u)/<*>)/o~r(u)\ i s s m a l l o r l a r g e . I n t h e l i m i t u> —> 0 t h e e f f e c t i v e p e r m i t t i v i t y ee^(uj) —• o o b u t |u>(e + (Ti(uj)/UJ)/ar(oj)\ i s s t i l l s m a l l f o r e a r t h m a t e r i a l s b e c a u s e o~i(u>) < <rr(uj). F o r t y p i c a l e a r t h m a t e r i a l s t h e <J;(u>) i s o n e t o t w o o r d e r s o f m a g n i t u d e s m a l l e r t h a n (7r(u>). T h u s t h e I P r e s p o n s e c a n b e e i t h e r r e p r e s e n t e d as f r e q u e n c y d e p e n d e n t c o m p l e x c o n d u c t i v i t y o r as a n e f f e c t i v e d i e l e c t r i c c o n s t a n t . A s W a r d a n d F r a s e r (1967) s t a t e "The use of the concept of frequency-dependent conductivity or the concept of abnormal dielectric constant seems to be a matter of personal preference". I n t h i s t h e s i s I a d o p t t h e c o m p l e x c o n d u c t i v i t y f o r m u l a t i o n t o r e p r e s e n t t h e f r e q u e n c y d i s p e r s i o n o f t h e m e d i u m c a u s e d b y I P e f f e c t s a t l o w f r e q u e n c i e s . A t l o w f r e q u e n c i e s (o> —> 0) t h e c o n t r i b u t i o n o f t h e t i m e v a r y i n g m a g n e t i c f i e l d t o t h e e l e c t r i c f i e l d i s n e g l i g i b l e a n d t h u s V x E = 0. T h i s i m p l i e s t h a t t h e e l e c t r i c f i e l d c a n b e e x p r e s s e d as E = — W w h e r e V i s a n y s c a l a r p o t e n t i a l . T a k i n g t h e d i v e r g e n c e o f t h e e q n (1.2) a n d n e g l e c t i n g t h e d i s p l a c e m e n t c u r r e n t t e r m d u e t o d i e l e c t r i c p e r m i t t i v i t y (e) Chapter 1. Introduction 7 of the medium, I obtain, V . ( C J ( Z ' U > ) W ( » ) = V . J s = -I(iu)S(r - r,). (1.5) where J3 is the source current density, rs is the location of the current source and 6 is a Dirac delta function. In eqn (1.5) a(iu>) is the complex conductivity of the medium and it is a function of the conductivity and the chargeability. Equation (1.5), or its equivalent versions, has been extensively used to model IP effects in the medium ( Seigel, 1959; Pelton et al., 1978b; LaBrecque, 1991; Oldenburg and L i , 1994; WeUer et. al, 1996; Shi et al., 1998; Yuval and Oldenburg, 1997). Inversion methods to extract chargeability information are formulated based on the forward modelling of the eqn (1.5) (Pelton et a l , 1978b; Rijo, 1984; Oldenburg and L i , 1994; Shi et a l , 1998; L i and Oldenburg, 1999). Many practical case histories exist in the literature to illustrate the usefulness of the approach in mineral exploration and in environmental problems (Frangos et al., 1999; Morgan et al., 1999; Sogade et al., 1999; Yuval and Oldenburg, 1997; Oldenburg et al., 1997). If the frequency is large then implying that E = — V V is no longer valid. In that case, the full Maxwell equations in (1.1) need to be solved for the electric field. Usually these are combined and expressed in terms of the electric field. This is given by V X V x E + ico/j,a(iio)E — u>2fieE = —iu>pjs. (1.6) When the frequency is large, there can be two other effects, in addition to the galvanic currents in the ground. They are: (a) induction effects due to time varying magnetic field and (b) displacement currents due to polarization effects in the dielectric materials. However, for the frequencies (0.1-1000Hz) that are commonly used in IP surveys, the second term in eqn (1.6) is dominant over the third term. This implies that \u>e/a(iu>)\ << 1, for representative values of e and <r(iu)) at these frequencies, so that the displacement Chapter 1. Introduction 8 currents can be ignored. This is called the quasi-static assumption for the fields inside the earth. Thus the equation for electric field in 1.6 is given by V x V x E + icofia(iuj)E = -iujfiJ,. (1.7) In the air, a —• 0, so the second term in eqn (1.6) approaches zero. Therefore for the fields in air, I compare the remaining two terms V x V x E and to2fieE in eqn (1.6). If the characteristic length scale of the problem is denoted by L (i.e. source-receiver separation), then the first term in eqn (1.6) can be represented by VxVxEzzE/L2. Thus if the ratio, LO2fieE/(l/L2)E = LO2L2/C2 « 1, where c is the velocity of light (3 x 108m/sec), then displacement currents can be ignored. In other words, if u> << c/L then displacement currents can be neglected in the air. This is called the quasi-static assumption for the fields in the air. Thus, for the frequencies used in the exploration problems (IP,CSAMT) presented in this thesis, I neglect displacement currents. Therefore at non-zero frequencies when the induction effects are present, the measured voltage V(iu)) = / E.dl will be different than that obtained by solving eqn (1.5) and the governing equation is 1.7. The electric field in this case is given by E = -itoA - W = time varying + static (1.8) = induced field + galvanic. where A is the vector potential and V is the scalar potential. The first part in eqn (1.8), which is due to inductive effects of the medium, is the E M coupling component. If the processing is to find information about chargeability using eqn (1.5), then the E M coupling first needs to be removed from the data. The motivation for attacking the problem in this manner is that it is easier to solve for the potential in eqn (1.5) than the Maxwell's equation in (1.7). Chapter 1. Introduction 9 1.3 O r g a n i z a t i o n o f t h e t h e s i s The general organization of the thesis is presented in the flow chart in Figure 1.3. The goal of my thesis is to investigate the phenomenon of E M coupling in frequency domain IP data and to develop a practical method to remove its effects. Having developed the removal methodology, I further investigate the E M coupling problem in terms of complex conductivity. I do this by formulating an inverse problem to recover a ID complex conductivity function from complex resistivity data at a single frequency. Two possibilities exist to attack this problem: (a) Invert the E M coupling contaminated data using a forward mapping based on eqn (1.7). (b) Remove the E M coupling responses from the data and invert the resulting data to recover complex conductivity using the forward mapping in eqn (1.5). Both of these approaches are shown in the middle column of Figure 1.3. IP data at multiple frequencies sometimes can provide additional information about the chargeable materials in the ground. I also study the possibility of recovering this additional information from multi-frequency data when they are not contaminated by E M coupling. This is indicated in the third column of Figure 1.3. The thesis is organized in the following manner: The forward problem for E M coupling requires the computation of the fields for a finite length electric dipole source. Having completed this, a natural project is to in-vert controlled source audio-frequency magnetotelluric (CSAMT) data as indicated in the first column of Figure 1.3. C S A M T has been used extensively in the mineral explo-ration problems (Ostrander et al., 1983; Zonge and Hughes, 1991; Boerner et al., 1993; Basokur et al., 1997). Traditionally C S A M T data are either corrected to an equivalent M T approximation and then inverted using M T inversion algorithm, or only the far-field Chapter 1. Introduction 10 IP data CSAMT data CSAMT Inverse Problem: Requires electric & magnetic field due to finite length source (Chapter 2) With EM coupling T Without EM coupling Requires: Solution of Electric fields due to finite length dipole sources Data at many frequency t Spectral parameters inversion (Cole-Cole parameters) (Chapter 7) Directly invert for complex conductivity (Chapter 5) Remove EM Coupling Chargeability/complex conductivity inversion of corrected data at a single frequency (Chapter 4 & 6 ) Figure 1.3: Organization of the thesis. The first column indicates the C S A M T problem presented in Chapter-2. The middle column shows the different approaches to extract chargeability/complex conductivity information when the IP data is contaminated with E M coupling. This is presented in Chapter-4,5 and 6. The third column indicates the inversion of spectral Cole-Cole parameters from multi-frequency IP data, when there is no E M coupling contamination. This is presented in Chapter-7. Chapter 1. Introduction 11 C S A M T data, which mimics the M T responses, are used in the inversion. Inappropri-ate correction of the C S A M T data can generate artifacts in the inverted model. Since no complete solution was available in the published literature, and because C S A M T is sometimes carried out in conjunction with IP surveys, I have developed an inversion al-gorithm to determine the conductivity information for a I D earth from C S A M T data. In the C S A M T problem the electric and magnetic field are measured and transformed into an apparent resistivity and phase as a function of frequency. Synthetic and field examples presented in the thesis advocate the need for full C S A M T inversion without any correction. I begin the thesis with this work in Chapter-2 since it presents the details for both E M modeling in I D and for the inversion methodology used in this thesis. In Chapter-3 I present a review of the IP responses in time and frequency domain and introduce the Cole-Cole model for complex conductivity. Following Pelton et al. (1978a) I review the Cole-Cole relaxation model and discuss the complex resistivity measurements commonly known as spectral IP surveys. To obtain chargeability information from the E M coupling contaminated data I for-mulate the problem in three ways, shown as a flow-chart in Figure 1.4. In Chapter-4 I develop the removal methodology and work with amplitudes and phase response of the IP signal. First the E M coupling effects are removed from the data, and the resulting data are inverted for either amplitude or phase response. I discuss this next. E M coupling has been a major problem in the interpretation of IP data from the early days of data collection to modern interpretative techniques. In the past three decades, various removal techniques were developed to mitigate this effect (Hallof, 1974; Zonge and Wynn, 1975; Wynn and Zonge, 1977; Pelton et a l , 1978a; Coggon, 1984; Song, 1985). Some of these methods are based on empirical approaches to fitting the data and do not take into account all of the physics associated with the process and some are proprietary. However, Wait and Gruszka (1986) have posed the removal problem from Chapter 1. Introduction 12 Field Data Conductivity Real Conductivity Complex Estimate o Calculate EM responses Correct the data Invert for r\ using V.(O(1TI) v<H = - I 5 ( r - r s ) (Chapter 4; Use full EM equation for Electric field V x V x E + icopa E = - icofj. Js Invert for complex conductivity o(ico) (Chapter 5 ] Estimate a Calculate EM responses Correct the data Invert for o(ico) using V. ( c (ico) V<p(ico)) = -1 (ico) § ( r - r s ) (Chapter 6 ) F i g u r e 1.4: T h r e e d i f f e r e n t f o r m u l a t i o n s t o o b t a i n c h a r g e a b i l i t y i n f o r m a t i o n f r o m E M c o u p l i n g c o n t a m i n a t e d f i e l d d a t a . Chapter 1. Introduction 13 the basic p h y s i c a l pr inc ip les for a homogeneous half-space. T h e y have shown tha t at low frequencies, the I P response can be ob ta ined by sub t r ac t ing the E M coup l ing i m p e d a n c e f r o m the observed impedance data . T h e approach adop ted i n this work is s imi l a r to tha t of W a i t a n d G r u s z k a (1986), bu t the fo rmu la t i on a n d the r emova l m e t h o d o l o g y is subs tan t i a l ly different f rom theirs . I express the observed i m p e d a n c e as p r o d u c t of an I P response func t ion and an impedance due to E M effects. G e n e r a l formulae , app l i cab le to I D , 2 D a n d 3 D da t a sets, are generated to show how E M coup l ing i n b o t h a m p l i t u d e and phase can be r emoved . T h e basic phys ics of the E M coup l ing p r o b l e m can be s tud ied i n the contex t of a one-d i m e n s i o n a l ear th . I n this work , the theore t ica l founda t ion of the E M c o u p l i n g r e m o v a l m e t h o d is deve loped us ing an electr ic field due to a h o r i z o n t a l e lec t r ic d ipole ( H E D ) over a half-space. Different types of fields are character ised based on the f requency of the t r a n s m i t t e d s ignal a n d the presence or absence of chargeabi l i ty . T h e so lu t ion for an e lec t r ic f ield due to a ho r i z on t a l e lectr ic d ipole on the surface of a conduc t ive m e d i u m is o b t a i n e d by so lv ing a vector po t en t i a l equa t ion i n I D us ing p ropaga to r ma t r i ces . T h e e lect r ic fields are t h e n ob ta ined by differentiat ing these vector po ten t ia l s . T h e e lec t r ic fields o b t a i n e d due to a d ipole source are t hen in tegra ted a long the lengths of the finite l eng th sources to o b t a i n the impedances measured on the surface of the ea r th . These responses are t h e n used to correct the da t a so tha t the f ina l d a t a are c o m p a t i b l e w i t h eqn (1.5). T h e correct ions are pe r fo rmed b o t h for the ampl i tudes a n d phases of these impedances . Syn the t i c examples for a I D ear th are presented i n the thesis. It is found tha t the c o n d u c t i v i t y i n f o r m a t i o n of the ea r th is c r u c i a l i n c o m p u t i n g the E M c o u p l i n g responses. T h e p roposed me thods to correct for E M coup l ing can be app l i ed w h e n the ea r th s t ruc ture is more compl i ca t ed . Ideal ly however I need to k n o w the t rue 2 D or 3 D con-d u c t i v i t y a n d to be able to car ry out r igorous fo rward m o d e l l i n g . I do not have tha t Chapter 1. Introduction 14 capability in 2D, so I propose the following. First the data at the lowest frequency are inverted to obtain a 2D conductivity distribution of the earth. Then for each transmitter position an averaged ID conductivity model is generated from the 2D conductivity map. Next I use the ID averaged model to compute the E M coupling responses and correct the data that are associated with this particular transmitter. The corrected data are then inverted using a 2D IP inversion algorithm to produce a map of the chargeability. This 2D approximate removal methodology is applied to two field examples. In practice the conductivity of the earth is likely to be three-dimensional. Moreover the analysis in ID showed that the conductivity information is important to compute the E M coupling responses and correct the data. Therefore I investigate the validity of the E M coupling removal method in 3D environments using a 3D staggered grid finite-difference E M forward modeling (Newman and Alumbaugh, 1995). Synthetic examples of a conductive and chargeable block in a half-space and layered medium are considered to test the method. In addition to 3D bodies in a layered medium, I also investigate the effects of topography on the E M coupling responses, since the topography alters the Conductivity structure. The tests are relegated to synthetic data sets only. Although E M coupling responses have been computed for 3D bodies using integral equation approach for flat earth geometry (Hohmann, 1975), the solution to the 3D E M coupling removal problem has not been addressed in the previous literature. The standard methodology for computing IP responses is to use a two-stage approach. First data are used to estimate the conductivity cr. Then, under the assumption that the chargeability rj is small, a linearized system is solved for v. However the conductivity and chargeability can be combined together to form a complex conductivity model of the earth (Wait, 1959a; Van Voohris et al, 1973; Pelton et a l , 1978a). Therefore an alternative approach for solving the inverse problem is to formulate the physical problem as one physical process in which both IP effect and the E M coupling contribution are Chapter 1. Introduction 15 s imul t aneous ly considered. T h i s is an en t i re ly new approach to the p r o b l e m i n w h i c h there is no requi rement to remove E M coup l ing . I n C h a p t e r - 5 , I fo rmula te the p r o b l e m i n t e rms of c o m p l e x c o n d u c t i v i t y and work w i t h the measured c o m p l e x impedances . T h e fo rward m o d e l l i n g a l g o r i t h m is t hen based on the so lu t ion of the fu l l E M equat ions for the e lec t r ic f ie ld . T h e da t a can be inve r t ed for a c o m p l e x c o n d u c t i v i t y m o d e l w h i c h conta ins i n f o r m a t i o n about the c o n d u c t i v i t y and the chargeab i l i ty of the m e d i u m . T h i s is ca r r i ed out for a I D ea r th and syn the t ic examples are shown to i l lu s t r a t e the v a l i d i t y of the approach . I n this C h a p t e r , I develop a general inve r s ion m e t h o d o l o g y to ex t rac t c o m p l e x c o n d u c t i v i t y m o d e l parameters f rom c o m p l e x res i s t iv i ty da ta . In C h a p t e r - 6 , I first remove E M coup l ing f rom the measured impedances a n d then inver t the resu l t ing d a t a for a I D c o m p l e x c o n d u c t i v i t y s t ruc ture w i t h the c o m p l e x D C res i s t iv i ty fo rward m o d e l l i n g i n eqn (1.5). T h i s is also a new approach for f requency d o m a i n I P surveys , i n w h i c h a single invers ion is ca r r ied out to ex t rac t b o t h chargeab i l i ty and c o n d u c t i v i t y i n f o r m a t i o n f rom the decoupled da ta . T h e goal up to th is po in t i n the thesis has focussed u p o n ge t t ing a d i s t r i b u t i o n of chargeabi l i ty . T h e next ques t ion that can be asked is: I f the two bodies i n the m e d i u m are chargeable, do they cor respond to the same mine ra l i za t ion? T h i s raises a l o n g s tand-i n g ques t ion i n this f ield: " C a n one use I P d a t a for m i n e r a l d i s c r i m i n a t i o n ? " If there is some a b i l i t y to d i s t ingu i sh between graphi tes , sulphides , clays i t w o u l d g rea t ly enhance the i m p o r t a n c e of this m e t h o d . T o do this we need to look in to the f requency behav iou r of the c o m p l e x c o n d u c t i v i t y for each region of the ea r th . T h e r e are at least two possible approaches to the p r o b l e m : (a) O b t a i n cr{iio) for a range of frequencies a n d a t t e m p t to recover i n f o r m a t i o n about the m i n e r a l i z a t i o n f rom a(iu>). Y u v a l and O l d e n b u r g (1997) app l i ed th is m e t h o d to ex t rac t C o l e - C o l e parameters f r o m m u l t i - c h a n n e l t i m e d o m a i n da ta . T h e p r o b l e m w i t h this m e t h o d is tha t , parameters for each ce l l i n the 2 D m o d e l are o b t a i n e d i ndependen t ly and there is no regu la r iza t ion i m p o s e d tha t w o u l d generate Chapter 1. Introduction 16 s m o o t h d i s t r i b u t i o n of the spec t ra l parameters , (b) Pa rame te r i ze the c o m p l e x conduc t -i v i t y m o d e l <j(iu)) = f(o~o,p) and o b t a i n i n f o r m a t i o n about these parameters (p) f r o m the da ta . T h e n use the values of p to infer someth ing about the m i n e r a l i z a t i o n . T h e l a t t e r approach , w h i c h is adop ted here, requires specif icat ion of f(cr0,p). A m o n g the several mode ls tha t have been tested, the C o l e - C o l e m o d e l is the most c o m m o n l y used a m o n g the workers i n the f ie ld. I n Chap te r -7, I consider mul t i - f r equency c o m p l e x r e s i s t i v i ty d a t a to recover the spec t ra l parameters of the C o l e - C o l e m o d e l . T h e ob jec t ive here is to fo rmula te the p r o b l e m as a nonl inear inverse p r o b l e m and o b t a i n a s m o o t h d i s t r i b u t i o n of the spec t r a l parameters f rom mul t i - f requency c o m p l e x res i s t iv i ty da ta . T h i s is ca r r i ed out i n a two-d imens iona l env i ronment w i t h syn the t ic and field d a t a examples . F i n a l l y , the ma jo r conclusions and future scope of this thesis are s u m m a r i z e d i n Chapter-8. Chapter 2 One-dimensional inversion of C S A M T data 2.1 Introduction I n t h i s C h a p t e r , I d e v e l o p a n i n v e r s i o n m e t h o d o l o g y t o o b t a i n a o n e - d i m e n s i o n a l ( I D ) c o n d u c t i v i t y s t r u c t u r e f r o m c o n t r o l l e d s o u r c e a u d i o - f r e q u e n c y m a g n e t o t e l l u r i c s ( C S A M T ) d a t a . I n p r a c t i c e , t h e s o u r c e - r e c e i v e r s e p a r a t i o n s i n C S A M T s u r v e y s a r e c o n s i d e r e d t o b e s u f f i c i e n t l y l a r g e s o t h a t t h e m a g n e t o t e l l u r i c ( M T ) m e t h o d c a n b e u s e d t o i n t e r p r e t t h e d a t a . H o w e v e r t h i s m e a n s t h a t s i g n a l s t r e n g t h i s s m a l l , a n d l a r g e r r e c o r d i n g t i m e s a n d b i g g e r d i p o l e t r a n s m i t t e r s a r e r e q u i r e d t o c o m p e n s a t e f o r t h i s . T h e i n s t a l l a t i o n o f a l a r g e - l e n g t h r e m o t e d i p o l e c a n a l s o a d d s i g n i f i c a n t l y t o t h e s u r v e y c o s t . M o r e i m p o r t -a n t l y , t h e r e i s v a l u a b l e i n f o r m a t i o n a b o u t t h e c o n d u c t i v i t y m e a s u r e d b y i n s t r u m e n t s t h a t a r e c l o s e t o t h e t r a n s m i t t e r . W h e n u s i n g t h e M T f o r m u l a t i o n f o r p r o c e s s i n g , t h e n t h e s e d a t a h a v e t o b e d i s c a r d e d , o r e l s e m o d i f i e d t o r e m o v e t h e n e a r - f i e l d e f f e c t s . A m e t h o d -o l o g y t o o b t a i n the, c o n d u c t i v i t y i n f o r m a t i o n d i r e c t l y f r o m C S A M T d a t a , i r r e s p e c t i v e o f t h e i r d i s t a n c e f r o m t h e t r a n s m i t t e r , w o u l d b e p r a c t i c a l l y u s e f u l . T h u s t h e g o a l h e r e i s t o d e v e l o p f o r w a r d m o d e l i n g a n d i n v e r s i o n t h a t t a k e s i n t o a c c o u n t a n y s o u r c e - r e c e i v e r s e p a r a t i o n a n d h a s t h e a b i l i t y t o e x t r a c t I D c o n d u c t i v i t y s t r u c t u r e . C S A M T i s a f r e q u e n c y d o m a i n e l e c t r o m a g n e t i c m e t h o d w h i c h u s e s a g r o u n d e d d i p o l e o r h o r i z o n t a l l o o p as a n a r t i f i c i a l s o u r c e . O v e r t h e y e a r s C S A M T h a s e m e r g e d as a p o w -e r f u l e x p l o r a t i o n t o o l a n d h a s f o u n d i t s a p p l i c a t i o n i n m i n e r a l e x p l o r a t i o n ( Z o n g e e t a l . , 17 Chapter 2. ID CSAMT inversion 18 1987; B a s o k u r et a l . , 1997), geo the rmal inves t iga t ion ( B a r t e l and J acobson , 1987; Sand-berg a n d H o h m a n n , 1982; W a n n a m a k e r , 1997a), h y d r o c a r b o n e x p l o r a t i o n (Os t r ande r et a l . , 1983) a n d ground-water c o n t a m i n a t i o n (Zonge et a l . , 1985) p rob lems . A n excel lent r ev iew of C S A M T and its app l i ca t i on is g iven by Zonge and Hughes (1991). T h e o r i g i n a l deve lopment of C S A M T was designed to i m p r o v e the s ignal s t reng th p r o b l e m tha t arises i n magne to t e l l u r i c ( M T ) m e t h o d ( G o l d s t e i n and St rangway, 1975). Howeve r the non-p lane wave na tu re of the source at ce r ta in f requency b a n d w i d t h l i m i t s the i n t e r p r e t a t i o n of d a t a by conven t iona l M T methods . Therefore the t r a d i t i o n a l p rac t i ce is to in te rpre t the far-f ield d a t a w h i c h are ob ta ined at d is tance of four to five s k i n depths (Sasak i et a l . , 1992), or correct the da t a for non-plane wave effect and use M T m o d e l i n g to in te rpre t the observat ions ( B a r t e l and Jacobson , 1987). T h e near-field correct ions are often based o n homogeneous half-space models and the i r v a l i d i t y is i n ques t ion i n c o m p l i c a t e d envir -onments . T h i s compromises the i n fo rma t ion about the subsurface c o n d u c t i v i t y tha t can be o b t a i n e d f rom t r ans i t i on zone da ta . B o e r n e r et a l . (1993), M a u r e r (1988) p o i n t e d out the necessi ty for C S A M T m o d e l i n g i n an invers ion a n d have w a r n e d against co r rec t ing the d a t a to the p lane wave a p p r o x i m a t i o n . T h e advances i n the fo rward m o d e l i n g for C S A M T became the next step towards the deve lopment of i n t e rp re t a t ion and he lped i n a id ing the i n t e rp re t a t i on of near-f ie ld a n d t r ans i t i on zone da ta . T h e m o d e l i n g results of Bosche t t o a n d H o h m a n n (1991) shows the effect of 3 D s t ructures on C S A M T da ta . M o s t of the previous work o n C S A M T d a t a i n t e rp re t a t i on i n the l i t e ra ture is based o n m o d e l i n g techniques ( W a n n a m a k e r , 1997a,b). He re I present an inverse f o r m a l i s m to a id in t e rp re t a t ion a n d app ly i t to one d i m e n s i o n a l ea r th s t ruc tures . E x i s t i n g I D invers ion techniques for C S A M T da t a ( e.g. Zonge and Hughes , 1991) a t t empt to es t imate the conduc t iv i t i e s and thicknesses of a few layers by f ind ing the least squares so lu t ion of an overde te rmined sys tem. S u c h a lgor i thms concent ra te u p o n r ep roduc ing the da t a but the resul tant m o d e l depends u p o n the n u m b e r Chapter 2. ID CSAMT inversion 19 of layers assumed, and the initial values of the conductivities and thicknesses. I present an alternative approach by parameterizing the earth into a large number of horizontal layers each of which has a constant, but unknown, conductivity and fixed thickness. The inverse problem becomes underdetermined and the solution is therefore non-unique. Mathematically in such a case there exists many solutions which satisfy the data, however a class of models is often sought by minimizing a particular model objective function. For a purely ID problem the different character of such models depends on the model objective function being minimized. The flexibility of minimizing different model norms and fitting the data to different misfits in the inversion algorithm helps to explore the model space which is otherwise restricted in parametric inversions. The data in C S A M T are apparent resistivity and phase as a function of frequency, and the data are computed from measured orthogonal components of E and H (Ex,Hy or Ey, Hx). In the next section the forward modeling of responses are outlined. 2.2 F o r w a r d P r o b l e m The forward modeling of C S A M T in one dimension requires the computation of electric and magnetic fields viz. Ex, Ey, Hx, Hy over a stratified earth due to a horizontal electric dipole (HED). The generic solution of HED is well known for a layered earth problem (Wait, 1982; Ward and Hohmann, 1988). The derivation is carried out in the frequency domain which follows from the work of Ward and Hohmann (1988). For a source free region the electric and magnetic field can be decomposed in terms of Schelkunoff poten-tials A and F, often referred as T M and T E potential. The potential A is considered to arise from electric sources (Js) and the F potential is due to magnetic sources (Ms). The total magnetic and electric field in terms of these potentials is given by (Ward and Chapter 2. ID CSAMT inversion 20 Hohmann, 1988), H = VxA-((7 + iwe)F + ^ V ( V.F) E = -VxF- iuu.A + V ( V . I ) Since the physical properties are varying in one direction only one component of the vector potential is of interest. I have A — Az(x,y, z)z and F = Fz(x,y, z)z and hence I need consider only scalar potentials Az and Fz. In each layer of constant conductivity (Figure 2.1) the scalar potential satisfies the following differential equation. Grounded Transmitter Wire <L> Y Receivers X • _ h, z = o z= z-, CT1 CT2 h 2 z= z 2 - I Z = Z j 2 = Z M - 1 CTM Figure 2.1: Conductivity structure for horizontally layered earth model V2A{ + fcjAJ = 0 V 2 F / + fcji^' = 0 (2.2) Chapter 2. ID CSAMT inversion 21 where k2 = cv2fijej — icofijcrj. The continuity of tangential components of E and H across each boundary leads to the following boundary conditions for the z component of Schelkunoff potential which is given by A?r\xiV,z = Zj-u") = AJz(x,y,z = Zj,u>) 1 dAj-1 _ 1 dAj <TJ_I + itoe^x dz CTJ + iioej dz ^ FrX(x,y,z = Zj-Uu>) = FJz(x,y,z = zhcu) 1 dF?-1 _ 1 dF> ico/Aj-i dz iwfij dz In order to solve the system in one variable (z) a 2D Fourier transform of the eqn(2.2) is taken which gives the following differential equation, d z „ (2.4) dF3 - • where u2 = k2 + k2 — k2 and A3Z — Az(kx,ky, z = ZJ,UJ) is the Fourier transform of A\. The Fourier transform pair is given by / R(x,y,z)e-i^x+k^dxdy (2.5) -oo J —oo -i y»oo poo R(x,y,z)=— / R{kx,ky,z)e*kM^dkxdky (2.6) ^ J-oo J-oo The solution to eqn(2.4) can be expressed in terms of upward (Uj) and downward (Dj) propagating coefficients. The solution at the jth layer is given by A{(kx, ky,z = ZJ,LO) = Df{kx,ky,w)e-U^-Z'-^ + Uf{kx,ky)co)eu^-z^ (2.7) F>(K, ky, z = Zj,u) = Df (kx, ky,«,)e-u*z-'i-^ + Uf(kx, ky,to)eu^-z>-^ (2.8) Chapter 2. ID CSAMT inversion 22 The upward and downward coefficients in the (j — l)th layer can be determined in terms of the coefficients of the jth layer through a propagator matrix by applying the boundary condition at z = The details of the derivation are shown in Appendix-A. The solution is propagated from the bottom half-space to the top of air-earth interface through a stack of layers. At a height h above the air-earth interface the solution for the potential is given by A;(z,y,z,u) = -^^f (e-U°lz+hl + ^ e " " ' ^ ) ^ l d \ (2.9) 4TT dy J0 ' u0\ where J 0 is the Bessel function of the zeroth order and Ids is the current element. The electric field (Ex) and magnetic field (Hy) recorded at the surface due to x-directed HED in terms of the potentials is given by (Ward and Hohmann, 1998), ^ = 0 , . ) ^ f £ - < p ( 2 . u ) ^u>e0 oxoz oy Hy(z = 0,c) = ? - ± - , (2.12) ox iu\i0 oyoz In practice a long grounded wire is used as a transmitter. It is typically l-2km long while the receiver dipole length varies from 50m to 200m which measures the electric field. The magnetic fields are measured by the magnetic sensors. Linear superposition of the fields due to a point dipole are used to represent the finite length transmitting dipole. The C S A M T data are generally converted to apparent resistivity (pa) and phase ($) as a function of frequency. The relationships are: Chapter 2. ID CSAMT inversion 23 pa(x,y,z = 0,w) UJfl Ex(z = Q,u>) Hy{z = 0,uj) (2.13) <j)a(x,y,z = 0,UJ) = 4>Ex(x,y,z = 0,u>) - <j)Hy(x,y,z = 0,u) (2.14) where Ex and Hy are the integrated fields due to finite length long grounded wire. 10° 101 102 10 3 10 4 Frequency (Hz) 10" 10" 10" 10° 10' 10 2 10 3 10 4 Depth (m) 10° 101 10 2 10 3 10" Frequency (Hz) Figure 2.2: (a) The transmitter and receiver geometry used to generate the synthetic data, (b) Conductivity model used to generate the data, (c) The C S A M T (dashed) and M T (solid) apparent resistivity curve for the five-layer conductivity model. The transition zone notch is observed around 100 Hz on the C S A M T curve, (d) The phase response for C S A M T is shown in dashed and the solid fine represents the M T phase response. C S A M T data for a five layer earth model are shown in Figure 2.2. The transmitter is Chapter 2. ID CSAMT inversion 24 1.5 km in length and the receiving dipoles are parallel to the transmitter but offset by 2 km. The conductivity structure used to compute the data is shown in Figure 2.2(b) and apparent resistivities and phases are given in Figure 2.2(c)-(d) respectively. It is instruc-tive to compare these responses with M T responses. M T responses are obtained from the orthogonal component of electric and magnetic field due to an incident plane wave on the surface of the earth. There is agreement at higher frequencies but disagreement is pronounced at lower frequencies when non-plane wave effects become important. The response from the non-plane wave effect depends on various factors: (a) frequency of the signal, (b) transmitter-receiver separation, (c) length of the transmitter, (d) orientation of the transmitter and receiver dipoles and (e) the overall conductivity which alters the skin depths at a particular frequency. The relative contribution of each of these factors to the non-plane wave effect is difficult to determine since the conductivity structure is unknown. The region around the transmitter is generally divided into three zones. In the region where r << 8, (8 being the skin depth) the magnitude of E decays as 1/r3 and that of H decays as 1/r2 resulting in an E / H which is independent of frequency but dependent on r. This results in the linear decrease in apparent resistivity with increasing frequency on a log-log plot as shown in Figure 2.2(c). The phase response shown in Figure 2.2(d), tends to zero with decreasing frequency. In the region where r f» 8, the decay of the H-field changes from 1/r2 to 1/r3 ; this is called the transition zone. For a layered earth the transition zone is characterized by a notch in the apparent resistivity and this is seen to he in the frequency range around 100 Hz for this example. The inflection of the phase curve is also indicative of the transition zone response shown in Figure 2.2(d). It has been pointed out (Zonge and Hughes, 1991, pg. 772; Boerner and West, 1989) that data acquired in the transition zone are generally sensitive to layered structure. The shape of the notch can change depending on the conductivity and source-receiver separation (Zonge and Hughes, 1991). The far-field region is characterized by plane wave behavior Chapter 2. ID CSAMT inversion 25 of the source and occurs when r » 8. In this region the C S A M T response is same as the M T response. 2.3 Inverse Problem A primary objective in an inverse problem is to recover a geologically interpretable model that can acceptably reproduce a finite set of observations. The data in a one-dimensional C S A M T problem are apparent resistivities and phases at frequencies fj,j = 1,N. At each sounding location the data vector can be represented as d = (pa(h),..., Pa(fN), 4>(h\<f>(fN))T (2.15) The medium is discretized into horizontal layers of constant conductivity with increasing layer thickness. To ensure positivity and also to allow large variations of conductivity I choose rrij — In(crj) as our parameters. The model for the inverse problem is m = (mi, m 2 , • • • , rriM) T where m2 = ln(cr,-). For a finite number of inaccurate observations there are infinite number of solutions that will reproduce the data to within their error. The inverse problem is solved in a standard way by minimizing a model objective function subject to adequately fitting the data. A regularized Gauss-Newton methodology ( Constable et al. , 1987; Oldenburg et al., 1993) is implemented to iteratively solve the nonlinear optimization problem. In this algorithm a generic model objective function is chosen that is a combination of smallest and flattest model penalty terms 4>m = OLs J ws(z) (m(z) - mrep[z)f dz + az J wz(z) (^^J^ d z (2-16) The smallest model objective function penalizes the deviation of the model from a reference model and the flattest model objective function penalizes roughness of the model Chapter 2. ID CSAMT inversion 26 w i t h dep th . In eqn (2.16) mref is the reference or the base m o d e l a n d ots a n d az are the con t ro l parameters tha t con t ro l the re la t ive i m p o r t a n c e of closeness to the reference m o d e l a n d the roughness. ws(z) and wz(z) are the a d d i t i o n a l we igh t ing funct ions w h i c h c o n t r o l the s t r u c t u r a l i n f o r m a t i o n i n the m o d e l . Incorpora t ing the l ayered ea r th represen ta t ion in to eqn(2.16) al lows to formula te the inverse p r o b l e m as m i n i m i z e : (f>m — as \ \Ws(m — m r e / ) | | 2 + az \flVzm\f (2.17) subject to: <f>d = \ \Wd (dobs - < F r e d ) \\ = <f>*d where Ws and Wz are f ini te difference m o d e l we igh t ing mat r ices , <f>d is the f ina l target misf i t to be a t t a ined i n the invers ion process. Wd is a 2N x 2N da t a we igh t ing m a t r i x . I f the d a t a errors are unbiased , independent and Gauss i an , t hen Wd is a d i agona l m a t r i x con ta in ing the rec iprocals of s t anda rd devia t ions of the da t a and <fid is a % 2 r a n d o m var iable . I f such is the case t hen the expec ted value of <f>d is equa l to 2 N . T h e inverse p r o b l e m g iven i n eqn(2.17) can be solved by m i n i m i z i n g a g loba l ob jec t ive func t i on g iven by <t>(m) = <f>m + [3-1(4>d-4>d) (2-18) where (5 is the r egu la r i za t ion or r ec ip roca l of Lagrange parameter . L e t F represent the non l inear m a p p i n g between the da t a and the m o d e l by d = F(m). T h e o p t i m i z a t i o n p r o b l e m is nonl inear and must be solved i te ra t ive ly . Le t mn be the current m o d e l and 8m be a p e r t u r b a t i o n . T h e fo rward m a p p i n g opera tor is l i nea r i zed by T a y l o r ' s series expans ion about m^n' w h i c h y ie lds , F(mn + Sm) = F(mn) +JSm + 0\\(Sm)2\\ (2.19) where O \ \{8m)2\\ contains the te rms i n c l u d i n g the second a n d higher order der ivat ives w i t h respect to 8m. J is the sens i t iv i ty m a t r i x of the order 2N x M w i t h elements Chapter 2. ID CSAMT inversion 27 Jij = ddi/drrij. T h e p e r t u r b e d objec t ive func t ion can be w r i t t e n as 4> ( m ( n ) + 8m) = ots 1 W. ( m ( n ) + 8m - mref) \\2 + az\\Wz ( m ( n ) + 8m) \\ + (2.20) / T 1 111 Wd (d°bs - d^ - J8m) 11" - </>* \ T h e m i n i m u m of this p e r t u r b e d objec t ive func t ion is found by different ia t ing eqn (2.20) w i t h respect to 8m a n d equa t ing to zero. T h i s generates a sys t em of M equat ions i n M u n k n o w n s g iven by (jTWjWdJ + BWlWm) 8m = JTWjWd8d (2.21) - BWZWmmW + 6WjWsmref where W^Wm = a,WjW, + azW^Wz. T h e sys t em i n eqn(2.21) is so lved us ing Q R d e c o m p o s i t i o n ( G o l u b and V a n L o a n , 1996). T h e new m o d e l at the (n + l)th i t e r a t i o n is g iven b y m ( n + 1 ) = m^ + 8m and the new misfi t is g iven by <tf[L = \ \Wd (dobs - d ( m ( n + 1 ) ) ) 112 (2.22) T h e inverse p r o b l e m is solved i te ra t ive ly . A t each i t e r a t i on a p re sc r ibed amoun t of misf i t is r educed . T h i s is done by adjus t ing the value of 3 us ing a fine search p rocedure , u n t i l the so lu t ion of eqn (2.21) generates a misfi t of (f>^L = (j)*}^ where <j>*}n^ is the target misf i t at the nth i t e r a t i on . T h e search is con t inued u n t i l <j>^L is close enough to (j)*^ or a m i n i m u m misfi t is found i f the target cannot be achieved. T h e leve l of des i red misf i t at each i t e r a t i on can be chosen based on a s imple scheme, < ^ n + 1 ^ = max[((f>dn\2N] , where 2Af is the t o t a l n u m b e r of observat ions and 0 < ( < 1.0. 2.4 C o m p u t a t i o n o f S e n s i t i v i t y T o solve the inverse p r o b l e m i t is requ i red to compu te the sensi t ivi t ies Jij = ddi/dmj i n eqn(2.21) . He re I discuss two methods to compute the sensi t ivi t ies . T h e first app roach is Chapter 2. ID CSAMT inversion 28 the adjoint Green's function method (Farquarson and Oldenburg,1993; McGillivray and Oldenburg, 1990) and second method is formulated by differentiating the propagator matrices. Adjoint Green's function method to compute sensitivity is an efficient method and has been extensively used in many EM problems (Farquarson and Oldenburg,1993; Smith and Booker, 1988; Oldenburg et al., 1993). Here I develop the necessary equations based on this approach. An alternative method to compute the sensitivity is based on simple differentiation approach called the differentiation of propagator matrices. This method is conceptually simpler than the adjoint method and can be modified to be of comparable efficiency to the adjoint approach. The method is robust since it does not involve any positive exponentials for the ID problem. For a generic ID problem this method can be easily extended for other physical properties like magnetic susceptibility and dielectric constant. First I discuss the adjoint method and then the differentiation approach. For a ID problem the medium can be discretized in terms of linear combination of pulse basis functions given by M a{z) = Y,aMz) (2-23) 3=1 where the coefficient CTJ is the conductivity of the jth layer and, 2.4.1 Adjoint Green's function method Consider that the forward problem for the potential can be written in the following operator form, 1 z. < Z < Zj (2.24) 0 elsewhere CR = S(u>) where (2.25) Chapter 2. ID CSAMT inversion 29 and R = R(kx, ky) z,u>) represents the A\ or F® potential and S(u/) is the source function which is independent of the conductivity. The boundary conditions are given by R —» 0 as z —>• ±oo and the interface conditions (i.e. boundary conditions between layers) for R are given by eqn (2.3). To formulate the differential equation for the sensitivity eqn(2.25) can be differentiated with respect to ak and noting that the source does not depend on the conductivity. This gives, M 2 dau dz2 dR d2 C-— = itofi0ipk(z)R where £ = —— — ul — iuft0 } ajipAz) (2.26) with the boundary condition dR(z)/dak —• 0 as z —> ±oo. The system of equations given in eqn (2.25) and eqn (2.26) are equivalent with the exception of the source term. The differential equation in dR/dak can be solved using an adjoint Green's function method. First I will consider the derivation for the sensitivity of the F potential. Differentiating the interface condition for the F potential in eqn (2.3) with respect to ak yields, dF''1 _ dF3' dak dak 1 02F3-1 1 82Fj K ' fij-i dzdak ftj dzdak The sensitivity equation in (2.26) with the interface conditions in eqn(2.27) can be solved using the adjoint Green's function method. This is given by = f°° iLo^k{z)F{z)G\z- ()dz (2.28) OVk J-oo where adjoint Green's function satisfies, d2 M C*G = S(z-C) where C = — - u20 + iujfi0 <rji/>j(z) (2.29) with the boundary condition that G(z; £) —> 0 as z —> ±oo and C* is the adjoint operator and is the complex conjugate of £ in eqn(2.25). The internal boundary conditions for the Chapter 2. ID CSAMT inversion 30 2 - u \ G = 0 (2.30) Green's function can be obtained by replacing OF3/d<Jk by G3 in eqn(2.27). Therefore if the solution to eqn(2.25) is known then the solution for the sensitivity equation in (2.26) can be easily obtained through the Green's function. In this problem the solution for the sensitivity at the surface, i.e., ( = 0 is of interest. In the region —oo < z < ( = 0 the Green's function is the solution to the homogeneous equation given by A 2 dz whose solution can be written in the form G(z; () = B((^)eMZ. And in the region 0 = £ < z < oo the solution is proportional to F*(z) since the operator for Green's function in eqn(2.29) is the complex conjugate of the operator in eqn(2.25). Combining the two solutions yields G-(Z;() = B(()eu°* z<( = 0 (2.31) G+(z;C) = C(C)F*(z) z>C = 0 At the discontinuity z — ( the Green's function is continuous i.e. G~(z;() = G+(z;() and the jump in the first derivative is equal to —1 which implies dG~ /dz — dG+ /dz = — 1. Therefore applying the above conditions at z = ( = 0 yields ^ ' ( o ) - ^ ( o ) ( 2 3 2 ) c(o)= 1 F*'(0) - uoF*(0) where the prime denotes the derivative. Thus substituting G in eqn(2.28) by C(0)F*(z) and noting that ipk(z) is unity in the kth layer and zero everywhere else yields the following equation for sensitivity, d*k - F'(0) - uoF(0) ] Z k J k { Z ) d Z (2.33) Next I will derive the sensitivity for the A potential. The derivation is very similar to that of the F potential shown above, except that the internal boundary conditions for Chapter 2. ID CSAMT inversion 31 the sens i t iv i ty of A po t en t i a l are different t h a n that of the F po t en t i a l . Di f fe ren t ia t ing the b o u n d a r y condi t ions for A po ten t i a l i n (2.3) w i t h respect to c o n d u c t i v i t y crk for the interface z = such that j ^ k and j ^ k + 1 gives dA*-1 dA> dcrk dak 1 d2A'~1 1 d2A> (2.34) <7j_i dzdcrk Uj dzdak A n d at the interface z = zk_\ the interface condi t ions are g iven by dAk~x _ SA*= dcrk dcrk 1 d2Ak~1 1 / d2Ak 1 dAk (2.35) <Tfc_i dzdak <7fc \dzdcrk <rk dz A n d at interface z = the interface condi t ions are, dAk _ dAk+l dak dak d2Ak i QAk\ 1 82Ak+1 (2.36) ak \dzdak <Jk dz ) a k + 1 dzdak T h u s the i n t e r n a l b o u n d a r y cond i t i on for the sens i t iv i ty equa t ion are different f r o m tha t of the fo rward p r o b l e m of A po ten t i a l on ly at z = zk_i and z = zk. So the Green ' s func t ion deve loped for the fo rward p r o b l e m cannot be read i ly used to solve the sens i t iv i ty equa t ion . O n e poss ib i l i t y is to convert the inhomogeneous b o u n d a r y c o n d i t i o n i n eqn (2.35) and (2.36) to homogeneous b o u n d a r y cond i t i on s imi la r to tha t of the fo rward p r o b l e m and in t roduce a new source t e r m i n eqn (2.26). I n eqn (2.35) a n d (2.36) the e x t r a t e rms i n the b o u n d a r y cond i t i on can be regarded as two b u r i e d sources, one o n ei ther side of the kth layer i .e. at z = zk_\ and z = zk. Therefore the o r i g i n a l equa t ion for sens i t iv i ty equa t ion g iven i n (2.26) can be changed to, dA 1 8Ak C-— = icofi0ipj(z)A — (S(z - zk) - 8(z - z f e_i)) (2.37) oak <rk Oz Chapter 2. ID CSAMT inversion 32 where C is g iven by eqn(2.26). S imi l a r s i tua t ion w i l l arise w h e n sens i t iv i ty for F p o t e n t i a l is r equ i red w i t h respect to magne t ic pe rmeab i l i t y /z. E q u a t i o n (2.37) has a ve ry s imi l a r f o r m w i t h the sens i t iv i ty equa t ion for e lectr ic f ield w i t h respect to magne t i c p e r m e a b i l i t y u. g iven i n Z h a n g and O l d e n b u r g (1997). N o w that the differential opera tor a n d the b o u n d a r y condi t ions of the sens i t iv i ty of A po ten t i a l are s imi l a r to tha t of the fo rward p r o b l e m , the adjoint Green ' s func t ion m e t h o d can used to evaluate the sens i t iv i ty . T h i s is g iven b y j z=zu dA(C) ~ dcrk /oo iu>fi0if>k(z)A(z)G*(z;C)dz --co G* dA o-k dz (2.38) where G(z; () satisfies the differential eqn(2.29) w i t h the b o u n d a r y c o n d i t i o n tha t G(z; () -0 as z —> ± o o . T h e b o u n d a r y value p r o b l e m for the Green ' s func t ion is so lved i n s imi l a r way as of the F po ten t i a l . A p p l y i n g the con t inu i ty and the j u m p c o n d i t i o n of the Green ' s f unc t i on at the a i r -ear th interface yields the sens i t iv i ty equa t ion for A po t en t i a l . T h i s is g iven b y 1 8A dak A'(0) - uoA{0) I iujpoAk\z)dz (Tk dz Z = Z k _ ! (2.39) Therefore eqn (2.33) and (2.39) can be used to compu te the sens i t iv i ty for the F a n d A p o t e n t i a l respect ively . Since the coefficients for the potent ia ls are a l ready c o m p u t e d i n the fo rward p r o b l e m , so eqn(2.33) and (2.39) can be easi ly ob ta ined . T o o b t a i n the sens i t iv i ty i n spa t i a l d o m a i n inverse F o u r i e r - H a n k e l t r ans fo rm is a p p l i e d to (2.33) a n d (2.39). A l t h o u g h there is no e x t r a c o m p u t a t i o n cost needed to evaluate Fk a n d Ak i n (2.33) a n d (2.39) respect ively, however for each layer the H a n k e l t r ans fo rm has to be ca r r i ed out separate ly since the in teg rand changes for every layer . H a v i n g o b t a i n e d the sens i t iv i ty for the potent ia ls , the sens i t iv i ty for the e lectr ic and magne t i c fields can be cons t ruc t ed b y different iat ing eqn (2.11) and (2.12) w i t h respect to the c o n d u c t i v i t y ak Chapter 2. ID CSAMT inversion 33 and substituting the expressions for the sensitivity of the potentials from eqn (2.33) and (2.39). In C S A M T the data are represented by apparent resistivity and phases given by eqn (2.13) and (2.14). Following Farquarson and Oldenburg (1997) the sensitivity for the apparent resistivity and phase can be determined from the sensitivity of the fields. This is given by ExdcrkJ \HydakJ\ (2.40) dcrk \EX dak J \Hy dak 2.4.2 Differentiation of Propagator Matrices A n alternative method to compute sensitivity is by direct differentiation of the propag-ator matrices. This method is conceptually very simple and with some modifications it is possible to compute the sensitivity of the F and A potential in about the same compu-tation time as that of the adjoint method. In this section I will outline the procedure for both A potential and F potential. The goal is to compute the sensitivity of the potential at the surface of the earth. The A potential at a height h above the air^earth interface is given by (Appendix-A) = -ids d r r (M+h) 5jie«.<.-fc)\ M ^ l d X (2 41) z 4TT dxj0 \ P ° ; A V ; where the ratio P 2 0 i /A° i =  rTM is the T M reflection coefficient. In the above integral (eqn 2.41) the coefficient is only function of layer conductivities (cr,) and thicknesses (hj). So for a discretized ID earth the sensitivity for the A potential can be obtained by differentiating eqn 2.41 with respect to layer conductivity (say) ak. This gives Chapter 2. ID CSAMT inversion 3 4 dAZ = [°° / P ° [dPSJdau] - i » mjdau] ( t _ M \ J 0 (A r ) 6vfc 4TT 6 W 0 v ( P I 3 ! ) 2 J A ^ • ; T h u s i t is requ i red to evaluate the derivat ives of the coefficients and P2\. R e c a l l tha t the p roduc t of propagator mat r ices is g iven by P ° = ( P l 1 p ° 2 | = r r r ^ . . . r ^ . . . r ^ _ 1 r ^ ( 2 . 4 3 ) DO pO r21  r22 T h e propaga to r mat r ices w h i c h con ta in the terms i n v o l v i n g crk are Tk and Tk+1. Differ-en t i a t ing the m a t r i x P w i t h respect to ak for k = 1, M — 1 y ie lds ^ p O _ / dP°Jd<Tk dP?2/d*k Q a k \ dP?Jd*k 8P°2/dak (2.44) f)VA f)TA TIAT^A TIA U L k R A R A R A , R A R A T~A " x fe+1 R A V A TIA 1 1 1 2 • • • 1 fc-l"a 1 fc+1 • ' ' 1 M - l 1 AT "T 1 1 1 2 • • • 1 fc o_ 1 fc+2 • • • 1 A f - 1 1 M F o r the b o t t o m half-space c o n d u c t i v i t y (<T M ) , f £ = i ^ . . r (*.«) a < J M C O M T o solve eqn (2.44) i t is requ i red to evaluate dlkjdcrk a n d dTk+1/dcrk. D i f fe ren t i a t ion of Tk m a t r i x w i t h respect to ak y ie lds P- = ( ai ai I <*•«) Chapter 2. ID CSAMT inversion 3 5 wh ere. « 2 « 3 04 = — 2uk^X(j\ 2uk_1<Tk 4« f eUfc_ 1Cr f e iu/fioo-k-i 2 4 t t f c « f c _ 1 cr f e Uk(Tk-l iuJU.CTk-1 UkO-k-i iu>po-k-\ K2uk-i<rl Aukuk-Xo-k and differentiating Tfc+i with respect to <7fe yields, cVfc -2u t_i/v f c_i a-2ufc_ 1fek_i where, /3i = 2ttfe(Tfc+i iinpo-kuk+i tojficrkuk+1 iwpakuk+1 = - ( _ ^ ± i V2tt f c<7 f c + 1 4ii |o- f c + 1 1 2ttfe(Tfc+i 2 ( U k + 1 _ ivpykUk+i \2uko-k+i 4w|cr f e + 1 2M f e«j f c + 1 2 ioofihk -2ukhk -2ukhk (2.47) (2.48) (2.49) To compute the sensitivity for any layer conductivity <jfc it is required to evaluate the two series given in eqn (2.44). This means that for each layer two forward modelings are required to evaluate the sensitivity. And if there are M layers in the model then 2M forward modelings would be necessary. However this can be avoided by using a forward and a backward propagation of the solution and storing the successive products of the propagator matrices. The forward propagation in which the product of these propagator Chapter 2. ID CSAMT inversion 36 matr ices are s tored can be expressed as FP, = I \ FP2 = i \ r 2 FPz = Ti T2 T$ I (2.50) FP^Ti r 2 r 3 . . . r , FPM = T1 r 2 r 3 r M where FPj denotes the p roduc t of the propagator mat r ices u p t o j t h layer f r o m the top . A n d the b a c k w a r d p ropaga t ion i n w h i c h the p roduc t of the p ropaga tor mat r i ces can be s tored is g iven by TM = BPi T M - I T M = BP2 TM-2 T M - I T M = BP3 (2.51) r , TM-2 T M - I T M — BPj f i r 2 r 3 TM = BPJ M where BPj is the p roduc t of the propagator mat r ices u p t o the j t h layer f r o m b o t t o m half-space. Therefore eqn(2.44) for any kth layer for k = 1 , . . . , M — 2 can be expressed as = F P k ^ B P k + 1 + FPk°^BPk+2 (2.52) Chapter 2. ID CSAMT inversion 37 a n d the sens i t iv i ty for the M — 1 and M layers are g iven by 0P° _ F p 9TAM_ dVAM •5 a < r M U ( 7 M T h u s for M layers on ly two forward model ings are requ i red to c o m p u t e the sens i t iv i ty . T h e two fo rward model ings are the fo rward and b a c k w a r d p ropaga t ion g iven i n eqn (2.50) and (2.51). In a s imi l a r manne r the sens i t iv i ty for the F po t en t i a l c an be c o m p u t e d b y differ-en t i a t i ng the propaga tor matr ices . T h e F po t en t i a l at a height h above the a i r -ear th interface is g iven b y ( A p p e n d i x - A ) 4TT dy JO \ Q11 J * Diffe ren t ia t ing eqn (2.54) w i t h respect to layer c o n d u c t i v i t y <rk gives, dF? = -iuplds 6 r / < & [dQQJdau] - Q°21 [dQlJdau] (a_h)\ J o ( A r ) j A dcrk 4TT dy J0 V [Q^f J A Therefore to evaluate the in tegra l i n (2.55) i t is requ i red to c o m p u t e dQ,°/dak. T h i s is g iven b y < 9 Q ° / dQ°Jdak dQ°12/d*k 9<Tk \ dQ°2Jdcrk dQ°22/d<rk (2.56) r F r F TI dVkrp r p r F 1 VFVF r< ^ k + 1 ~T F V F V F 1 1 1 2 • • • 1 'Q^k L fc+l • • • 1 M - l 1 M ' 1 2 • • • 1 fe 1 fc+2 • • • 1 M-l 1 M T o solve eqn (2.56) i t is requ i red to evaluate dTF/dcrk a n d 8TE+1/dcrk. D i f fe ren t ia t ion of T f m a t r i x w i t h respect to crk y ie lds Chapter 2. ID CSAMT inversion 38 w h e r e , Ci = ILOfl AUkUk-! —ius/j, AUkUk-l 1 < j J ^ c - 2 m _ i f c t - i a n d d i f f e r e n t i a t i n g T f + 1 w i t h r e s p e c t t o cr f e y i e l d s w h e r e , 04 = dcr f c — ICUflUk+l v 4 4 ; / iujfiuk+1 V 4 ^ + 2it f e iujfih iLouh k  ^ e-2ukhk k \ e~2ukhk (2.57) (2.58) (2.59) (2.60) T h u s t o c o m p u t e t h e s e n s i t i v i t y o f t h e F p o t e n t i a l w i t h r e s p e c t t o c o n d u c t i v i t y s i m i l a r m e t h o d t h a t i s s h o w n f o r A p o t e n t i a l i s u s e d . S i n c e t h e r e a r e n o p o s i t i v e e x p o n e n t i a l i n v o l v e d i n e v a l u a t i n g t h e s e n s i t i v i t y , i t i s c o m p u t a t i o n a l l y s t a b l e . S i m i l a r t o a d j o i n t m e t h o d t h e H a n k e l t r a n s f o r m f o r t h e s e n s i t i v i t y i s c a r r i e d o u t f o r e a c h l a y e r s e p a r a t e l y . H a v i n g o b t a i n e d t h e s e n s i t i v i t y f o r t h e p o t e n t i a l s , t h e s e n s i t i v i t y f o r t h e e l e c t r i c a n d Chapter 2. ID CSAMT inversion 39 magne t i c fields can be cons t ruc ted by different iat ing eqn(2.11) a n d (2.12) w i t h respect to the c o n d u c t i v i t y cr*.. A n d sens i t iv i ty for the apparent res i s t iv i ty a n d phase d a t a can be ob t a ined by eqn (2.40). 2.5 S y n t h e t i c E x a m p l e T o test the invers ion a l g o r i t h m , syn the t ic C S A M T da t a are genera ted f r o m a 5 layer c o n d u c t i v i t y m o d e l shown i n F i g u r e 2.2(b). T h e t r ansmi t t e r a n d receiver dipoles are shown i n F i g u r e 2.2(a) and apparent res i s t iv i ty and phase d a t a were c o m p u t e d at 14 l o g a r i t h m i c a l l y spaced frequencies (1-8192 H z ) y i e ld ing 28 d a t a po in ts . T h e apparent r e s i s t iv i ty d a t a were con t amina t ed w i t h Gauss i an noise h a v i n g a s t a n d a r d d e v i a t i o n of 5% of the d a t u m value (apparent res i s t iv i ty ) and 2° for the phase da ta . These error levels can be achieved w i t h m o d e r n ins t ruments tha t measures the e lectr ic a n d magne t i c fields. T h e m o d e l ob jec t ive func t ion is , w h i c h is d i sc re t i zed as a layered m o d e l i n F i g u r e 2.1. T h e in t eg ra l equa t ion i n (2.61) is t a k e n w i t h respect to ln(z). T h i s is a n a t u r a l choice for e lec t romagne t ic diffusive p rob lems i n w h i c h the frequencies for the da t a are l o g a r i t h m i c a l l y spaced. T h e above eqn(2.61) is same as eqn(2.16) w i t h ws(z) = 1/z, wz(z) = z and m — ln(cr(z)) . D i s c r e t i z i n g the first t e r m i n eqn(2.61) results i n a M x M we igh t ing m a t r i x for the smallest m o d e l s h o w n i n A p p e n d i x - B . T h i s is g iven by where Z{ is the dep th to the ith interface and hi is the thickness of ith layer . z0 = ( * hx w i t h 0 < ( < 1. D i s c r e t i z i n g the second t e r m i n eqn(2.61) results i n (M — 1) x M 4>m = a (2.62) Chapter 2. ID CSAMT inversion 4 0 w e i g h t i n g m a t r i x f o r t h e f l a t t e s t m o d e l s h o w n i n A p p e n d i x - B . T h i s i s g i v e n b y / _ 7 l 7 l ... o \ Wz = (2.63) \ 0 0 - 7 M - I 7 M - I / w h e r e *jj = y/(zj + Zj_i)/(hj+ / i j + i ) . T h e c h o i c e o f z0 i s t h e s a m e f o r t h e s m a l l e s t m o d e l . T h e e a r t h w a s p a r a m e t e r i z e d i n t o 50 l a y e r s w i t h t h e l a y e r t h i c k n e s s i n c r e a s i n g ex-p o n e n t i a l l y w i t h d e p t h , s i n c e t h e f r e q u e n c i e s w h i c h c o r r e s p o n d t o t h e s k i n d e p t h s a r e l o g a r i t h m i c a l l y s p a c e d . T h e r e f e r e n c e m o d e l i s c h o s e n t o b e 7 x 1 0 - 4 S / m w h i c h i s t h e s a m e as t h e t r u e b a c k g r o u n d c o n d u c t i v i t y . T h e t a r g e t m i s f i t a t e a c h i t e r a t i o n w a s h a l v e d ( i . e . C — 0-5) a s t h e i t e r a t i v e p r o c e s s c o n t i n u e d . I c a r r y o u t t h e i n v e r s i o n s u s i n g t w o d i f f e r e n t o b j e c t i v e f u n c t i o n s ( a ) cts = 1.0, az = 0 ( s m a l l e s t ) a n d ( b ) as = 0, az = 1.0 ( f l a t t e s t ) . T h e f i t t o t h e o b s e r v e d a p p a r e n t r e s i s t i v i t y a n d p h a s e f o r b o t h s m a l l e s t a n d f l a t t e s t m o d e l s a r e s h o w n i n F i g u r e 2.3(b) a n d ( c ) . T h e f i n a l m i s f i t f o r t h e s m a l l e s t m o d e l a n d f l a t t e s t m o d e l w e r e 27.9 a n d 28.0 r e s p e c t i v e l y . T h e s m a l l e s t m o d e l s h o w n i n F i g u r e 2.3(a) d e l i n e a t e s t h e b o u n d a r i e s o f t h e c o n d u c t i v e b l o c k b u t i t e x h i b i t s m o r e s t r u c t u r e t h a n t h e f l a t t e s t m o d e l w h i c h i s s m o o t h v e r s i o n o f t h e t r u e m o d e l . T h e d i f f e r e n c e i n t h e t w o m o d e l s i s s e e n i n t h e r e c o v e r y o f t h e b a c k g r o u n d c o n d u c t i v i t y a t l a r g e r d e p t h . T h i s o c c u r s b e c a u s e t h e r e f e r e n c e c o n d u c t i v i t y i s c h o s e n t o b e t h e t r u e b a c k g r o u n d c o n -d u c t i v i t y . T h e f l a t t e s t m o d e l p e n a l t y t e r m d o e s n o t h a v e a n y i n f o r m a t i o n a b o u t t h e r e f e r e n c e m o d e l a n d t h e r e f o r e t h e m o d e l flattens o u t a t d e p t h s w h i c h a r e n o t s e n s e d b y d a t a . I n s o m e c a s e s w h e n t h e b a c k g r o u n d c o n d u c t i v i t y i s k n o w n a p r i o r i i.e, c r r e / , i t i s o f t e n d e s i r a b l e t o h a v e a s m a l l e s t m o d e l c o m p o n e n t a d d e d t o t h e flattest m o d e l n o r m . T h i s e n s u r e s t h a t a t g r e a t e r d e p t h t h e m o d e l r e t u r n s t o t h e r e f e r e n c e c o n d u c t i v i t y w h e n n o l o n g e r i n f l u e n c e d b y d a t a . T h e n e x t o b j e c t i v e w a s t o i n v e r t o n l y o n e s e t o f i n f o r m a t i o n i.e. e i t h e r a p p a r e n t Chapter 2. ID CSAMT inversion 41 (a) 10° 101 10 2 10 3 10° 101 10 2 10 3 F r e q u e n c y ( H z ) F r e q u e n c y ( H z ) Figure 2.3: (a) Inversion of apparent resistivity and phase data. The block model is the true model. The flattest and the smallest model are shown by solid and dotted line respectively, (b) The apparent resistivity data with 5% Gaussian random noise. The predicted data from the smallest and flattest model are shown by continuous and dashed curves respectively and cannot be distinguished, (c) The phase data with Gaussian random noise having 2° standard deviation. The predicted data from the smallest and flattest model are shown by continuous curves and cannot be distinguished. Chapter 2. ID CSAMT inversion 42 (c) (d) i 1 1 1 i 1 1 1— 10° 101 10 2 103 10° 101 10 2 10 3 Frequency (Hz) Frequency (Hz) Figure 2.4: (a) The recovered model from the inversion of apparent resistivity data. The blocky model is the true model. The flattest and smallest models are shown by solid and dashed lines respectively, (b) The recovered model from the inversion of phase data. The flattest and smallest models are shown by solid and dashed line respectively, (c) The fit to the apparent resistivity and (d) phase data from smallest and flattest models are indicated by continuous curves that are not distinguishable Chapter 2. ID CSAMT inversion 43 resistivity or phase. I n practice this is sometimes necessary because one of these data sets does not exist, or one is contaminated with severe noise and must be discarded. I first invert only the apparent resistivity data contaminated with noise. T h e desired misfit in this case is 14. T h e constructed smallest and flattest models are shown in F i g u r e 2.4(a) and the fit to the data is shown in F i g u r e 2.4(c). T h e final misfit for the smallest and the flattest model is 13.9 and 14.0 respectively. T h e flattest model has slightly less structure than that in F i g u r e 2.3(a) but the smallest model has considerably more structure. I nve r t i ng only phase data produces the models in F i g u r e 2.4(b) and similar conclusions hold. T h e fit to the data is shown in F i g u r e 2.4(d). I n the inversion with the flattest model objective function where aa = 0 there is no influence of the reference Conductivity on the inversion. However it is still possible to recover the conductivity. T h i s arises because the phase data obtained in near-field and transition zone change with the frequency and conductivity. T h e results in F i g u r e 2.4 show that conductivity information can be obtained from either amplitude or phase information but having both data sets improves the inversion result. T h e degree of improvement depends on the data errors and is also problem dependent. In all three cases the flattest model recovers a smoother version of the true model whereas the smallest model shows more structure. A flattest model may be more appropriate in cases where reference conductivity information is not known a priori. 2.6 M T inversion of Near-Field Corrected Data It is a common practice to invert C S A M T data by first removing the near-field effects and then inverting the data using an M T inversion code. I carry out this procedure to illustrate the advantage of working with fully modelled C S A M T data. I n this sec-tion I present the inversion of near-field corrected data using the scheme outlined by Chapter 2. ID CSAMT inversion 44 Bartel and Jacobson(1987) (hereinafter called BJ). Following B J , two half-space conduc-tivities <Ti and <T2 are chosen such that at frequency / the apparent resistivity of the C S A M T response is bounded between the two homogeneous half-space responses, that is, PaiCSAMT(f) < PamCSAMT(f) < pa2CSAMT(f) The next step is to use equation (3) from B J to compute the near-field corrected apparent resistivity given by w (nNFc(f)) _ [^ g (PZAMT/PCJAMT) log (P%T/P%T)} g [ P W ) ^g(P^MT/pCSAMT) ( 2 6 4 ) + logoff) where the subscript ' N F C denotes the near-field corrected value of apparent resistivity. Unfortunately there is no correction for phase data. This occurs for two reasons. First, the phase at low frequency for C S A M T tends to be small because the notion of frequency sounding breaks down and apparent resistivity depends on the geometry. Secondly the half-space technique which works for correcting the apparent resistivity is not applicable since the phase of the M T response for a homogeneous half-space is 7r/4 irrespective of the frequency or conductivity. The important question to be asked in most near-field corrections is which data points belong to the near-field and which belong to the transition zone? This is often difficult to judge since the relative contributions of the various factors are unknown. However from theoretical modeling and characteristic nature of the curves ( like the transition zone notch), one can select the frequencies which seem to be affected by the non-plane wave nature of the source. The apparent resistivity data generated by the model in the last section shows a transition zone notch around 100 Hz. Therefore the data obtained for frequencies lower than 128Hz were corrected using eqn (2.64). The near-field corrected apparent resistivity, the C S A M T data and the computed M T response for the true model are shown in Figure 2.5(b). The data for inversion consist of 14 apparent resistivities including the 10 near-field corrected data, and 4 phase data in the far-field region. The Chapter 2. ID CSAMT inversion 45 errors in apparent resistivity data are assumed to be 5% of the datum value and 2° for the phase. The data are inverted using an M T algorithm with a target %2 of 18. The resulting model is shown in Figure 2.5(a). Next the C S A M T inversion was carried out with 14 apparent resistivity data and 14 phase data. The resulting model for the flattest model penalty is superimposed on the M T inversion results for comparison. Although the model objective function for the two inversion is identical there is an erroneous higher conductive feature observed at greater depth in the near-field corrected inversion. This is probably due to under-corrections of the data in the transition zone. In a more extreme example I consider a conductive overburden model shown in Figure 2.5(c). I keep the same geometry of receiver and transmitter as that of previous case. The C S A M T response, the M T response and near-field corrected response are shown in Figure 2.5(d). Unlike the previous case (Figure 2.5(b) ) the response is mostly obtained in the near-field zone and perhaps in the transition zone. This is probably due to a more resistive basement than in the previous example. The inverted models obtained by minimizing a flattest model objective function are shown in Figure 2.5(c). The deeper conductive layer is totally absent in the near-field corrected inversion whereas it is clearly visible in the C S A M T inversion. 2.7 I n v e r s i o n o f e l e c t r i c field d a t a a c q u i r e d i n C S A M T s u r v e y s In M T the earth is excited by the natural fields, so the nature of the source is not known in general. To circumvent this problem the data are considered to be the ratio of electric and magnetic field. Traditional practice is to consider the data as the apparent resistivity and phase given by eqn's (2.13) and (2.14), which are nonlinear transform of the fields. The same practice is also followed to represent the C S A M T data, since the data interpreting tools for M T are routinely applied to C S A M T data, with the assumption that C S A M T Chapter 2. ID CSAMT inversion 4 6 (a) (b) Depth (m) Frequency (Hz) Figure 2.5: (a) The recovered model from the C S A M T (solid line) and M T inversion of near-field corrected data (dashed line). The block model is the true model, (b) The computed M T apparent resistivity for the true model in Figure 2.5(a) is shown as a solid line. The C S A M T responses are shown with error bars and the near-field corrected data by dashed lines, (c) The C S A M T inversion (solid line), the M T inversion of near-field corrected data (dashed line) and the true model (block solid line) for the second example, (d) The M T data for the true model in Figure 2.5(c) is shown by a solid line. The C S A M T responses are shown with error bars and the dashed line indicates the near-field corrected data. Chapter 2. ID CSAMT inversion 47 data are corrected for the near-field effects. The effect of such corrections were discussed in the previous section. In practice however the nature and geometry of the C S A M T source is available. Commonly used source is a finite length grounded electric dipole. Generally electric fields are recorded at each receiver location and magnetic field data are recorded at larger intervals. This is because the magnetic field does not vary significantly in close intervals. So I consider the electric field data and invert it to obtain a I D conductivity structure of the earth. The data are considered to be the amplitude and phase of the electric field (Ex) represented by d = ( |^ (^ i ) l , • • • , \E*("N)\, < ^ > i ) , • • • , <f>-E,Mf (2-65) The formulation for the inverse problem is similar to that in section (2.3). The sensitivity for the amplitude and phase of the electric field can be easily obtained from dEx(u>)/daj. This is given by d ^ - \ E x H \ R e J 1 0E-M da, 1 - " \E.(u) da, , 6 M U ) Imag^ 1 d E ' W da, \Ex(u) da, I consider the same example given in Figure 2.3(a) and generate the electric field data (Ex(u>)) at 14 discrete frequencies (0.5-4096 Hz) yielding 28 data points. The amplitude data were contaminated with Gaussian noise having a standard deviation of 5% of the datum value and 2° for the phase data shown by error bars in Figure 2.6(b) and (c) respectively. I carry out the inversion using a flattest model objective function with a3 = 0.0001 and az = 1.0. The recovered model shown in Figure 2.6(a) indicates a smooth representation of the true model. The fit to amplitude and phase data is shown in Figure 2.6(b) and (c) and the final misfit from the inversion is equal to 27.98. In this example I have shown that in principle the electric field data from a C S A M T survey can Chapter 2. ID CSAMT inversion 4 8 10-4 H 10": 10° 10C 101 102 103 Depth (m) (b) (c) 101 102 103 Frequency (Hz) 101 102 103 Frequency (Hz) Figure 2.6: (a) The recovered model from the C S A M T inversion of electric field data. The blocky model is the true model. The model from C S A M T inversion is shown by solid lines, (b) The fit to the amplitude data of the electric field, (c) The fit to the phase of the electric field. Chapter 2. ID CSAMT inversion 49 be inverted to obtain the conductivity structure and it is not necessary to transform the data to apparent resistivities. 2.8 Field Examples A goal of this research is to develop an algorithm which produces an interpretable elec-trical conductivity model when applied to field data. In practice, geophysical surveys are likely affected by three dimensional conductivity. Therefore the validity of using ID algorithm to invert 3D data can be in question, but this is not the issue I am trying to address in this work. Wannamaker (1997b) and Boschetto and Hohmann (1991) have examined the discrepancy between the ID and 3D modeling results with C S A M T . My goal here is to present a robust inversion algorithm that can be applied to field data when the earth is approximately ID. I make this assumption here and present two examples of inverting field data and compare the results obtained from C S A M T inversion with the near-field corrected M T inversion obtained from a contractor. The first example is a data set obtained in a resistive environment and second data set is collected in a conductive environment. 2.8.1 Example from a resistive environment The field data were acquired during a mineral exploration survey. A transmitter of length 3700ft was placed 15700ft from the sounding location 4100S. Data at 14 frequencies (0.5-4096)Hz were collected at 56 locations along the profile and the station separation was 200ft. The profile was parallel to within 5 degrees of the transmitting dipole. The high frequency data at 4096Hz were affected by instrument saturation and those data were not considered for the inversion. Therefore 26 data were inverted at each location for ID conductivity structure. The observed apparent resistivity is shown in Figure 2.7(a) and Chapter 2. ID CSAMT inversion 50 the observed phases in Figure 2.7(c). Near field effects, which cause resistivity to increase and phase to decrease with decreasing frequency, dominate the data images. Therefore it is necessary to invert the data to obtain the conductivity information of the subsurface. Regional geology suggests the subsurface is primarily composed of silica rich quartzites and a likely background conductivity is about 10"4S/m. I use a half-space of this value as a reference model and choose a model objective function that is a combination of flattest and smallest component ( as = 0.001 and az = 1 ). Error assignment is problematic since the region does not conform to the ID assumption. First an error estimates of 15% were assigned to the apparent resistivities and 3° to the phases. The target misfit for the inversion was set to 26. The approach was as follows. At some stations the algorithm was unable to achieve this misfit. So the inversion was run with a revised target misfit of X2 = 78 which could easily be reached at all stations. This is approximately equivalent to assigning an error of 25% to the apparent resistivity data and 5° to the phase and then inverting the data to the expected chisquare value of 26. The fit to the data for apparent resistivity is shown Figure 2.7(b) and for phase in Figure 2.7(d). In general it was observed that the fit to the apparent resistivity data in Figure 2.7(b) was better than that of phase data presented in Figure 2.7(d). The resulting model is shown in Fig 2.8(a). South is indicated by the negative station numbers and the north by positive. The model shown in Figure 2.8(a) suggests that there is a resistive layer on the surface, a major conductor near 3100S and a layer of high conductivity between station 4400S and 10000S at about 500 ft depth. The anomaly at 3100S appears to be closed shown in Figure 2.8(a). Next I compare the C S A M T results with those obtained by inverting data with an M T inversion code after the data have been corrected for near-field effects. The near-field correction is necessary because highly resistive environment made most of the data to he in the near-field and in the transition zone. This is exhibited by the linear decrease in the Chapter 2. ID CSAMT inversion 51 Observed Apparent Resistivity (a) Predicted Apparent Resistivity (b) -8000 -4000 0 -8000 -4000 Distance (ft) Distance (ft) log 1 0 (p ) ohm-m 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 Observed Phase (c) Predicted Phase (d) I I I I I I -8000 -4000 0 -8000 -4000 0 Distance (ft) Distance (ft) <|> (degree) -2.0 3.7 9.4 15.1 20.8 26.5 32.2 37.9 43.6 49.3 55.0 F i g u r e 2.7: (a) T h e observed apparent res i s t iv i ty da t a for f ield d a t a co l lec ted i n a resist ive env i ronment , (b) T h e p red ic ted apparent res i s t iv i ty d a t a f rom C S A M T invers ion , (c) T h e observed phase data , (d) T h e p red ic t ed phase da t a f rom the C S A M T invers ion . Chapter 2. ID CSAMT inversion 52 apparent resistivity value with frequency (Figure 2.7(a)). The results shown in Figure 2.8(c) were supplied by a commercial contractor. The near-field corrected inversion shows conductive anomalies of equal amplitude beneath several stations on both the north and south sides of the profile. These anomalies extend to the surface, and they are not seen in the C S A M T inversion. Instead in Figure 2.8(b) there is a thin resistive cover on the top of conductive anomalies. A major difference between the two inversions is the strength and size of the conductive anomaly at 3100S. This conductor is dominant feature in the C S A M T inversion. Unfortunately ground-truth is not available and I cannot conclude that Figure 2.8(b) is a better representation of the earth than Figure 2.8(c). Sometimes due to the small conductive inhomogeneities near the surface the currents gets channelled into these conductors resulting in an overall amplitude shift in apparent resistivities at all frequencies. This is called static shift. Recognizing this the vertical elongated conductivity structure may be result of static shift which are prevalent in magnetotellurics (Jones, 1988). So before leaving this example I carry out, independent inversions of apparent resistivity and phase. Previous studies (Jones, 1988) indicate that static-shift affects only the apparent resistivity curve and not the phase curve. Therefore inverting two data sets independently would indicate the effect due to static-shift on the recovered model. The resulting model from apparent resistivity inversion ( shown in Figure 2.9(a) ) indicates similar conductivity structure shown in Figure 2.8(b), but the amplitude of the conductive anomaly is reduced. Inversion of phase-only data shown in Figure 2.9(b) also indicates the presence of the conductive body near station 3100S. The consistency from the inversions therefore adds confidence that the observed features are demanded by the data if the earth is adequately approximated as ID. The broad structural features are prominent in both phase inversion (Figure 2.9b) and resistivity-phase inversion(Figure 2.8b). Chapter 2. ID CSAMT inversion 53 (a) C S A M T Invers ion (Greater Depth Extent) (c) MT i n v e r s i o n o f near - f ie ld c o r r e c t e d data -10000 -8000 -6000 -4000 Distance (ft) 2 0 0 0 -3.9 -3.7 -3.5 -3.3 -3.1 -2.9 -2.7 -2.5 -2.3 -2.1 i o g 1 0 ( o ) S/m Figure 2.8: (a) Conductivity model obtained from simultaneous inversion of ap-parent resistivity and phase data for a depth range of 9000 ft. Zero on the depth axis indicates the mean sea level, (b) The recovered model shown in Figure 2.8(a) replotted with a smaller depth scale to compare with the near-field corrected MT inversion, (c) Recovered conductivity structure from inversion of near-field corrected data obtained by commercial processing. Chapter 2. ID CSAMT i n v e r s i o n 54 -10000 -8000 -6000 -4000 -2000 0 Distance (ft) iog10 (a) -4.1 -3.9 -3.7 -3.5 -3.3 -3.1 -2.9 -2.7 -2.5 -2.3 -2.1 Figure 2.9: (a) Conductivity model obtained from independent inversion of apparent resistivity data, (b) Conductivity model obtained from independent inversion of phase data. Chapter 2. ID CSAMT inversion 55 2.8.2 Example from conductive environment In C S A M T , the data at low frequencies that contain the deeper information are often associated with the near-field effects. Therefore the goal of this example is to investigate the usefulness of C S A M T inversion in a conductive environment when information about the deeper subsurface is desired. The second data set was acquired during a mineral exploration survey in Nevada, USA. The observed apparent resistivity and phase data are presented in Figure 2.10(a) and Figure 2.10(b) respectively. The data were collected using a transmitter of length 7450 ft placed 14800 ft from the receiving dipoles. The receiving dipoles are separated by 100 ft. along the survey line. Data at 13 frequencies (0.5 Hz - 2048 Hz) were acquired at 60 locations. Near-field effects, which cause the resistivity to increase and phase to decrease with decreasing frequency are observed in the low frequency regime shown in Figure 2.10(a) and (c). To invert the data 10% error were assigned to the apparent resistivities and 5° error on the phases. The target misfit was set to 26 which is equal to the number of data. In this example the desired target misfit of 26 was achieved for all the stations. The model objective function chosen for inverting the data is a combination of flattest and smallest component (as = 0.0001 and az = 1). The conductivity model obtained from inversion is shown in Fig. 2.11(a). The recovered conductivity model indicates a resistive block near x=2800 ft. The regional structure indicates a near surface conducting layer with a resistive basement. The shallow depth information is mostly obtained from the high frequency data, which lies in the far-field region. The deeper information is obtained from the near-field data collected at low frequencies. Figures 2.10(b) and (d) indicate a good match between the predicted and the observed data. Next the C S A M T inversion is compared with the M T inversion of near-field corrected. Chapter 2. ID CSAMT inversion (a) Observed Apparent Resistivity (b) Predicted Apparent Resistivity 1000 2000 3000 4000 5000 1000 2000 3000 4000 5000 Distance (ft) Distance (ft) l og 1 0 (p ) ohm-m 1.00 1.18 1.36 1.54 1.72 1.90 2.08 2.26 2.44 2.62 2.80 F i g u r e 2.10: (a) T h e observed apparent res i s t iv i ty d a t a for f ield da t a col lec ted i n a conduc t ive env i ronment , (b) T h e p red ic t ed apparent res i s t iv i ty d a t a f rom C S A M T invers ion , (c) T h e observed phase data , (d) T h e p red ic t ed phase da t a f rom the C S A M T invers ion . Chapter 2. ID CSAMT inversion 57 da ta . F o r c o m p a r i s o n the top 4000 ft of recovered m o d e l ob t a ined f r o m C S A M T inver -s ion is cons idered and i t is compared w i t h the near-field cor rec ted inve r s ion shown i n F i g u r e 2.11(c). T h e c o n d u c t i v i t y s t ruc ture i n the upper par t of the m o d e l ind ica tes shal-low conduc t ive layer w h i c h agrees we l l w i t h tha t ob ta ined f rom C S A M T inve r s ion ( F i g . 2.11(b)) . Howeve r there are several differences i n the deeper s t ruc tures as w e l l as i n the recovered ampl i t udes . T h e a m p l i t u d e of the resist ive a n o m a l y ob t a ined at x = 2 8 0 0 ft f r o m C S A M T invers ion is higher t h a n that ob ta ined f rom the near-f ield cor rec ted da ta . T h e C S A M T invers ion shown i n F i g u r e 2.11(b) indica tes a gent ly d i p p i n g resis t ive basement where as the near-f ield correc ted da t a shows more s t ruc ture i n the deeper sec t ion . I n general there is a reasonable agreement be tween the two invers ions ( F i g u r e 2.11(b) a n d 2.11(c)) at shal lower depths . T h i s is because the da t a ob ta ined at h igher frequencies he i n the far-f ield zone where the plane wave a p p r o x i m a t i o n is va l i d . Howeve r the agreement worsens at d e p t h w h i c h is p robab ly due to the near-field correct ions app l i ed to the low frequency da ta . Some qua l i t a t ive i n f o r m a t i o n about the c o n d u c t i v i t y can be ob t a ined f r o m dr i l lho le logs B H 1 a n d B H 2 shown i n F i g u r e 2.12(b) and (c) respect ively. T h e loca t ions of the d r i l lho le logs o n the i nve r t ed m o d e l are presented i n F i g u r e 2.12(a). B H 1 penetra tes the resis t ive b lock a n d B H 2 intersects the surface conduc t ive layer . T h e B H 1 log shown i n F i g u r e 2.12(b) indica tes tha t the top a l l u v i u m layer is u n d e r l a i n b y more res is t ive s i l ts tone and sandstone uni t s . T h e rock type where the resis t ive b lock is encounte red , is p r e d o m i n a n t l y s i l ts tone w i t h fine gra ined sandstone. A few qua r t z veins w i t h s i l ic i f ied s i l ts tone are also found i n this region and this is consistent w i t h the h i g h r e s i s t iv i ty o b t a i n e d f rom the invers ion . T h e dr i l lho le i n f o r m a t i o n i n B H 2 ind ica tes tha t the top layer is a l l u v i u m fol lowed by c lay and si l ts tone. In the d e p t h range of 470-710 ft there is c lay r i c h layer w i t h occurrences of s i l ts tone and i r o n oxide w h i c h is r e l a t ive ly conduc t i ve . A l t h o u g h there are no geophys ica l logs avai lable i n this region the h i g h resis t ive b lock Chapter 2. ID CSAMT inversion 58 a n d the sha l low conduc t ive layer ob ta ined i n our invers ion are a qua l i t a t i ve i n d i c a t o r of the rock t y p e assemblage found i n the two dr i l lho le logs B H 1 a n d B H 2 . B o t h the examples i l lus t ra te the need to inver t C S A M T da t a w i t h o u t any cor rec t ion . Inappropr i a t e cor rec t ion of the near-field and t r ans i t i on zone d a t a can manifest ar t i facts i n the inve r s ion results . M o r e o v e r the correct ions are on ly app l i ed to r es i s t iv i ty da t a , and , phase d a t a are d i sca rded wh i l e c a r r y i n g out M T invers ion . In C S A M T inve r s ion b o t h apparent r es i s t iv i ty a n d phase da t a are u t i l i z e d and do not suffer f r o m any a m b i g u i t y of r educ ing the d a t a to plane wave a p p r o x i m a t i o n . 2.9 C o n c l u s i o n s T h e r e are two reason for not in t e rp re t ing "correc ted" C S A M T d a t a w i t h a M T inve r s ion a l g o r i t h m : ( 1 ) i nappropr i a t e cor rec t ion of the near-field and t r a n s i t i o n zone d a t a can generate ar t i facts i n the invers ion results , ( 2 ) the correct ions are o n l y app l i ed to r e s i s t iv i ty d a t a a n d the phase da ta , w h i c h cannot be correc ted , must be d i sca rded wh i l e c a r r y i n g out M T invers ion . In this work I have developed an inver s ion a l g o r i t h m to recover a I D c o n d u c t i v i t y s t ruc ture f rom C S A M T da t a w i thou t a p p l y i n g any co r rec t ion p r io r to inve r s ion . T h e fo rward m o d e l i n g is ca r r ied out i n the frequency d o m a i n . T h e sensi t iv i t ies for the fields are de r ived us ing two approaches (a) adjoint Green ' s f o r m u l a t i o n a n d (b) dif ferent ia t ing the propagator mat r ices . Fo r this work the adjoint equat ions are used to c o m p u t e the sens i t iv i ty . T h e invers ion m e t h o d finds a pa r t i cu l a r m o d e l by m i n i m i z i n g a m o d e l ob jec t ive func t ion subject to adequate ly f i t t ing the da ta . He re I confine m y s e l f to the mode ls tha t are s m o o t h ve r t i ca l ly and are also close to a reference m o d e l . W i t h th is m e t h o d the res i s t iv i ty and phase da t a can be inve r t ed j o i n t l y or separately. Resu l t s f r o m the C S A M T invers ion are compared w i t h those ob ta ined by a p p l y i n g an M T inve r s ion a l g o r i t h m to near-f ield correc ted da ta . Signif icant differences i n the c o n d u c t i v i t y mode ls Chapter 2. ID CSAMT inversion 5 9 are ob t a ined i n b o t h syn the t i c and field da t a examples and these differences i l lu s t r a t e the need to inver t C S A M T da t a w i thou t any cor rec t ion . Chapter 2. ID CSAMT inversion (a) CSAMT Inversion ( Greater Depth Extent) c/> E CD > o CO CL CD Q 3000 -3000 -6000 -9000 (b) CSAMT Inversion 4500 4000 3500 3000 2500 co E CD > o -O « , 2000 4^ 1500 CD Q 1000 500 (c) MT inversion of near-field corrected data 1000 2000 3000 4000 5000 Distance (ft) I I -3.00 -2.75 -2.50 -2.25 -2.00 -1.75 -1.50 -1.25 -1.00 -0.75 -0.50 log 10(CT) S/m Figure 2.11: (a) Conduc t iv i ty model obtained from C S A M T inversion of the data collected i n conductive environment. Zero on the depth axis indicates the mean sea level, (b) The recovered model shown i n Figure 2.11(a) replotted w i t h a smaller depth scale for comparison, (c) Inverted mode l obtained from M T inversion of near-field corrected data obtained from commercial processing. Chapter 2. ID CSAMT inversion 61 C S A M T I n v e r s i o n ^ 4600 —. 4500 «? 4400 E 4300 0) 4200 O 4100 m 4000 3900 — 3800 S 3700 g" 3600 Q 3500 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 5500 Distance (ft) —mm^^m iog10(o) S/m -3.00 -2.75 -2.50 -2.25 -2.00 -1.75 -1.50 -1.25 -1.00 -0.75 -0.50 (b) (c) 0 240 275 g 385 4= 400 <D o 600 795 Alluvium Sand stone (sst) medium grained Silt + fine grained sst Silt + chert Silt + silicified silt + quartz veins + iron oxide 260 370 470 Silt + chert 710 795 Alluvium Clay + silt Sand stone + iron oxide Clay + siltstone iron oxide Silt + iron oxide BH1 LOG BH2 LOG Figure 2.12: (a) Conductivity model obtained from CSAMT inversion for top 1150 ft above the mean sea level. The drillhole logs are superimposed on the inverted section, (b) The rock type assemblage in BH1. The drillhole is 795 ft long and penetrates the resistive block, (c) The rock type assemblage in BH2. Chapter 3 Induced Polarisation and Complex Resistivity Method 3.1 Introduction In this chapter I review the IP signals to establish the connection between the responses measured in the field and the chargeability that is of interest. First I discuss the time-domain and frequency domain signals in IP and later introduce the Cole-Cole model which is commonly used to interpret the complex resistivity measurements (Zonge et al., 1975; Sumner, 1976; Pelton et a l , 1978a). 3.2 Time Domain IP Time domain IP consists of injecting currents into the ground, and analysing the transient discharge of the voltage after the current is shut off. Since this decay is observed in time, it is called time-domain IP. Typically in time domain equipment the current is a on-off pulse usually in the form of a square wave form shown in Figure 3.1(a). During the on-time the voltage slowly builds up by charging the polarizable material in the ground. Ideally if the ground was non-polarizable then the voltage curve would be flat measuring only Va which is due to the true conductivity of the medium. In the presence of chargeable material the voltage measured is V^. The extra voltage i.e Vv — Va is called the overvoltage effect. In essence the effective conductivity of the medium during the charging process is decreased from cr to cr(l — rj). After the current is switched off the charges decay resulting in the voltage discharge curve shown in Figure 3.1(b). Thus, during the charging process, 62 Chapter 3. IP and complex resistivity Method 63 the I P effect opposes the bu i l d -up of the po ten t i a l and d u r i n g the d i scharg ing process i t opposes the collapse of the po ten t i a l . Idea l ly i f the g round is chargeable, and i f effects due to E M coup l ing are absent, t hen the ra t io of decay vol tage i m m e d i a t e l y after the shut-off to the vol tage jus t before the shut-off is defined as chargeabi l i ty . T h i s can be w r i t t e n as Vs V«=y (3.1) where na is ca l led apparent chargeabi l i ty and Vs = V, — Va. I f we have a fo rward m o d e l i n g w h i c h can c o m p u t e the D C potent ia ls due to c o n d u c t i v i t y a w h i c h is g iven b y V . ( o - V I 4 ) = —I8(r — r3), t hen the same fo rward m o d e l i n g can be used to c o m p u t e VJ, w h i c h is due to the mod i f i ed c o n d u c t i v i t y <r(l — 77). T h e po t en t i a l Vv c an be d e t e r m i n e d b y so lv ing the equa t ion V . ( c r ( l — T/JVK,) = —I6(r — rs) and the apparent chargeab i l i ty d a t a can be generated us ing two fo rward model ings ( O l d e n b u r g a n d L i , 1994). T h i s is g iven by V A ~ FDC(<T(1-V)) ' 1 J where Tuc is the D C fo rward m o d e l i n g operator . E q u a t i o n (3.2) shows tha t the I P d a t a depends o n the i n t r i n s i c chargeabi l ty and the c o n d u c t i v i t y of the m e d i u m . In p rac t i ce potent ia l s are measured at n u m b e r of t imes after current cessat ion and therefore apparent chargeab i l i ty is a func t ion of t ime , g iven by Va{tk) = —rr- (3.3) "v where Vt(tk) is vol tage at t i m e tk. T h i s can also be an in tegra ted average over a s m a l l t i m e w i n d o w at tk. T h e uni ts of apparent chargeabi l i ty are m i l l i v o l t / v o l t . Chapter 3. IP and complex resistivity Method 64 I(t) F i g u r e 3.1: T i m e d o m a i n I P signals. Panels (a) On-off current waveform and (b) the vol tage charg ing and d ischarg ing process. Chapter 3. IP and complex resistivity Method 65 3.3 F r e q u e n c y D o m a i n I P In the frequency domain IP two types of measurements are usually made: (a) Percent frequency effect (PFE) and (b) phase response. Figure 3.2 illustrates the principle of P F E measurement. Currents at two frequencies (tohigh and u>iow, usually a decade apart) are transmitted into the ground and the corresponding amplitudes of the potentials are recorded. If the inductive effects are negligible and if the ground is polarizable, then the amplitude of the voltage measured at low frequency is greater than that at high frequency. This is because at high frequency the chargeable material does not get enough time to build up charges, and this results in a smaller IP effect. If the ground is non-polarizable then the ratio of \Z(iu>high) I Z{i<jJiow)\ will be equal to unity, where Z(iui) = V(iu>)/1(iu>). Therefore any departure from unity will indicate the presence of chargeable material. This is called the frequency effect and is defined by F E = 1 - ^hi9h}} (3.4) \Z(iulow)\ y ' Since the quantity in eqn (3.4) is small, it is multiplied by 100 and called Percent Fre-quency Effect (PFE). This is given by PFE = (l - l§^4) * 100. (3.5) The frequency dependence of IP is also manifested as a phase difference between the receiver voltage waveform and the transmitted current waveform shown in Figure 3.3. For making phase measurements, a sinusoidal current is transmitted into the ground and the in-phase (real) and quadrature (imaginary) voltage components are recorded. The arctangent of the ratio of quadrature to in-phase components of the measured impedance Chapter 3. IP and complex resistivity Method 66 (b) V F i g u r e 3.2: F r e q u e n c y d o m a i n I P s i g n a l s f o r P F E m e a s u r e m e n t s . C u r r e n t w a v e f o r m a t ( a ) l o w f r e q u e n c y ( b ) h i g h f r e q u e n c y . V o l t a g e w a v e f o r m a t ( c ) l o w f r e q u e n c y a n d ( d ) h i g h f r e q u e n c y . Z(iuj) g i v e s t h e p h a s e a n g l e . T h e u n i t s a r e e x p r e s s e d i n m i l h r a d i a n s . T h i s i s g i v e n b y 3.4 C o m p l e x R e s i s t i v i t y M e a s u r e m e n t s T h e a p p a r e n t c h a r g e a b i l i t y m e a s u r e m e n t s i n t h e t i m e d o m a i n , t h e P F E a n d p h a s e s h i f t m e a s u r e m e n t s i n f r e q u e n c y d o m a i n , p r o v i d e a f i r s t o r d e r i n d i c a t i o n o f c h a r g e a b i l i t y i n t h e m e d i u m . I n o r d e r t o s t u d y t h e d e c a y c h a r a c t e r i s t i c s o f t h e v o l t a g e d i s c h a r g e c u r v e i n t h e t i m e d o m a i n , v o l t a g e m e a s u r e m e n t s a t m u l t i p l e t i m e c h a n n e l s a r e n e c e s s a r y . A n a l -o g o u s l y i n t h e f r e q u e n c y d o m a i n m u l t i p l e f r e q u e n c y m e a s u r e m e n t s a r e r e q u i r e d t o s t u d y Chapter 3. IP and complex resistivity Method 67 Figure 3.3: Frequency domain IP signals for phase measurement, (a) Harmonic current waveform and (b) voltage waveform. The phase lag between the two is phase response. the frequency dispersion of the medium. These goals are achieved in a complex resis-tivity survey where the ground impedance Z(iuj) = V(iu>)/I(iu>) is measured at discrete frequencies over a wide range (usually 0.1 Hz - 100 Hz). These data are typically rep-resented by an argand diagram, (similar to Cole-Cole plot in dielectrics) in which the real component of the impedance is plotted against the negative imaginary component shown in the schematic diagram in Figure 3.4. The datum measured at each frequency is a point on this complex plane. The frequency domain signals discussed in the previous section can be easily obtained from these measurements. For example the P F E data at two separated frequencies of 0.1 Hz and 10 Hz is given by (1 — IM2I/IM11) * 100 where M\ and Mi are the vectors from origin to the respective data points on the complex plane obtained at 0.1 Hz and 10 Hz. The phase data at 0.1 Hz is equal to the angle subtended by the vector M\ from the real axis. Chapter 3. IP and complex resistivity Method 68 c CD c O a E o O Normalised Real Component F i g u r e 3.4: D a t a r e p r e s e n t a t i o n f o r m u l t i p l e f r e q u e n c y m e a s u r e m e n t s i n a c o m p l e x r e s i s -t i v i t y s u r v e y . T h e a b o v e d i a g r a m i s a c o m p l e x i m p e d a n c e p l o t ( s i m i l a r t o C o l e - C o l e p l o t i n d i e l e c t r i c s ) i n w h i c h t h e r e a l c o m p o n e n t o f i m p e d a n c e i s p l o t t e d a g a i n s t t h e n e g a t i v e i m a g i n a r y c o m p o n e n t . Chapter 3. IP and complex resistivity Method 6 9 Initially the complex resistivity measurements were carried out in the laboratory to identify the first and second order IP effects. The first theoretical treatment of complex resistivity method was given by Wait (1959a). A thorough account on the theory of complex resistivity is given in Wait (1984) which is fundamental to the understanding of the multi-frequency responses. The instrumentation and field tests to measure the multiple frequency responses was pioneered by Zonge et al.(1975). Here I present a brief review of this topic. The starting point is Ohm's law in the frequency domain which relates the current density J to the electric field E. It is experimentally observed that both conductivity and electrical permittivity of the medium are frequency dependent and are complex quantities (Wait, 1959a; Ward and Fraser, 1967; Fuller and Ward, 1970; Shuey and Johnson, 1973; Knight, 1983). Thus the relation between the electric field are current density for a general dispersive media is given by J(iui) — (o~(iui) + iu>e(iu>))E(iuj) (3-7) where a(iuj) and e(iu>) are the complex conductivity and complex electrical permittivity of the medium respectively. Thus if we denote the complex conductivity of the medium by a(iu>) = <rr +i(Ti and complex dielectric permittivity by e{iu>) — er + ze;, then the effective conductivity of the medium is given by creff = aT — a>e; and the effective dielectric permittivity by eeff — eT + o~i/u>. So from a conventional physical interpretation the conduction current is caused by the effective conductivity <7e// and the displacement current by the effective dielectric permittivity teff of the medium. Thus irrespective of what the polarization process is, i.e. whether the polarization is due to induced-polarization effect or due to dielectric polarization effect, it is possible to interpret in terms of effective conductivity and effective dielectric permittivity. Such a practice is followed by Lesmes and Morgan (1999). Because this leads to abnormally high dielectric constant of the medium, I adopt the complex conductivity formulation to explain the Chapter 3. IP and complex resistivity Method 70 frequency dispersion of the medium at low frequencies. In a typical IP survey the frequencies of interest rarely exceed 1000 Hz. For general earth materials at low frequencies the displacement current due to dielectric polarisation is negligible implying that a(iu>) » iwe(iu>) which is called the quasi-static assumption. This leads to, In eqn (3.8) the effect due to dielectric properties have been neglected in the low fre-quency range. Thus any frequency dispersion that is measured is considered as "induced polarisation effect". The induced polarization phenomena that is prevalent in the low frequency regime is due to electrode or membrane polarisation mechanisms (Ward and Fraser, 1967), which causes the conductivity to be complex and frequency dependent. The macroscopic explanation of the electrode and membrane polarization process led to the development of electrical circuit models to model the observed responses (Ward and Fraser, 1967; Fuller and Ward, 1970; Shuey and Johnson, 1973; Pelton et a l , 1978a; Wait, 1984). These models are based on the relaxation process which constitutes a charge flow that can be measured as a decaying potential after the charging process is interrupted. Relaxation theory for electrical properties originated to explain the polarization phe-nomena in dielectrics (Cole and Cole, 1941; Jonscher, 1983). Later this was applied to the conductors to explain the frequency dispersion effects at low frequencies due to the IP effect. The relaxation behaviour, or the rate of decay, varies for different materials with different time constants. The need to properly model the decay characteristics led to the emergence of various models (Debye, Warburg, Cole-Cole, flat spectrum). Wait (1984) gives a comprehensive treatment of these models based on the relaxation theory. Among the various models in literature the Cole-Cole model is commonly used among the workers in this field. The Cole-Cole model strictly applies to frequency dispersion in J(iw) = a{i(jj)E(iui) (3.8) Chapter 3. IP and complex resistivity Method 71 die lect r ics ( C o l e and C o l e , 1941), however the same f o r m of f requency dependence exists for the i m p e d a n c e (Z). T h i s is g iven by ^ ^ ( ' - ' ( ' - i T R ) ) ( 3 J ) where Ro is the D C res i s t iv i ty of the m e d i u m , 77 is the in t r in s i c chargeabi l i ty , r is the t i m e constant a n d c is the r e l axa t ion constant . F i g u r e 3.5(a) and (b) shows the geologic and the cor respond ing c i rcu i t m o d e l for the C o l e - C o l e representa t ion ( P e l t o n et a l . , 1978a). T h e c i r cu i t m o d e l i n 3.5(a) exh ib i t s an I P r e l axa t ion behav iour expec ted f r o m the geologic m o d e l i n 3.5(a). T h e impedance [itoX)~c s imulates a me ta l l i c g ra in e lec t ro ly te interface. T h e resis tance Ro represents the resistance of the unb locked pore and R\ is the resistance of the b l o c k e d pore p a t h . T h e impedance of the c i rcu i t is g iven by eqn (3.9) . T h e t i m e constant r w h i c h can vary over order of magni tudes is g iven by r = x(^j". (3.10) T h e r e l a x a t i o n pa ramete r c varies i n the range of (0 ,1] . T h e in t r i n s i c cha rgeab i l i t y 77 is g iven by " = inb; ( 3 U ) where 77 varies be tween [0,1]. T h e frequency d o m a i n response of the C o l e - C o l e m o d e l is shown F i g u r e 3.5(c). A t low frequencies the a m p l i t u d e response asympto tes to R0 w h e n the current flows t h r o u g h the unb locked pore paths . A t h igh frequencies i t a sympto tes to Ro(l — 77). T h e r e is m o n o t o n i c decrease i n the a m p l i t u d e f r o m Ro to i ? o ( l — v) a s the f requency increases. T h e phase curve shows a peak at the poin t of m a x i m u m a m p l i t u d e curva tu re . T h e peak occurs w h e n to = l / [ r ( l — ?7)1//2c] ( M a j o r and S i l i c , 1981). T h e phase curve is s y m m e t r i c w i t h slopes —c and c a round the m a x i m u m . F i g u r e 3.5(d) ind ica tes the t rans ient vol tage decay i n t ime d o m a i n . Chapter 3. IP and complex resistivity Method 72 Since the mic roscop ic theory for I P is not fu l ly unde r s tood the v a l i d i t y of us ing C o l e -C o l e m o d e l can be i n ques t ion. However i t has been ex tens ive ly used to in te rpre t the spec t r a l I P responses and w i t h some success. P e l t o n et a l . (1978a) showed tha t for the l a b o r a t o r y a n d field measurements the spec t ra l I P da t a can be closely m a t c h e d w i t h the C o l e - C o l e m o d e l . T h e y used i t for m i n e r a l d i s c r i m i n a t i o n p rob lems a n d to iden t i fy E M coup l ing r emova l effects i n the da ta . J o h n s o n (1984) and Seigel et a l . (1997) used the C o l e - C o l e m o d e l to ana lyze the vol tage discharge curve i n t i m e d o m a i n for m i n e r a l d i s c r i m i n a t i o n p rob lems . In this thesis I adopt the C o l e - C o l e m o d e l to m o d e l the c o m p l e x re s i s t iv i ty behav iour of the ear th . Chapter 3. IP and complex resistivity Method 73 F i g u r e 3.5: C o l e - C o l e m o d e l for complex res is t iv i ty . P a n e l (a) is the schemat ic represen-t a t i o n of the geologic rock m o d e l , (b) T h e equivalent c i rcu i t m o d e l , (c) T h e f requency a n d (d) t i m e d o m a i n responses (Af t e r P e l t o n et a l . , 1978a) Chapter 4 Electromagnetic Coupling and a Method for its Approximate Removal 4.1 Introduction E M coup l ing can be defined as the i n d u c t i v e response of the ea r th w h i c h manifests i t se l f as a response over the I P s ignal . E M coup l ing is a ma jo r i m p e d i m e n t i n the i n t e r p r e t a t i o n of c o m p l e x res i s t iv i ty and I P da t a ( W y n n and Zonge, 1975; W y n n a n d Zonge , 1977; W a i t a n d G r u s z k a , 1986) and makes the in t e rp re ta t ion difficult w h e n the c o n t r i b u t i o n is large c o m p a r e d to the I P s ignal . T h e effect is increased w h e n the survey is ca r r i ed over a conduc t ive ea r th or w h e n the d ipole l eng th and d ipole spacings are large. T h i s has p r a c t i c a l consequences since the dep th of exp lo ra t i on depends u p o n the largest d ipo le l eng th a n d d ipole spacings. In m a n y surveys i t is necessary to remove E M c o u p l i n g f r o m I P d a t a i n order to re t r ieve meaningfu l i n f o r m a t i o n about the chargeable zones i n the subsurface. T h e r e m o v a l of E M coup l ing has been a long-s tand ing p r o b l e m i n I P d a t a in te rpre t -a t ion . A s i n most geophys ica l exp lo ra t i on surveys the i n i t i a l step is to remove the noise at the d a t a acqu i s i t i on leve l . A t the da t a acqu is i t ion stage a compromise is m a d e i n the s ignal s t reng th to avo id E M coup l ing . T h i s is done by s a m p l i n g the la ter t i m e channels i n t i m e d o m a i n or us ing on ly low frequencies i n the f requency d o m a i n . H o w e v e r even at l ow frequencies w h e n the m e d i u m is conduc t ive or w h e n the d ipole l eng th is large the ob jec t ive is not fulf i l led. T h e c o n t a m i n a t i o n can be significant and can p roduce ar t i facts 74 Chapter 4. EM CoupHng and a method for its approximate removal 75 Calculate EM responses V x V x E + icou.a E = - icou Js Calculate E M responses Correct the data Invert for r\ using V . ( a ( 1 - n ) V<|>) = • I 5 ( r - r s ) (Chapter 4 ) Invert for complex conductivity o(ico) (Chapter 5 ] Correct the data Invert for o(ico) using V. ( a (ico) V<j)(ico)) = - I (ico) 5 ( r - r s ) ; Chapter 6; Figure 4.1: Three different formulations to recover the chargeability information from the data. The highlighted section indicates the formulation to be investigated in this Chapter. in the inversion results and this has prompted me to try and develop a practical meth-odology to remove E M coupling so that the resultant data can be inverted with standard IP inversion algorithm. The basic procedure to recover the chargeability structure, in this Chapter, is shown in Figure 4.1. Research in the past three decades on E M coupling removal has evolved principally in three directions. The first method is based on the assumption that IP phase is constant (Hallof, 1974; Coggon, 1984) or a linear function of logarithm of frequency (Song, 1984), and that the phase due to E M coupling is represented by a power series in frequency. Chapter 4. EM Couphng and a method for its approximate removal 76 H a l l o f (1974) used a constant phase for I P , and l inear or quadra t i c f requency var ia t ions for E M couphng , to fit curves to the phase measurements at two or three frequencies respect ively . T h e observed phases at two or three frequencies were used to solve a set of s imul taneous equat ions to de termine the coefficients of the l inear or q u a d r a t i c equa t ion . Since the I P phase was assumed constant , the ze ro th power coefficient was then the meas-ure of decoup led I P phase. T h i s is sometimes referred to as the three-point decoup l ing scheme. C b g g o n (1984) also represented the I P phase as constant , and represented the E M couphng phase as a t r unca t ed non-integer power series of frequency. T h e coefficients of th is non-integer p o l y n o m i a l were t hen de te rmined us ing phase measurements at three frequencies. T h e above men t ioned techniques assume a pa r t i cu l a r p o l y n o m i a l f o r m of f requency to represent the E M coup l ing responses and treat the I P phase as constant or l inear . H o w e v e r previous invest igat ions suggests tha t I P phase is not a lways cons tant , nor l inear , and can va ry depend ing on the various factors such as m i n e r a l t ex tu re , g r a i n size, t y p e of m i n e r a l ( P e l t o n et a l , 1978a; V a n V o o r h i s et a l , 1973; Seigel et a l . , 1997). S ince these me thods do not take in to account a l l of the phys ics associated w i t h the process, the i r a p p l i c a t i o n can be quest ionable . T h e i r advantage however , is tha t they are s imple to use. T h e second m e t h o d is based on a p p r o x i m a t i n g the low frequency I P d a t a c o n t a m -i n a t e d by E M couphng response by a double C o l e - C o l e m o d e l ( P e l t o n et. a l , 1978a; B r o w n , 1985). In th is m e t h o d the E M couphng response at low f requency is a p p r o x i m -a ted b y a C o l e - C o l e d ispers ion m o d e l h a v i n g h i g h r e l axa t i on constant (c) be tween 0.9 a n d 1.0. Invest igat ions w i t h this m e t h o d have revealed that the t i m e constant for po lar -i z a t i o n effect TJP is greater t h a n the t i m e constant ob ta ined for E M couphng response TEM a n d the r e l axa t i on constant for I P c/p is less t h a n the r e l axa t i on constant for E M couphng CEM- M a j o r and Si l i c (1981) have po in t ed out the l i m i t a t i o n of us ing this m o d e l , w h i c h is based o n an e m p i r i c a l approach to fit the da ta , and they have i n d i c a t e d tha t the Chapter 4. EM Couphng and a method for its approximate removal 77 i n t e r a c t i o n between the i n d u c e d po l a r i za t i on and i n d u c t i v e effects is , i n general , more c o m p l i c a t e d t h a n a m u l t i p l i c a t i o n of two C o l e - C o l e models . T h e t h i r d approach is to assume the ea r th is I D a n d c o m p u t e the E M c o u p h n g response by eva lua t ing the m u t u a l impedance of the g rounded c i rcu i t s at the des i red f requency ( M i l l e t t , 1967; H o h m a n n , 1973; D e y and M o r r i s o n , 1973; H o h m a n n , 1975; W y n n a n d Zonge, 1977; W a i t and G r u s z k a , 1986). W a i t and G r u s z k a (1986) c o m p u t e the i m p e d a n c e due to E M couphng for a homogeneous half-space m o d e l a n d sub t rac t th is f r o m the observed impedance that has b o t h I P and E M couphng response. T h e y show tha t this is v a l i d at low frequencies where the in t e r ac t ion of E M effects a n d the I P effects are negl igible , or equ iva len t ly w h e n the p ropaga t ion constant of the diffusive E M waves are not g rea t ly a l tered by the chargeabi l i ty of the m e d i u m . T h e w o r k here follows f rom the t h i r d approach but differs i n n u m b e r of ways . I fo rmula te the p r o b l e m i n te rms of e lectr ic field and show tha t the observed e lec t r ic f ie ld of a half-space can be expressed a p p r o x i m a t e l y as a p roduc t of an I P response func t i on a n d an e lec t r ic f ie ld due to E M couphng response. E i t h e r the phase or the a m p l i t u d e of the I P response func t ion are ind ica t ions of chargeabi l i ty . T h i s enables m e to " r emove" E M c o u p l i n g f rom P F E and phase da ta . T h e m e t h o d is general and c a n be a p p l i e d to I D , 2 D or 3 D da t a sets. I i l lus t ra te this us ing I D and 3 D syn the t i c examples . F o r col inear arrays I develop an app rox ima te r emova l me thodo logy to hand le 2 D I P d a t a sets. I first generate a 2 D c o n d u c t i v i t y s t ruc ture cr(x, z) by i n v e r t i n g the D C poten t ia l s at the lowest frequency. W i t h the c o n d u c t i v i t y s t ruc ture cr(x,z) I cons t ruc t a m u l t i -l aye red I D c o n d u c t i v i t y by t a k i n g a l o c a l average of the 2 D c o n d u c t i v i t y d i s t r i b u t i o n . A t each t r a n s m i t t e r pos i t i on the E M response f rom a finite l eng th d ipole is c o m p u t e d at the des i red f requency and this is then used to correct the observed da ta . T h e cor rec ted responses are t h e n inve r t ed us ing a 2 D I P invers ion a l g o r i t h m to recover 2 D cha rgeab i l i t y d i s t r i b u t i o n . Chapter 4. EM Coupling and a method for its approximate removal 78 4.2 E f f e c t s o f E M C o u p l i n g o n I P d a t a T h e p r o b l e m w i t h E M c o u p l i n g i s t h a t , b o t h I P a n d E M c o u p l i n g e f f e c t s o c c u r s i m u l t a n e -o u s l y i n a c e r t a i n f r e q u e n c y b a n d w i d t h . T o b e a b l e t o d i s t i n g u i s h t h e d i f f e r e n t r e s p o n s e s I f i r s t d e f i n e t h e v a r i o u s e l e c t r i c f i e l d s t h a t a r e a s s o c i a t e d w i t h t h e s e e f f e c t s . I c o n s i d e r a h a l f - s p a c e c o m p l e x c o n d u c t i v i t y m o d e l o f t h e e a r t h . T h e e l e c t r i c f i e l d i n t h e x - d i r e c t i o n d u e t o a n x - d i r e c t e d h o r i z o n t a l e l e c t r i c d i p o l e ( H E D ) a t n o n - z e r o f r e q u e n c y i s g i v e n b y ( W a r d a n d H o h m a n n , 1988, p g . 2 3 3 ) , F i g u r e 4.2: X - d i r e c t e d h o r i z o n t a l e l e c t r i c d i p o l e o n a h o m o g e n e o u s c o m p l e x c o n d u c t i v e h a l f - s p a c e . 2x2-y2 (4.1) 2irr3a(iuj) r v~ ' '~ J ' 2Tr<r(ico) w h e r e k2 = iu)fia(iuj) i s t h e p r o p a g a t i o n c o n s t a n t o f t h e m e d i u m , LO i s t h e a n g u l a r f r e -q u e n c y , fi i s t h e m a g n e t i c p e r m e a b i l i t y , a(iu>) i s c o m p l e x c o n d u c t i v i t y o f t h e m e d i u m , r — -\/x2 + y2 i s t h e d i s t a n c e f r o m t h e d i p o l e a n d Ids i s t h e c u r r e n t e l e m e n t . T h e c o m -p l e x c o n d u c t i v i t y criito), o r t h e c o m p l e x r e s i s t i v i t y p(iu>), c a n h a v e m a n y f o r m s . A g e n e r a l Chapter 4. EM Couphng and a method f o r its approximate removal 79 representa t ion can be w r i t t e n as p(iuj) = p0 (1 — m(i<jj)) (4.2) where p0 is the D C res i s t iv i ty of the m e d i u m i.e. r es i s t iv i ty at zero frequency, m(iu>) is the c o m p l e x chargeabi l i ty w h i c h depends on the in t r in s i c chargeabi l i ty , f requency a n d other a d d i t i o n a l parameters . For example , i n the C o l e - C o l e m o d e l , m{iuj) is g iven by " ^ " O - T T f W ) ( 4 3 ) where rj is the in t r in s i c chargeabi l i ty (0 < n < 1), u> is the frequency, r is the t i m e constant a n d c is the r e l axa t ion constant . For the C o l e - C o l e c o m p l e x chargeab ih ty g iven i n eqn(4.3) i t is seen that l i m ^ ^ o o m(iu>) = n and hm^^o m(iu>) = 0. T h e D C res i s t iv i ty can be w r i t t e n as Po = y ^ (4-4) 1 - 7/ where p0 = l i m ^ o p(iw) and p^ = l i m ^ o o p(ioo). Therefore the D C re s i s t i v i ty p0 or D C c o n d u c t i v i t y 0~o is dependent u p o n the in t r in s i c chargeabihty . T h i s means tha t i f the m e d i u m is chargeable t hen at zero frequency the measured response w i l l have effects due to chargeabihty . Fo r c l a r i ty i n future discussions I w i l l refer to p^ a n d respec t ive ly as the res i s t iv i ty a n d c o n d u c t i v i t y of the m e d i u m i n absence of chargeabihty . T h e e lect r ic field due to the source w i l l va ry w i t h frequency a n d w i t h i n t r i n s i c chargeabihty . It is convenient to e x p l i c i t l y i n t roduce the four different e lec t r ic fields shown i n the F i g u r e (4.3). F o r no ta t i ona l s i m p l i c i t y the e lec t r ic field Ex w i l l be deno ted by E hencefor th . T h e four different e lectr ic fields shown i n F igure (4 .3 ) can be a n a l y z e d as fol lows. Chapter 4. EM CoupHng and a method for its approximate removal 80 CHARACTERISTICS CD = 0 = 0 co = 0 *\ * 0 co * 0 *1 = 0 co ^ 0 i l * 0 EFFECTS V E D C DCIP/ EM EM IP/ Figure 4.3: The characterization of various effects based on an electric field generated by a point horizontal electric dipole. 4.2.1 D C Effect: EDC EDC is defined to be the electric field that exists when to — 0 but there are no effects of chargeability (w = 0). The resistivity of the medium is therefore equal to p(i(o) = P o o = 1 / f J c o . Setting the propagation constant = 0 in eqn(4.1), and replacing <T(ILO) by cr^, yields E D C = Ids 2z2 - y2 (4.5) 4.2.2 D C I P Effect: E D C I P JJJDCIP - g e l e c t r i c field that exists when LO = 0 and the medium is chargeable (77 ^  0). The resistivity of the medium is therefore equal to p(i(o) = Po = l/cr 0 . Setting the Chapter 4. EM Couphng and a method for its approximate removal 81 propagation constant k = 0 in eqn(4.1), and replacing a(w) by <J0, yields E 4.2.3 E M I P Effect: EEMIP DCIP Ids 27T<70 2 * 2 - y 2 (4.6) pjEMiP j s electric field in eqn (4.1) when a(iui) is any complex conductivity function. It includes electromagnetic induction and IP effects at non-zero frequency. The propa-gation constant kEMIP — ^iuipa{iu3) depends on conductivity and chargeabihty of the medium and the electric field is written as E EMIP -Ids 2-Kr3o~{iu>) \-{l + ikEMIPr)e -ik1 + Ids 2ircr(iu>) [ r 2x2 - y2 (4.7) 4.2.4 E M Effect: EEM EEM is the electric field obtained at non-zero frequency (u> / 0) and when the chargeabil-ity effects are not considered to alter the conductivity (77 = 0). As in the definition of EDC, I have p(iu) = p^ = l/<Xoo- However since u> ^ 0 the wavenumber kEM = y/iupa^ and the electric field is given by EEM(iu) -Ids 2irrzcr0 l _ ( l + i ^ M r ) e - ^ + Ids 2'KO~~L 2x2 - y2 (4.8) The expression for EEM is similar to eqn(4.7) except that the complex conductivity a(iw) is replaced by and the propagation constant is calculated using instead of a(iu>). It is to be noted that EEM —> EDC as frequency approaches zero (a; —> 0). In an IP survey, finite dipole lengths are used and the measured quantity is impedance rather than electric field. Effect due to finite length grounded source can be obtained by linear superposition of small but finite dipole sources. The finite length source is divided into N dipoles of length As such that As << r (distance to receiver dipole). The voltage at the receiver is obtained by first integrating the electric field due to the small dipole Chapter 4. EM Couphng and a method for its approximate removal 82 source of length As. Next, I use linear superposition of the source dipoles to obtain the voltage due to finite length grounded source. Finally the voltage is normalized by the current, to generate the impedance. For example, the measured EMIP impedance can be written as ^ » = y E A ' i / ffMIP{iw)dr (4.9) 1 j=l  J R X where dr is the elemental length of receiver, ASJ is the length of the jth elemental trans-mitter dipole and EEMIP(iu>) = IAsfEMIP(iu>) is the electric field due to a horizontal electric dipole of length A s . Typically the IP data are represented in terms of percent frequency effect (PFE) and/or phase responses. The P F E responses are generated from amplitude measurements at two widely separated frequencies (u>i and o>2). If o>i < u>2, then the observed P F E response is given by pFE°B$=(' - g^Sj) * io°-The phase response is defined as the phase of measured impedance ZEMIP(iu>). This is a single frequency measurement and is given by * H = t a n U ( £ ™ ' M ) ) ' ( } In order to gain some insight into how the E M coupling responses affect the IP data I compute the P F E and phase responses for a half-space model in the absence of chargeabil-ity. The corresponding electric fields are given by eqn(4.8) and the factors affecting the E M couphng responses are: (a) dipole length (b) dipole spacing (c) conductivity structure of the medium and (d) frequency of the transmitting signal. I consider a dipole-dipole survey over a homogeneous half-space conductivity of 0.05S/m. The basic parameters are shown in Figure 4.4(a). Figure 4.4(b) and 4.5(b) shows that both P F E and phase response increase with the dipole length and dipole spacing. With the increase in dipole Chapter 4. EM Couphng and a method for its approximate removal 8 3 s p a c i n g s t h e i n d u c t i v e c o m p o n e n t o f t h e e l e c t r i c f i e l d d o m i n a t e s o v e r t h e g a l v a n i c c o m -p o n e n t . F o r a g r o u n d e d w i r e , t h e e n d p o i n t s o f t h e w i r e c o n t r i b u t e t o g a l v a n i c c u r r e n t s i n t h e g r o u n d , a n d t h e r e s t o f t h e w i r e a c t s as a n i n d u c t i v e s o u r c e . T h u s w i t h l a r g e r d i p o l e l e n g t h s t h e m a g n i t u d e o f t h e i n d u c t i v e s o u r c e i n c r e a s e s r e s u l t i n g i n t h e i n c r e a s e o f E M . c o u p h n g r e s p o n s e . T h i s i m p l i e s t h a t f o r t h e p o l e - d i p o l e a n d p o l e - p o l e s u r v e y g e o m e t r y t h e E M c o u p l i n g c o n t r i b u t i o n w i l l b e l a r g e c o m p a r e d t o d i p o l e - d i p o l e s u r v e y s , s i n c e t h e l e n g t h s o f t h e g r o u n d e d w i r e a r e m u c h l a r g e r . F i g u r e s 4.4, 4 . 5(c) a n d ( d ) s h o w t h a t E M c o u p h n g i n c r e a s e s w i t h i n c r e a s i n g c o n d u c t i v i t y a n d f r e q u e n c y . B e c a u s e t h e d i f f u s i v e p r o p a g a t i o n c o n s t a n t i s p r o p o r t i o n a l t o t h e p r o d u c t o f f r e q u e n c y a n d c o n d u c t -i v i t y (k = y^w/iUoo), t h e i n d u c t i v e e f f e c t i n c r e a s e s w h e n e i t h e r c o n d u c t i v i t y o r f r e q u e n c y i n c r e a s e s . T h e s e c o n c l u s i o n s a r e i n a g r e e m e n t w i t h c o m m e n t s m a d e b y W a r d ( 1 9 9 0 ) a n d D e y a n d M o r r i s o n ( 1 9 7 3 ) . T h e s e h i g h p h a s e r e s p o n s e s , w h i c h a r i s e d u e t o E M c o u p h n g , c a n b e m i s - i n t e r p r e t e d as I P r e s p o n s e s a n d c a n m a s k t h e I P a n o m a l y . F o r e x a m p l e , a n I P a n o m a l y o f 50 m r a d s i n a c o n d u c t i v e m e d i u m o f l S / m i s s m a l l c o m p a r e d t o 80-320 m r a d s f o r N = 2 - 6 . T h u s t o b e a b l e t o e x t r a c t t h e I P a n o m a l y f r o m t h e d a t a t h e r e i s a n e e d t o r e m o v e E M c o u p h n g . I n t h e n e x t s e c t i o n I d i s c u s s t h e r e m o v a l i s s u e a n d d e r i v e a n e x p r e s s i o n f o r d e c o u p l i n g t h e E M c o u p h n g r e s p o n s e s f r o m t h e d a t a . 4.3 T h e R e m o v a l o f E M C o u p l i n g I n t h i s s e c t i o n I d e r i v e t h e E M c o u p h n g r e m o v a l e q u a t i o n s b a s e d o n t h e e l e c t r i c f i e l d s d e f i n e d i n t h e p r e v i o u s s e c t i o n . T h e f o r m u l a t i o n c a n b e e x t e n d e d t o i n c l u d e t h e e f f e c t s o f finite l e n g t h g r o u n d e d w i r e s b y i n t e g r a t i n g t h e e l e c t r i c field a l o n g t h e t r a n s m i t t e r a n d r e c e i v e r d i p o l e l e n g t h s . S u b s t i t u t i n g e q n ( 4 . 2 ) i n t o e q n ( 4 . 7 ) , a n d u s i n g t h e r e l a t i o n Chapter 4. EM Couphng and a method for its approximate removal 84 (a) 300 m 300 m cr= 0.05 S /m f = 0.1 H z 2 3 4 N-Spacing 5 6 (d) 5 Hz y 1 H z _ -"cCl Hz \ 2 3 4 N-Spacing 2 3 4 N-Spacing F i g u r e 4.4: ( a ) S y n t h e t i c h o m o g e n e o u s e a r t h m o d e l u s e d t o s t u d y v a r i o u s f a c t o r s w h i c h c o n t r i b u t e t o E M c o u p h n g r e s p o n s e s f o r t h e P F E d a t a . D a t a a r e c o m p u t e d f o r 6 d i p o l e s p a c i n g s i n a d i p o l e - d i p o l e c o n f i g u r a t i o n . T h e e f f e c t o f v a r y i n g ( b ) t h e d i p o l e l e n g t h a n d d i p o l e s p a c i n g , ( c ) c o n d u c t i v i t y o f t h e h a l f - s p a c e a n d ( d ) t h e f r e q u e n c y o f t h e t r a n s m i t t e d s i g n a l . <r0 = ^ ( l - n), I o b t a i n (1 — m(iijj)) ( —Ids E»M1F(iu>) \2irr3a, l-(l+ikEMIPr)e-ikBMIpT + Ids 27T<Xor 2x2 - y2 (4.12) I f I a s s u m e t h a t t h e p r o p a g a t i o n c o n s t a n t i s n o t g r e a t l y a l t e r e d b y t h e c h a r g e a b i l i t y o f t h e m e d i u m t h e n I c a n w r i t e kEMIP(tr(iu),u,) « fc*M(<7oo,") (4.13) Chapter 4. EM Couphng and a method for its approximate removal 8 5 (a) 300 m 300 m 140 N-Spacing N-Spacing F i g u r e 4.5: ( a ) S y n t h e t i c h o m o g e n e o u s e a r t h m o d e l u s e d t o s t u d y v a r i o u s f a c t o r s w h i c h c o n t r i b u t e t o E M c o u p h n g r e s p o n s e s f o r t h e p h a s e d a t a . D a t a a r e c o m p u t e d f o r 6 d i p o l e s p a c i n g s i n a d i p o l e - d i p o l e c o n f i g u r a t i o n . T h e e f f e c t o f v a r y i n g ( b ) t h e d i p o l e l e n g t h a n d d i p o l e s p a c i n g , ( c ) c o n d u c t i v i t y o f t h e h a l f - s p a c e a n d ( d ) t h e f r e q u e n c y o f t h e t r a n s m i t t e d s i g n a l . A q u a n t i f i c a t i o n o f t h i s a p p r o x i m a t i o n i s p r e s e n t e d i n A p p e n d i x - C . S u b s t i t u t i n g t h e a p p r o x i m a t i o n g i v e n i n e q n ( 4 . 1 3 ) i n t o e q n ( 4 . 1 2 ) I o b t a i n a r e l a t i o n b e t w e e n t h e m e a s u r e d field E E M I P a n d t h e c o m p u t e d f i e l d EEM(iu) g i v e n b y EEMIP(iu) = ( 1 - m W ) E B M ( i u ) = *{iw)EEM{iu). (4.14) (1 - 7 7 ) T h i s i s t h e f u n d a m e n t a l r e l a t i o n s h i p b e t w e e n t h e fields w i t h , a n d w i t h o u t , t h e c h a r g e a b i l -i t y . I t s h o w s t h a t t h e m e a s u r e d r e s p o n s e EEMIP, w h i c h i s a c o m b i n a t i o n o f I P e f f e c t s a n d E M i n d u c t i o n e f f e c t s , c a n b e a p p r o x i m a t e l y e x p r e s s e d i n a p r o d u c t f o r m i n w h i c h Chapter 4. EM Couphng and a method for its approximate removal 86 the quantity (1 — m(iu)) / ( l — rj) is the IP response, and EEM is the response due to E M couphng arising from induction effects and DC effects. So, with assumptions in eqn(4.13), it is possible to write the measured response EEMIP(iu>) as a product of a complex scalar ib(iw) and the electric field that would exist in absence of any polarization. 1 1.05 1.1 1.15 1.2 1.25 Real (y(ico)) Figure 4.6: Plot for the complex function ib(iu) using the Cole-Cole model with •q = 0.2,r = 1.0sec,c = 0.5. (a) Amplitude of l/\ib(iw)\ as a function of frequency, (b) Phase of ib(iu) as a function of frequency, (c) Argand diagram (Cole-Cole plot) in which the negative of imaginary part is plotted as function of real part. Figure 4.6 shows the nature of the function ib(iu) for a Cole-Cole model with rj = 0 . 2 , r = 1 . 0 and c = 0 . 5 . Figure 4.6(a) indicates that the amplitude l/|^>(iu;)| monoton-ically increases from ( 1 — 7 7 ) , its low frequency limit, to unity at high frequencies. The function ib(iu) is unity if there is no IP response and hence any deviation from Chapter 4. EM Couphng and a method for its approximate removal 87 u n i t y is an i n d i c a t i o n of chargeable ma te r i a l . It can also be u n i t y i f the f requency is too h i g h even i f the m e d i u m is chargeable. T h i s is expec ted , since at h igher frequencies the charg ing process is i n t e r rup t ed so q u i c k l y tha t the chargeable m a t e r i a l does not get sufficient t i m e to b u i l d up the charges. T h i s observa t ion can also be e x p l a i n e d us ing the e lec t r ic f ield EEMIP i n eqn (4.7). I f the m e d i u m is chargeable a n d f requency is h i g h enough such that <r(iu>) t hen the electr ic f ield E E M I P -» EEM. T h u s at h i g h enough frequencies, the measured response does not have any I P effect. I f the rea l par t , or the a m p l i t u d e , is measured then i ts greatest m a g n i t u d e is at zero frequency. T h e phase of the func t ion ip(iu>) i n F i g u r e 4.6(b) indica tes tha t I P phase is zero as f requency approaches zero or in f in i ty and peaks near the f requency 1/r . T h u s i f the phase measurement is made to de te rmine the I P effect, the measurement is best m a d e at an in t e rmed ia t e f requency 1/r . In teg ra t ing the fields for the effects of f ini te l eng th t r ansmi t t e r a n d receiver i n eqn(4.14) I o b t a i n ( 1 - m M ) ZEMIP{iw) \ZE^p(iu>)\e^EMIP^ W " ( 1 - 7 / ) ~ ZEM{iw) ~ | Z ^ ( i o ; ) | e ^ M H ' 1 j A s an es t imate of I P response I can choose any measurement tha t es t imates how far ib{iu>) deviates f r o m uni ty . T h i s can be represented i n a n u m b e r of ways , bu t the two tha t are useful here are I P a m p l i t u d e response = 1 — , ,} . (4-16) I P phase response = Ph.a.se(rb(iu>)) It is to be no ted that as u> —»• 0, the I P a m p l i t u d e defined i n eqn(4.16) (1 — j ^ ^ y r ) ~^ Vi w h i c h expla ins the choice of this pa r t i cu l a r fo rm. In the next subsect ions I consider the a m p l i t u d e a n d phase responses due to I P effects a n d discuss h o w t h e y compare to the field measurements . Chapter 4. EM Couphng and a method for its approximate removal 88 4.3.1 Amplitude Response The IP amplitude response at any frequency o> can be obtained by substituting eqn(4.15) into (4.16). This yields IP amplitude = \l - ^ p ^ j * 100 (4.17) Multiplication by 100 converts the response to a percentage and is refered to as IP amplitude in percentage and denoted by JPAMP. The IP amplitude in eqn(4.17) is obtained at a single frequency. We note that the impedance \ZEMIP(iuj)\ in eqn(4.17) can be measured in the field at frequency u>, whereas the impedance |zT£ M(ia>)| cannot be measured in the field. However |^' B A f(za;)| can be numerically computed at frequency uj if the conductivity structure cr^ is available. The implication of this formulation is that, if we collect amplitude data at a single frequency and have knowledge about the conductivity of the ground, then eqn(4.17) can be used to obtain an estimate of the IP response of the chargeable material in the medium. 4.3.2 Phase Response In phase domain IP surveys, the measured signal is the phase of the impedance at a partic-ular frequency. The IP phase response according to eqn(4.16) is <f)IP(uj) = Phase(V>(«^)). Using eqn(4.15) this can be written as = argtyM) = arg ( ^ f f i ) = ^ " " M " <^V)- (4-18) Equation (4.18) is used to correct for the E M couphng effects in the data. The phase due to E M coupling is computed at the desired frequency u> using the conductivity structure Coo-Chapter 4. EM Couphng and a method for its approximate removal 89 4.3.3 T h e P F E r e s p o n s e a n d i t s r e l a t i o n t o IPAMP In this section I discuss the P F E response that is generated from amplitude measurements at two frequencies and outline its relationship with the IP amplitude response (JPAMP) defined in eqn (4.17). Traditional field practice is to acquire data at sufficiently low frequencies so that the E M couphng is not important. This frequency range typically lies between 0.1-100Hz and it is sometimes refered to as the exploration frequency range. If E M couphng effects are negligible, then at low frequency, the amplitude of the impedance contains a greater IP signature than at higher frequency. Thus at higher frequencies when the chargeable material is not sufficiently charged, the impedance response ZDC will be due to a^. At low frequencies the response will be due to tr0, and the measured impedance ZDCIP, has both effects due to conductivity and chargeability. Ideally field data are routinely collected at two different frequencies that are widely separated within the exploration frequency range (at least by one or two decades). Thus if the ratio of the high frequency to low frequency amplitude departs from unity, then it indicates the presence of chargeable material in the ground. This is given by PFEIP{u2 = oo : u>x = 0) = f 1 - J * 100. (4.19) Substituting the expression of ZDC and ZDCIP for a homogeneous half-space from equations (4.5) and (4.6) I obtain that the PFE = n * 100. Next I compare the P F E in eqn(4.19) with the IP amplitude in eqn (4.17). We note that if the frequency u> -> 0 then ZEM ZDC and Z E M I P -> ZDCIP. This shows the relation between the two formulas. However, there is a fundamental difference between equations (4.19) and (4.17). The equation for the IP amplitude in (4.17) is obtained at a single frequency, whereas the P F E equation in (4.19) is obtained by taking measurements at two frequencies. It is because of this difference that I denote the response in eqn (4.17) as IP amplitude instead of P F E response. Chapter 4. EM Couphng and a method for its approximate removal 90 The PFE response in eqn (4.19) is theoretically generated from amplitude measure-ments at zero and infinite frequency. However, in practice, the amplitudes are measured at finite non-zero frequencies. Then the PFE response is given by / I yDCIP (• \|\ PFEZP(U = c 2 : w = c*) = ( l - \ z 5 c n ^ ) * 1 0 0 (4-20) where u)\ > o>2 and ZDCIP(iu)) is the impedance measured in the field when EM couphng effects are negligible. As to, —> 0, a(iuj) —> a0 and ZDCIP(u>i) —> ZDCIP given in eqn (4.6). For u>2 -» oo, a(iuj) -+ and ZDCIP(u2) -* ZDC. This gives back the PFE response in eqn (4.19). The impedance ZDCIP(iu>) can be obtained by solving V.(a(iw)W(iw, r)) = -I(iu)8(r - rs) (4.21) where V(iou) is the complex scalar potential and /(zu;) is the harmonic current. Equation (4.21) is the usual DC resistivity equation except that the conductivity is complex and therefore the potentials are also complex quantities. The potential is evaluated by solving eqn (4.21) subject to the usual boundary conditions that no current flows out of the surface of the earth and that the potential approaches zero at infinite distance from the source. For a complex conductive half-space the electric field in the x-direction in the absence of EM couphng is given by EDCIP(iu,) = Ids r5 = i){iw)EDC (4.22) where ib(iw) is the IP response function defined in eqn(4.15). Integrating eqn (4.22) along the transmitter and receiver dipole lengths I obtain ZDCIP{iu;) = i){iu)ZDC. (4.23) Chapter 4. EM Couphng and a method for its approximate removal 91 4.3.4 Removal of E M coupling from the P F E responses I n f r e q u e n c y d o m a i n I P m o s t o f t h e w o r k o n E M c o u p l i n g r e m o v a l h a s b e e n f o c u s s e d o n t h e p h a s e d a t a a n d t h e r e i s l i t t l e w o r k t h a t a d d r e s s e s t h e r e m o v a l p r o b l e m f o r t h e P F E d a t a ( W a n g e t a l . , 1 9 8 5 ) . R e c o g n i z i n g t h a t t h e P F E d a t a a r e g e n e r a t e d f r o m t h e a m p l i t u d e m e a s u r e m e n t s a t t w o f r e q u e n c i e s , I c o n s i d e r a p r a c t i c a l s c e n a r i o w h e n E M c o u p h n g i s p r e s e n t i n t h e d a t a . T h u s i f t h e d a t a a r e c o l l e c t e d a t t w o s e p a r a t e d f r e q u e n c i e s , u>i a n d o>2 w i t h o>2 > t h e n t h e P F E i s g i v e n b y PFE°BS={lJ^m)*m- (4-24) T h e g o a l i s t o c o r r e c t e q n (4.24) a n d o b t a i n t h e P F E e q u a t i o n g i v e n i n ( 4 . 2 0 ) . I u s e e q n (4.23) a n d s u b s t i t u t e t h e e x p r e s s i o n o f i/;(iu)) f r o m e q n (4.15). T h i s i s g i v e n b y yEMIP(- ,\ T h u s t o o b t a i n t h e P F E r e s p o n s e d u e t o I P e f f e c t , I s u b s t i t u t e t h e e x p r e s s i o n oiZDCIP(iu>) g i v e n i n e q n (4.25) i n t o e q n (4.20) a n d d e n o t e t h e r e s p o n s e as c o r r e c t e d p e r c e n t f r e q u e n c y e f f e c t (PFECORR). T h i s i s g i v e n b y PFE^nn, _u.u_u)_(1 \ Z m P ( ^ ) \ \ Z E U M \ \ PFE (u, - a* . « - c*) - 1^ - l z E M I P { i u J i ) l l Z E M { i u j 2 ) l ) * 1 0 ° - (4-26) E q u a t i o n (4.26) i s t h e f u n d a m e n t a l e q u a t i o n t o c o r r e c t t h e P F E r e s p o n s e s . W e n o t e t h a t i f -• 0 t h e n ZEMlp{i^) -> Z D C I P , ZEM(iwJ -+ ZDC a n d i f u2 -> o o t h e n ZEMIP{iui-t) —> ZEM{itA>2) s i n c e t h e r e i s n o I P e f f e c t a t v e r y h i g h f r e q u e n c i e s . T h i s g i v e s b a c k e q n (4.19) w h i c h v e r i f i e s t h e v a l i d i t y o f t h e e q u a t i o n . T o c o m p u t e t h e e x p r e s s i o n i n (4.26) I n e e d t o c o m p u t e \ZEM{iui)\ a n d \ZEM(iui2)\ u s i n g t h e c o n d u c t i v i t y s t r u c t u r e fJooj t h e r e s p o n s e s \ZEMIP(iu>1)\, \ZEMIP(iuj2)\ a r e a v a i l -a b l e f r o m t h e f i e l d m e a s u r e m e n t s . A n e s t i m a t e o f c a n b e o b t a i n e d b y i n v e r t i n g t h e Chapter 4. EM Couphng and a method for its approximate removal 92 amplitude data collected at the lowest frequency using a DC resistivity inversion algo-rithm. The conductivity model obtained from the inversion is then used to compute the E M couphng responses. 4.4 Removal methodology for inhomogeneous earth The E M couphng removal equation in Section 4.3 is derived using a homogeneous earth model. The IP response function ijj{iu>) was expressed as a ratio of EMIP and E M impedances for a homogeneous earth model. For a general earth model we can extend the expression in eqn (4.15) in terms of apparent quantities. An apparent response is a response due to an equivalent half-space. This is given by where iba(iuj) is an apparent IP response function, ZEMIP(icu) and ZEM{iui) are the apparent impedances due to EMIP and E M effects for an equivalent half-space. The ap-parent IP response function iba(iu>) depends on intrinsic chargeability function ip(iu>) and the conductivity of the medium a^. In a general mathematical framework the apparent impedances ZEMIP(ico) and ZEM(iu) can be written as ZEMIP(iu>) = TEMW^)) and ZEM{iu>) = J-EM{&oo)- J~EM is a nonlinear E M operator which depends on Maxwell's equations for the electric field. The conductivity o~(i(jj) and in eqn (4.27) can be either ID, 2D or 3D conductivities with, or without, topography. Thus the removal equations for the P F E data in eqn (4.26) and phase data in eqn (4.18) can be obtained for an inhomogeneous earth. This is given by Chapter 4. EM Coupling and a method for its approximate removal 93 <f>ipn = 4>EMIPn - <t>rn- ( 4 . 2 9 ) 4.5 Removal methodology in ID: Synthetic Examples In this section I consider a ID synthetic example to test the E M coupling removal meth-odology. A 3-layer earth model with the parameters for each layer is shown in Figure 4.7(a). The data are generated using a dipole-dipole array with the dipole length of 200m for N = 1, - • • , 10. The forward computation is carried out by computing the electric field due to a point horizontal electric dipole. The electric field in x-direction for a x-directed HED over a layered earth can be represented by the Ex component. This is given by (Ward and Hohmann, 1988) , E°""M = f [a - - (i -47T Ox2 J0 I iu>e0 / - [1 + rTE] J0(\r)d\ Jo uo J0(Xr)dX (4.30) 47T where TIE and rxM are the T E and T M reflection coefficients. The reflection coeffici-ents are functions of the complex conductivities of the layered earth model. These are computed using a propagator matrix method in which the solution is propagated from the bottom half-space to the surface. Jo is the Bessel function of the zeroth order of first kind. Equation (4.30) is used to generate the EMIP electric field. The impedance is then computed by integrating the electric field along the transmitter and receiver dipole lengths given in eqn (4.9). Equation (4.30) is an exact formulation for computing the electric field (or the poten-tial by integrating the electric field) when both E M coupling and IP effects are present. However in the absence of E M couphng an alternative method to compute the IP re-sponses is obtained by solving Equation (4.21). This equation is essential to compare Chapter 4. EM Couphng and a method for its approximate removal 94 the corrected responses after E M couphng removal with the true IP responses. The test for E M couphng removal developed in the previous section is applied to both P F E and phase data. 4.5.1 Synthetic test for P F E data The model in Figure 4.7(a) is used to generate the P F E data due to the IP effect using eqn (4.24) at u>i = 0.1Hz and o>2 = 5Hz using eqn (4.20). This is denoted by IP in Figure 4.7(b) which indicates a decrease in P F E with N-spacing. This is expected because only the first layer is chargeable and large dipole spacings are sensitive to material below this layer. Next I generate the P F E data using eqn (4.24) at the two frequencies 0.1 Hz and 5Hz. This is plotted as EMIP in Figure 4.7(b). We note that the amphtude of the EMIP curve increases with the dipole spacing; this is an indication that the responses are contaminated with E M couphng. The correction for E M couphng contribution in the P F E data is carried out using eqn (4.26). The impedance due to E M couphng was computed using the true conductivity structure (cr^). The corrected P F E denoted by CORR is plotted along with the P F E due to IP effect in Figure 4.7(b). The corrected P F E is in good agreement with the true IP response and thus the E M couphng response has been removed from the observed responses. 4.5.2 Synthetic test for phase data The synthetic model in Figure 4.7(a) is used to compute the phase response at 0.5 Hz. This is indicated as the EMIP response ( Figure 4.7(c)) which contains both IP response and the E M couphng effects. The true IP phase, shown in Figure 4.7(c) is computed by solving eqn (4.21) at 0.5Hz and taking the argument of the impedance function ZDCIP(iu>). To do the correction, the phase due to E M couphng is computed at 0.5 Hz by using the true conductivity model (<Too)- The corrected phase response <f>CORR Chapter 4. EM Couphng and a method for its approximate removal 95 200m 200m (a) a = 0.1 S/m m = 0.2 x = 10 c = 0.3 h = 200 m a = 0.01 S/m h = 200 m cr = 0.02 S/m i — i — i — r 6 7 8 9 10 N-Spacing 1 2 3 4 5 6 7 N-Spacing Figure 4.7: (a) 3-layer model used for testing the E M couphng removal method, (b) P F E data generated from amplitude data at 0.1Hz and 5Hz. The P F E due to IP and E M couphng is indicated by EMIP curve. The P F E due to the IP effect is indicated by the IP curve. The corrected P F E (CORR) shows a reasonable match with the IP curve, (c) The phase of the corrected IP response is compared with the true IP response generated at 0.5Hz. The E M couphng phase (EM) is subtracted from the observed phase (EMIP) to the give the corrected phase (CORR). Chapter 4. EM Coupling and a method for its approximate removal 96 is o b t a i n e d b y s u b t r a c t i n g t h e E M c o u p h n g p h a s e f r o m t h e E M I P p h a s e f o l l o w i n g e q n (4 . 1 8 ) . T h e c o r r e c t e d p h a s e r e s p o n s e (j)CORR i s a c l o s e m a t c h w i t h t h e I P p h a s e <f>IP w h i c h i n d i c a t e s t h a t t h e E M c o u p h n g r e s p o n s e h a s b e e n e f f e c t i v e l y r e m o v e d . 4.5.3 Effect of conductivity on E M coupling removal T h e k n o w l e d g e o f c o n d u c t i v i t y s t r u c t u r e i s i m p o r t a n t i n c o m p u t i n g t h e E M c o u p h n g r e s p o n s e s . I c o n s i d e r t w o t e s t c a s e s f o r t h i s l a y e r e d e a r t h e x a m p l e t o e m p h a s i z e t h e n e c e s s i t y o f c o n d u c t i v i t y i n f o r m a t i o n . I n t h e f i r s t c a s e I a s s u m e t h e m o d e l t o b e a h o m o g e n e o u s h a l f - s p a c e w i t h c o n d u c t i v i t y o f 0.01 S/m. W i t h t h i s m o d e l b o t h P F E a n d t h e p h a s e s a r e u n d e r - c o r r e c t e d as s h o w n i n F i g u r e 4.9(a) a n d ( b ) r e s p e c t i v e l y . I n t h e s e c o n d e x a m p l e I i n c r e a s e t h e h a l f - s p a c e c o n d u c t i v i t y t o 0.1 S/m. T h e c o r r e c t e d P F E a n d t h e p h a s e s w i t h t h i s c o n d u c t i v i t y a r e o v e r - c o r r e c t e d s h o w n i n F i g u r e 4.9(a) a n d ( b ) r e s p e c t i v e l y . F i g u r e s 4.9(a) a n d ( b ) r e s p e c t i v e l y i n d i c a t e t h a t s o m e n e g a t i v e P F E a n d p h a s e v a l u e s a r e o b t a i n e d d u e t o i n a p p r o p r i a t e c o r r e c t i o n . T h i s i n d i c a t e s t h a t a d e q u a t e c o n d u c t i v i t y i n f o r m a t i o n i s n e c e s s a r y t o c o m p u t e t h e r e s p o n s e d u e t o E M c o u p h n g . I n p r a c t i c e t h e t r u e c o n d u c t i v i t y (<Too) i s n o t k n o w n a p r i o r i . H o w e v e r a n e s t i m a t e o f t h e c o n d u c t i v i t y c a n b e o b t a i n e d b y i n v e r t i n g t h e a m p l i t u d e d a t a a t l o w e s t f r e q u e n c y . T o i n v e s t i g a t e t h i s I f i r s t i n v e r t t h e a m p l i t u d e d a t a a t 0.1 H z i n F i g u r e 4.8(a) u s i n g a I D D C r e s i s t i v i t y i n v e r s i o n a l g o r i t h m a n d o b t a i n a c o n d u c t i v i t y . m o d e l ( c r d c i n „ ) . U s i n g t h i s c o n d u c t i v i t y m o d e l i n F i g u r e 4.8(b) I c o m p u t e t h e E M c o u p l i n g r e s p o n s e s . T h e c o r r e c t e d P F E r e s p o n s e c u r v e d e n o t e d b y CORRdc i n F i g u r e 4.9(a) i s a g o o d m a t c h w i t h t h e P F E d u e t o I P e f f e c t s a n d s h o w s s i g n i f i c a n t i m p r o v e m e n t c o m p a r e d t o t h e c o r r e c t i o n s u s i n g t h e h a l f - s p a c e c o n d u c t i v i t i e s . T h e s a m e D C c o n d u c t i v i t y m o d e l i s u s e d t o c o m p u t e t h e E M c o u p h n g p h a s e s a t 0.5 H z . T h e c o r r e c t e d p h a s e s s h o w a c l o s e m a t c h w i t h I P p h a s e s s h o w n i n F i g u r e 4 . 9 ( b ) . Chapter 4. EM Couphng and a method for its approximate removal 97 1 2 3 4 5 6 7 8 9 10 N-Spacing 10° 101 10 2 10° Z(m) F i g u r e 4.8: D C r e s i s t i v i t y i n v e r s i o n o f ( a ) a m p l i t u d e d a t a a t 0.1 H z . ( b ) T h e r e c o v e r e d m o d e l f r o m t h e i n v e r s i o n 40 -20 -0 ^ o -UJ LL. D_ -20 --40 --60 -2 3 4 5 6 7 8 9 10 N-Spacing 2 3 4 5 6 7 8 9 10 N-Spacing F i g u r e 4.9: E M c o u p l i n g c o r r e c t i o n s u s i n g d i f f e r e n t c o n d u c t i v i t y m o d e l s . A h o m o g e n e o u s h a l f - s p a c e o f O . O l S / m (CORR0m) s h o w s a n u n d e r - c o r r e c t i o n a n d h o m o g e n e o u s m o d e l w i t h O . l S / m s h o w s a n o v e r - c o r r e c t i o n . C o r r e c t i o n w i t h t h e D C c o n d u c t i v i t y m o d e l d e -n o t e d b y CORRdc o b t a i n e d b y i n v e r t i n g t h e a m p l i t u d e d a t a a t 0.1Hz, g e n e r a t e s a c l o s e m a t c h w i t h t h e I P c u r v e . T h i s i s s h o w n f o r ( a ) P F E a n d ( b ) t h e p h a s e d a t a . Chapter 4. EM Couphng and a method for its approximate removal 98 4.5.4 S y n t h e t i c t e s t f o r t h e p o l e - d i p o l e a n d p o l e - p o l e d a t a E M coup l ing increases w i t h d ipole lengths and d ipole spacings. Here I test the E M c o u p h n g r e m o v a l for a pole-dipole and pole-pole survey geometry . T h e m o d e l i n F i g u r e 4.7(a) is used to generate the P F E and phase d a t a for a pole-d ipole survey. I n a pole-d ipo le survey one of the t r ansmi t t e r electrodes is p laced far enough f r o m the a rea o f interest so tha t i ts c o n t r i b u t i o n to the e lectr ic po t en t i a l can be neglected . In this e x a m p l e I consider a col inear 8000m t r ansmi t t e r d ipole and a 200m receiver d ipole . T h e cor rec ted P F E a n d the phase d a t a i n F i g u r e 4.10(a) and (b) respec t ive ly are i n good agreement w i t h the t rue I P responses. F o r the pole-pole survey, I consider a col inear 8000m t r a n s m i t t e r a n d receiver d ipole w i t h a separat ion of a=200m. T h e E M I P responses i n F i g u r e 4.10(c) a n d (d) shows large E M couphng c o n t a m i n a t i o n i n the da ta . T h e cor rec ted P F E a n d the phase responses i n F i g u r e 4.10(c) a n d (d) respec t ive ly i nd i ca t e tha t s ignif icant E M couphng has been r emoved f rom the da ta . T h i s example reinforces the c o m m e n t m a d e by prev ious workers to use d ipo le -d ipo le geometries to m i n i m i z e E M couphng c o n t a m i n a t i o n i n the da ta . 4.6 E M C o u p l i n g r e m o v a l i n 3 D I n p rac t i ce the d a t a are l i k e l y affected by three d imens iona l na tu re of the geologica l bodies . A l s o , f r o m the syn the t ic examples i n I D i n Sec t ion 4.5.3, i t was shown tha t the knowledge of the c o n d u c t i v i t y s t ruc ture is i m p o r t a n t to c o m p u t e the E M couphng responses. Therefore the goal here is to invest igate the v a l i d i t y of the E M couphng r emova l m e t h o d w h e n app l ied to 3 D da ta . Here I consider two examples to i l l u s t r a t e the a p p l i c a b i l i t y of the E M couphng r emova l m e t h o d i n 3 D . Chapter 4. EM Couphng and a method for its approximate removal 99 75 -70 -65 -60 -55 -50 -45 -40 -LU 35 -LL 30 -D_ 25 -20 -15 -10 -5 -0 --5 -N-Spacing 5 6 7 N-Spacing (c) EMIP' IP CORR ~i 1 1 1 1 1 r 2 3 4 5 - 6 7 400 350 300 3 250 as ^ 2 0 0 "aT 1 5 0 to g 1 0 0 C L 50 0 -50 10 - l r 3 4 N-Spacing 5 6 7 N-Spacing (d) EMIP IP C O R R i r 8 9 10 F i g u r e 4.10: E M c o u p l i n g c o r r e c t i o n s f o r t h e p o l e - d i p o l e a n d p o l e - p o l e s u r v e y g e o m e t r i e s . P F E c o r r e c t i o n s f o r ( a ) p o l e - d i p o l e a n d ( c ) p o l e - p o l e s u r v e y . P h a s e c o r r e c t i o n s f o r ( b ) p o l e - d i p o l e a n d ( d ) p o l e - p o l e s u r v e y s . Chapter 4. E M Couphng and a method for its approximate removal 100 4.6.1 E x a m p l e - 1 : C o n d u c t i v e b l o c k i n h a l f - s p a c e T h e first example is a 3 D conduc t ive and chargeable b lock i n a non-po la r i zab le half-space shown i n F i g u r e 4.11. D ipo l e -d ipo l e da t a are generated f rom a single t r an smi t t e r pos i t i on and m a n y receiver s tat ions at a frequency of 3 H z . T h e d ipole lengths of the t r an smi t t e r and receiver are equa l to 100m. T h e chargeabihty of the b lock is assigned us ing the C o l e - C o l e m o d e l w i t h the spec t ra l parameters n = 0.2, r = 0.2.sec and c = 0.5. T o a p p l y the E M c o u p l i n g r emova l m e t h o d to phase da ta , responses at o n l y a single f requency need to be generated. 40m o =0.02 S/m 120m « =0.5 S/m n = 0.2 x = 0.1 s e c c =0.5 —V 300m x = 550 200 m * x = 7 5 o F i g u r e 4.11: 3 D conduc t ive a n d po la r izab le b lock i n a non-po la r izab le half-space. T h e first step is to generate the E M I P da ta by so lv ing M a x w e l l ' s equa t ion for the e lec t r ic field i n 3 D c o m p l e x c o n d u c t i v i t y m e d i u m . T h u s I evaluate J-EM[o~{iw);uj), where o~(iuj) is the c o m p l e x c o n d u c t i v i t y a n d TEM is the fo rward m o d e l i n g EM opera tor . T h e 3 D m o d e l i n g is ca r r ied out us ing Sandia ' s 3 D staggered g r i d finite difference code ( N e w -m a n a n d A l u m b a u g h , 1995). T h e E M I P response i n F i g u r e 4.12 indica tes tha t phase Chapter 4. EM Couphng and a method for its approximate removal 101 increases with the separation between the source and receiver dipole approximately lin-early. There is a reduction in the phase value over the chargeable block but overall the phase data are dominated by E M couphng responses. Next I compute the E M couphng responses using the same forward modeling with true conductivity (T^ of the medium. This can be denoted by J-EM^OO'^}- The phase responses for the E M couphng effect in Figure 4.12 indicate a drop in value near the conductor, and in the rest of the re-gion phase increases with N-spacing. The corrected phases, obtained by subtracting the phases due to E M couphng from the EMIP phases, show a phase high in the region of chargeable block indicative of the presence of chargeability block (Figure 4.12). To generate the true IP responses for the model in Figure 4.11 it is instructive to solve eqn (4.21) in 3D at the desired frequency. I do not have an algorithm yet to do that. A n alternative approach to compute the IP responses with the 3D E M forward modeling code. Ideally if I simulate the electric fields at low enough frequency with the E M forward modeling code, then it should generate DC potentials. This principle is used to generate the IP responses. In terms of the forward modeling operator, the procedure can be described by J-EM[c(iio); UJI^]. This is executed in the following way: first the complex conductivity (i.e. the real and imaginary part of <T(ZU>)) values are generated at the frequency ui and this is fed into the E M forward modeling code. The next step is to compute the electric fields at much lower frequency than u> so that there is no contribution due to E M couphng. In essence the E M forward modeling solves the equation V.(a(iu>)W(iu>)) = —I(iw)8(r — rs). In this example the frequency used to evaluate the IP responses is 0.1 Hz. Figure 4.12 shows that the corrected phases are in good agreement with the true IP responses. It is important to note that the corrections are carried out using the true conductivity structure (cr^). Next I compute the E M couphng responses using the DC conductivity of the medium shown in Figure 4.13 that is <r0 = 0"oo(l — rj). The corrected phases shown in Figure 4.13 Chapter 4. EM Couphng and a method for its approximate removal 102 200 6 8 1 0 N-spacing 200 m Figure 4.12: E M coupling removal for phase data generated from the model in Figure 4.11. The E M couphng responses are computed using the true conductivity (Too of the medium. indicate a reasonable match with the true IP phases. Comparing the corrected phases in Figure 4.12 and 4.13, shows that the overall correction is slightly better with the true conductivity than the DC conductivity. This is in accordance with the theory. So small deviations in the conductivity from can be tolerated. To investigate large changes, I look at what happens when a half-space conductivity is used. The correction using the half-space conductivity value of the background shows under-corrections in Figure 4.14; indicating the importance of conductivity information to compute E M coupling response. Next I apply the E M couphng removal method to the PFE data. The data are generated with the same model in Figure 4.11. To generate the P F E data the EMIP Chapter 4. EM Couphng and a method for its approximate removal 103 6 8 1 0 N-spacing 40m 120 m 200 m F i g u r e 4.13: E M coup l ing remova l for phase da t a generated f r o m the m o d e l i n F i g u r e 4.11. T h e E M c o u p l i n g responses are c o m p u t e d us ing the D C c o n d u c t i v i t y a0 = <Joo(l — T/). i m p e d a n c e are c o m p u t e d at two separate frequencies of 3 H z a n d 3 0 H z . T h e observed PFE (PFEEMIP) shown i n F i g u r e 4.15 are ob ta ined us ing the def in i t ion i n eqn (4.10) where u>2 = 3 0 H z and ui, = 3 H z . T h e increase i n the PFEEMIP w i t h d ipole spac ing is i n d i c a t i v e o f E M couphng con t amina t i on . T o correct for the E M c o u p h n g c o n t r i b u t i o n i n P F E E M I F , the i m p e d a n c e due to E M effects are c o m p u t e d us ing the t rue c o n d u c t i v i t y ((Too) s t ruc tu re at 3 0 H z a n d 3 H z . T h e correc ted P F E da ta , c o m p u t e d us ing eqn (4.28) are shown i n F i g u r e 4.15. T h e y are i n good agreement w i t h the t rue I P responses. T h e m e t h o d used to c o m p u t e the t rue P F E responses due to the I P effect is s i m i l a r to the one used to c o m p u t e the t rue I P phases. Chapter 4. EM Couphng and a method for its approximate removal 104 200 150 Tn 100 CC E , CU CO CO -c= 50 o_ 0 -50 120 m 200 m Figure 4.14: E M coupling removal for phase data generated from the model in Figure 4.11. The E M couphng responses are computed using the half-space background conductivity. 4.6.2 Example-2: Conductive block in a layered medium Here I consider another example to illustrate the E M couphng removal in 3D. The model is a conductive and chargeable block in a non-polarisable three-layered medium with the conductivity of the layers decreasing with depth shown in Figure 4.16. For this example I consider the E M couphng removal for the phase data at 10Hz. The EMIP phases shown in Figure 4.17 are generated using the model in Figure 4.16 at 10Hz. The E M I P phases indicate a linear increase of phase with N-spacing and this is indicative of significant E M couphng contamination. There is a drop in phase values in the region above the anomalous block. The true IP responses, shown in Figure 4.17, are generated using the Chapter 4. EM Couphng and a method for its approximate removal 105 2 4 6 8 1 0 1 2 1 4 N-spacing I " x z 4C m 120 m 200 m Figure 4.15: E M coupling removal for P F E data generated from the model in Figure 4.11. The PFEEMIP is generated by computing the amplitude of the impedance at two frequencies (30 Hz and 3 Hz) for a dipole-dipole geometry with dipole spacing of 100m. procedure described in Section (4.6.1). It indicates a large anomaly of 80 mrads over the chargeable block. The phases due to E M couphng responses shown in Figure 4.17(a) are computed using the true conductivity structure. The corrected phases shown in Figure 4.17(b) show good agreement with the IP phases. The corrections with the DC conductivity are shown in Figure 4.18. The corrected IP responses in Figure 4.18(b) are somewhat under-corrected. However, a significant amount of E M couphng is removed from the data enhancing the signal to noise ratio. If a simple baseline correction is carried out with the E M I P curve in Figure 4.19(a), it will result in a negative IP response. However, this negative feature observed in Chapter 4. EM Coupling and a method for its approximate removal 106 -50,0) (50,0) o =0.02 S/m 30m a = 0.01 S/m r,= 0.5 S/m l 170m r = 0.3 j t =0.016 sec , c =0.5 _ 4 0 0 m x = 700nrT 300m ; = 1 0 0 0 m a =0.001 S/m F i g u r e 4.16: 3 D conduc t ive b lock i n a layered ear th the i n i t i a l E M I P phase plot i n F i g u r e 4.19(a) is conver ted to a pos i t ive I P response. T o inves t iga te whether the backg round m o d e l is adequate for E M coup l ing r emova l , I c o m p u t e the E M responses us ing the three-layer m o d e l . F i g u r e 4.19 shows the result of us ing on ly b a c k g r o u n d c o n d u c t i v i t y i n fo rma t ion . T h e correc ted response shown i n F i g u r e 4.19(b) indica tes negat ive I P phases over the po la r izab le body, that can lead to, w r o n g in t e rp re t a t i on of the da ta . T h i s example shows, tha t the b a c k g r o u n d m o d e l is grossly inadequa te to do the correct ions and reinforces the necessity of appropr ia te c o n d u c t i v i t y i n f o r m a t i o n to do the E M couphng removal . 4.7 E M c o u p l i n g i n t o p o g r a p h i c t e r r a i n s I n m a n y surveys the flat ear th geomet ry m a y not be v a l i d w h e n significant e levat ion dif-ferences are present a long a profile. These topograph ic features u l t i m a t e l y affect the d a t a tha t are col lec ted on these terrains . T h e goal here is to s tudy the effect of t opography on Chapter 4. EM Couphng and a method for its approximate removal 1 0 7 F i g u r e 4.17: E M c o u p l i n g r e m o v a l f o r p h a s e d a t a g e n e r a t e d f r o m t h e m o d e l i n F i g u r e 4.16. T h e r e s p o n s e s a r e c o m p u t e d f o r a d i p o l e - d i p o l e g e o m e t r y w i t h d i p o l e s p a c i n g o f 1 0 0 m . ( a ) E M I P p h a s e s s h o w n b y s o l i d l i n e s a n d E M p h a s e s ( d a s h e d l i n e s ) a r e c o m p u t e d u s i n g (Too a t 10 H z . ( b ) T h e t r u e I P ( s o l i d l i n e ) a n d c o r r e c t e d I P r e s p o n s e s ( d a s h e d l i n e s ) . Chapter 4. EM Coupling and a method for its approximate removal 108 Figure 4.18: E M coupling removal for phase data generated from the model in Figure 4.16. The responses are computed for a dipole-dipole geometry with dipole spacing of 100m. (a) EMIP phases are shown by solid lines. The E M phases (dashed lines) are computed using crdc at 10 Hz. (b) The true IP (solid line) and corrected IP responses (dashed lines). Chapter 4. EM Couphng and a method for its approximate removal 109 300m F i g u r e 4.19: E M c o u p l i n g r e m o v a l f o r p h a s e d a t a g e n e r a t e d f r o m t h e m o d e l i n F i g u r e 4.16. T h e r e s p o n s e s a r e c o m p u t e d f o r a d i p o l e - d i p o l e g e o m e t r y w i t h d i p o l e s p a c i n g o f 1 0 0 m . ( a ) E M I P p h a s e s a r e s h o w n b y s o l i d l i n e s . T h e E M p h a s e s ( d a s h e d l i n e s ) a r e c o m p u t e d u s i n g a I D b a c k g r o u n d ( l a y e r e d ) c o n d u c t i v i t y s t r u c t u r e a t 10 H z . ( b ) T h e t r u e I P ( s o l i d h n e ) a n d c o r r e c t e d I P r e s p o n s e s ( d a s h e d l i n e s ) . Chapter 4. EM Coupling and a method for its approximate removal 110 the E M couphng responses. Before going into the application of the E M couphng removal method, I examine the E M couphng response from a conductive (non-polarizable) half-space, with and without, topography for simple topographic terrains. This can perhaps give some insight into, how topography affects the E M couphng responses. The models used to compute the E M couphng responses are shown in Figure 4.20. In each case, the response from a flat earth is compared with that of a topographic terrain. Figure 4.20 indicates that the discrepancy between the E M couphng response for a terrain with, and without, topography increases with frequency. It will also occur when the conductivity increases. This is expected because at higher frequencies and larger conductivities, the E M couphng effect becomes dominant. Thus with the increase in frequency and/or con-ductivity of the medium, the effect of topography becomes important in the IP data. However at low frequencies the effect of topography on IP data is not significant. Next I consider the E M couphng removal problem for a topographic terrain. The model shown in Figure 4.21, is a conductive polarizable prism buried beneath a hill which sits on a uniform half-space. The EMIP phases shown in Figure 4.22(a) are generated at 3.0 Hz. The E M coupling responses, indicated by E M in Figure 4.22(a), are computed at 3 Hz using the true conductivity structure with topography. The true IP responses, shown in Figure 4.22(b), are computed using the same methodology described in Section (4.6.1). The corrected phases obtained by subtracting the E M phases from E M I P phases are in good agreement with the true IP phases in Figure 4.22(b). To iUustrate the importance bf topography I re-examine the problem with a flat earth model. The E M couphng phases are computed using the same conductivity values used in Figure 4.21 but without the topography. The E M responses computed using this model is shown in Figure 4.23(a). The corrected response in Figure 4.23(b) indicate over-corrections at large N-spacings resulting in negative IP phases. Note that although the right conductivity values are used in the computation of E M responses, the effect of topography plays a significant Chapter 4. EM Couphng and a method for its approximate removal 111 F i g u r e 4.20: E M c o u p l i n g p h a s e s f o r t o p o g r a p h i c t e r r a i n s a r e s h o w n b y d a s h e d l i n e s . C o n d u c t i v i t y o f t h e e a r t h i s 0.02 S/m. T h e p h a s e d a t a f r o m t h e f l a t e a r t h m o d e l a r e s h o w n b y a s o l i d l i n e f o r ( a ) d o w n s l o p e ( b ) u p s l o p e ( c ) v a l l e y a n d ( d ) a h i l l g e n e r a t e d a t 3 a n d 10 H z . T h e p o s i t i o n o f t h e t r a n s m i t t e r i s i n d i c a t e d b y T X . T h e d a t a a r e f r o m a d i p o l e - d i p o l e s u r v e y w i t h d i p o l e l e n g t h o f 1 0 0 m . Chapter 4. EM Couphng and a method for its approximate removal 112 role. This example again re-inforces the need to know the conductivity structure to compute E M couphng effects. 100m 250m * " TX if • 140m 550m a = 0.02 S/m 200m Y = 0 120m a =0.5 S/m ^=0.2 x = 0.1 sec 0.5 200m 200m Figure 4.21: The model used to test the E M couphng removal method for phase data on a topographic terrain. 4.8 R e m o v a l m e t h o d o l o g y i n 2 D : A n a p p r o x i m a t e m e t h o d In practice IP and DC resistivity data are acquired simultaneously in a complex resistivity survey. Both amplitude and phase of the voltage are measured at each location over a frequency range. The amplitude at the lowest frequency is converted to apparent resistivity data. The IP data are either the P F E responses generated from the amplitude data at two frequencies, or the phase difference between the transmitted current and the Chapter 4. EM Couphng and a method for its approximate removal 113 200 (a) -o 100 w EM -50 200 150 H -o 100 CD j§ 50 — 1 1 1 1 1 1 1 — 2 4 6 8 10 12 14 N-spacing ^ 0 + -CORR -50 i 1 1 1 — 8 10 12 14 N-spacing Figure 4.22: E M coupling removal for phase data on a topographic terrain, (a) The EMIP response is shown by solid line. The E M response is calculated with the true conductivity structure is shown by dashed line, (b) The true IP response (solid line) and the corrected response (dashed line). Chapter 4. EM Couphng and a method for its approximate removal 114 2 4 6 8 10 12 14 N-spacing ( b) -o 100 F i g u r e 4.23: E M coup l ing r emova l for phase da t a o n a topograph ic t e r ra in , (a) T h e E M I P response is shown by sol id l ine . T h e E M response shown by dashed l ine is ca l cu la t ed w i t h the flat ea r th c o n d u c t i v i t y m o d e l , (b) T h e t rue I P response (sol id l ine) a n d the cor rec ted response (dashed l ine) . Chapter 4. EM Couphng and a method for its approximate removal 115 receiver voltage at a single frequency. For a 2D survey the data are generally collected using a pole-dipole or dipole-dipole array geometries. For interpretation purposes the apparent resistivity data at the lowest frequency are inverted to recover a 2D conductivity structure (T(X, Z) and that conductivity is then used to generate the sensitivity for the IP inversion problem (Oldenburg and L i , 1994). The IP data are then inverted to recover chargeability. Our goal in this work is to remove the effects of E M couphng and obtain a response that can be inverted for a chargeability structure of the subsurface. This requires the computation of the E M couphng responses ZEM(iu/) with a known conductivity struc-ture (Too a t the desired frequency u>. From the previous section it was evident that the conductivity information is vital to the computation of the couphng response. So using a simple homogeneous earth model, or a single layered earth model, might lead to an inappropriate correction of the couphng contaminated data. Therefore an approximate technique is used which utilizes the ID assumption but also takes into account the lateral variation in conductivity. The 2D conductivity structure obtained from the inversion is averaged over a region which spans a particular transmitting dipole and the associated receiving dipoles. The averaging is carried out in the horizontal direction such that for each transmitter position Tj I obtain an averaged conductivity structure o-j(z) shown in Figure 4.24. The thickness of the layers are equal to the vertical dimensions of the rectangular cells used in the 2D inversion. Since the conductivity can vary over large orders of magnitude the logarithm of conductivity is averaged in each layer. This is given by k=l where crlJvg is the averaged conductivity of the ith layer in the ID conductivity model for Chapter 4. EM Coupling and a method for its approximate removal 116 2D M O D E L -4 Ncells 1D M O D E L F i g u r e 4.24: I D l o c a l average of 2 D c o n d u c t i v i t y m o d e l . T h e 2 D c o n d u c t i v i t y is averaged over a region w h i c h spans a pa r t i cu l a r t r ansmi t t e r and associated rece iv ing dipoles . T h e c o n d u c t i v i t y values of the cells spanning this region are l o g a r i t h m i c a l l y averaged i n the h o r i z o n t a l d i r ec t ion to p roduce I D c o n d u c t i v i t y m o d e l . Chapter 4. EM Couphng and a method for its approximate removal 117 the jih transmitter position, k denotes the number of cells in the horizontal direction. The cells included in the average extend from the transmitting dipole to the farthest receiver dipole. The averaging process is then rolled along as the transmitter moves to the next position. Therefore for N transmitter positions I have N ID conductivity models. A ID forward modeling is carried out to compute the E M coupling response. First the electric field is computed using a point dipole on a layered earth at the desired frequency using eqn(4.30) assuming the medium is not chargeable. The impedance ZEM(iu) is then computed using eqn(4.9). For P F E data the amplitude |Z £ M ( zo ; ) | is used to correct the observed responses given in eqn(4.26) and the phases are corrected using eqn(4.18). In the next step the corrected responses are inverted using a hnearized 2D IP inversion algorithm. This is applicable when the chargeabihty effects are considered small so that hnearized methods can be used. The derivation for the hnearized system for the P F E responses and the phase responses are shown in appendix-A. The inverse problem is formulated following Oldenburg and L i (1994) to recover 2D chargeabihty distribution. The inverse problem is solved by minimizing a model objective function 4>m subject to the data constraints <pd = tfd*9^'• This is mathematically represented by min <f>m= \\Wfjirj-rj0)\\2 (4.32) subject to <f>d = 1  Wd (d°b> - Jrj) 112 = ^ r 9 e t where rj is the chargeabihty either in P F E (for amphtude data) or phase, d°ha is the vector of observed data and J is the sensitivity matrix derived from the conductivity structure. Wd is the data weighting matrix containing the standard deviations of each datum and is the model weighting function. A global objective function given by <j)(m) = 4>{m) + p{4>d ~ (j>dar9et) is minimized, where a is the Lagrange parameter. The system of equations obtained by minimizing the global objective function is solved using a subspace method with positivity (Oldenburg and L i , 1994). Chapter 4. EM Coupling and a method for its approximate removal 118 4.9 F i e l d E x a m p l e s T h e E M couphng remova l a l g o r i t h m developed i n Sec t ion (4.8) is app l i ed to f ield d a t a sets f r o m A u s t r a l i a . T h e first example is f rom A c r o p o l i s area i n Sou the rn A u s t r a l i a and the second example is f rom the E l u r a ore deposit f rom C o b a r area i n N e w S o u t h Wales shown i n F i g u r e 4.25. Melbourne F i g u r e 4.25: M a p of A u s t r a l i a . T h e E M coup l ing con t amina t ed I P d a t a sets used i n this w o r k are f r o m A c r o p o l i s area i n Sou the rn A u s t r a l i a and E l u r a ore deposit f rom C o b a r area i n N e w S o u t h Wales shown as stars. 4.9.1 D a t a f r o m A c r o p o l i s Here I test the E M couphng r emova l m e t h o d on a field da t a set ob ta ined f r o m a m i n e r a l e x p l o r a t i o n survey i n the A c r o p o l i s area near O l y m p i c D a m i n Sou the rn A u s t r a l i a . T h e Chapter 4. EM Couphng and a method for its approximate removal 119 O l y m p i c D a m ore deposi t is C u - U - A u deposi t , whereas the A c r o p o l i s area does not have any significant mine ra l i s a t i on . H o w e v e r the I P d a t a co l lec ted i n th is region were severely c o n t a m i n a t e d b y E M couphng p r i m a r i l y because of the large d ipole spac ing used i n the survey and the conduc t ive na ture of the g round . T h e region has a great var ie ty o f g ran i t i c , h e m a t i t i c a n d siliceous breccias . T h e da t a were acqu i red us ing a d ipo le -d ipo le ar ray w i t h a d ipole l eng th of 600 m . T h e l eng th of the survey l ine was 18 k m a n d 26 current e lect rode loca t ions were used. T h e r e are three dr i l l -hole geologic logs a long this l ine . T h e d a t a were co l lec ted at several frequencies w i t h a lowest f requency of 0.125 H z . T h e apparent res i s t iv i ty pseudosect ion at 0.125 H z (F igu re 4.26(a)) is i n v e r t e d us-i n g a 2 D D C res i s t iv i ty invers ion a l g o r i t h m . T h e 2 D c o n d u c t i v i t y is shown i n F i g u r e 4.26(b) . T h e c o n d u c t i v i t y s t ruc ture indica tes a t h i n resis t ive cover o v e r l y i n g a back-g r o u n d c o n d u c t i v i t y of 20 m S / m (or 50 o h m - m ) . Sens i t iv i t ies for the I P inve r s ion are genera ted us ing the 2 D c o n d u c t i v i t y s t ruc tu re i n F i g u r e 4.26(b) . T h e pseudosec t ion of the observed phase at 0.125 H z is shown i n F i g u r e 4.27(a). Phases up to 125 m r a d s are observed i n the da ta . These d a t a were i n p u t to the I P inve r s ion a l g o r i t h m a n d the recovered 2 D chargeab i l i ty s t ruc ture r](x,z) is shown i n F i g u r e 4.27(b) . T h e m o d e l i n d i -cates h i g h chargeable zones near x = 4 0 0 0 m and x = 1 2 0 0 0 m i n the d e p t h range o f 1.5-2.0 k m . T h e h i g h recovered chargeabi l i ty , of the order of 200 mrads , suggests tha t there is a ma jo r c o n t r i b u t i o n due to E M couphng . T h i s is we l l e x h i b i t e d by the l i n e a r l y inc reas ing t r e n d of the observed phases w i t h N - s p a c i n g i n the undecoup led pseudosec t ion shown i n F i g u r e 4.27(a). T h e c o n d u c t i v i t y s t ruc ture ob ta ined i n F i g u r e 4.26(b) is used to generate the I D av-eraged c o n d u c t i v i t y models . T w e n t y - s i x I D averaged c o n d u c t i v i t y mode ls are genera ted for 26 t r a n s m i t t e r loca t ions us ing the mu l t i - l aye red averaging m e t h o d s h o w n i n F i g u r e 4.24. T h e phases due to E M coup l ing , shown i n F i g u r e 4 .28(b) , are c o m p u t e d at 0.125 Chapter 4. EM Coupling and a method for its approximate removal 120 (b) Conductivity Section ) 4 0 0 0 8000 Distance (m 12000 16000 Log 1 0 cj (S/m) -2.10 -1.99 -1.8 -1.77 -1.66 -1.55 -1.45 -1.34 -1.23 -1.12 -1.01 F i g u r e 4.26: (a) T h e observed apparent c o n d u c t i v i t y pseudosect ion at 0.125 H z for the d ipo le -d ipo le survey, (b) T h e recovered c o n d u c t i v i t y m o d e l after invers ion H z us ing the I D averaged models . T h e y ind ica te tha t E M couphng is a ma jo r con t r i bu -t i o n to observed phases. C o m p a r i n g the observed phases i n F i g u r e 4.28(a) w i t h the E M coup l ing phases i n F i g u r e 4.28(b) i t is seen that b o t h exh ib i t an increas ing l inear t r e n d w i t h N - s p a c i n g . T h e a m p l i t u d e of the phases suggests tha t the I P phase is d o m i n a t e d by the E M coup l ing con t r i bu t i on . T h e observed phases i n F i g u r e 4.28(a) are correc ted by sub t r ac t ing the phase due to E M couphng i n F i g u r e 4.28(b). T h i s removes the l inear t r e n d and reduces the da t a to Chapter 4. EM Couphng and a method for its approximate removal 121 (a) Apparent Chargeability Pseudosection 6 1 1 1 = 1 1 1 0 4 0 0 0 8 0 0 0 1 2 0 0 0 1 6 0 0 0 Distance (m) mrads 0.0 1 2 . 5 2 5 . 0 3 7 . 5 5 0 . 0 6 2 . 5 7 5 . 0 8 7 . 5 1 0 0 . 0 1 1 2 . 0 1 2 5 . 0 (b) Q 2 0 0 0 Chargeability Section 0 i 4 0 0 0 1 1 8 0 0 0 1 2 0 0 0 Distance (m) I 1 6 0 0 0 I—I— mrads 2 2 4 4 66 88 1 1 0 1 3 2 1 5 4 1 7 6 1 9 8 2 2 0 F i g u r e 4.27: (a) T h e observed pseudosect ion for the undecoup led phase for the d i -pole-d ipole a r ray at 0.125 H z . (b) T h e recovered 2 D chargeabi l i ty m o d e l after invers ion . about one - th i rd of thei r o r ig ina l magn i tude . T h e correc ted phases i n F i g u r e 4.29(a) are i nve r t ed us ing the 2 D I P invers ion a l g o r i t h m and the recovered chargeabi l i ty is shown i n F i g u r e 4 .29(b) . Signif icant differences are observed between the models i n F i g u r e 4.27(b) and 4.29(b) . O v e r a l l , h igh chargeabi l i ty regions for the decoupled d a t a have a lower a m p l i t u d e and are closer to the surface t h a n the h i g h chargeabi l i ty regions generated b y the undecoup led da ta . T h e borehole locat ions are i nd i ca t ed on the recovered sect ion i n F i g u r e 4.29(b) . T h e Chapter 4. EM Couphng and a method for its approximate removal 122 (a) Observed Undecoupled Pseudosection 0.0 12.5 25.0 37.5 50.0 62.5 75.0 87.5 100.0 112.0 125.0 (b) Computed EM Coupling Response Distance (m) | | | | | mrads 0.0 12.5 25.0 37.5 50.0 62.5 75.0 87.5 100.0 112.0 125.0 F i g u r e 4.28: T h e compar i son of the undecoup led phase shown i n (a) a n d E M c o u p l i n g phase c o m p u t e d us ing our m e t h o d ( shown i n (b)) . T h e l inear t r e n d of the increas ing phase w i t h N - s p a c i n g is a character is t ic of E M coup l ing c o n t a m i n a t i o n . borehole loca t ions are i n d i c a t e d on the recovered sect ion i n F i g u r e 9(b) . U n f o r t u n a t e l y no geophys ica l logging is avai lable and so the i n fo rma t ion is o n l y f rom geologic logging . T h e r e is however a qua l i t a t ive agreement between the logs and the invers ion results . O f the three borehole logs, B H 6 has the largest ind ica tors of chargeable mater ia l s and i t coincides w i t h region o f highest chargeabih ty p red ic t ed f rom the inve r s ion . T h e B H 6 log indica tes a l tered and ve ined granites tha t can be po la r izab le . T h e log also indica tes cha lcopyr i t e and p y r i t e veins w i t h some d i ssemina t ion ex tend ing f rom 600-720m. B H 1 coincides w i t h a flank of a chargeable h igh seen i n the invers ion . T h e B H 1 indica tes Chapter 4. EM Couphng and a method for its approximate removal 123 D e c o u p l e d P s e u d o s e c t i o n BH1 B H 9 B H 6 0 4000 8000 12000 16000 Distance (m) m r a d s 0.0 3.1 6.2 9.3 12.4 15.5 18.6 21.7 24.8 27.9 31.0 I n v e r t e d M o d e l v°> BH1 B H 9 BH6 2500 0 4000 8000 12000 Distance (m) 16000 m r a d s 0.0 3.1 6.2 9.3 12.4 15.5 18.6 21.7 24.8 27.9 31.0 F i g u r e 4.29: (a) T h e correc ted pseudosect ion ob ta ined by sub t r ac t i ng the E M coup l ing phase f rom the undecoup led phase, (b) T h e recovered chargeabi l i ty m o d e l ob t a ined after inve r s ion of cor rec ted da ta . hemat i t e -magne t i t e breccias ( massive i r o n oxides ) w h i c h are general ly po la r i zab le a n d have low res is t iv i t ies depend ing on the degree of a l te ra t ion . It also indica tes the presence of cha l copyr i t e i n vugs a long w i t h hemat i t e -magne t i t e breccias at dep th o f 720-950m. T h e B H 9 log coincides w i t h a region of no chargeabi l i ty i n F i g u r e 4.29(b) . T h i s seems to be consis tent w i t h the log i n f o r m a t i o n w h i c h indica tes felsic volcanics w i t h sparse m i n e r a l i z a t i o n . Felsic volcanics i n this region t end to have low p o l a r i z a b i l i t y c o m p a r e d to hemat i t e -magne t i t e breccias . In s u m m a r y a l l three logs ind ica t e m i n o r d i s semina t ion bu t none of t h e m con ta in significant sulfides. It is l i ke ly tha t the p o l a r i z a b i l i t y m a y be Chapter 4. EM Couphng and a method for its approximate removal 124 caused by a l t e ra t ion . T h e a l te ra t ion produc ts seen i n the geological logs ind ica t e tha t B H 6 , B H 1 and B H 9 might progressively be associated w i t h lower amount of chargeable m a t e r i a l . T h i s conjecture is i n accordance w i t h the invers ion result , bu t b e y o n d this ra ther weak qua l i t a t ive agreement there is l i t t l e tha t can be said. 3 point Decoupled Pseudosection ( A ) BH1 B H 9 B H 6 •P. 3 I \ :'~ => C3t 7 4000 8000 12000 Distance (m) 3.1 6.2 9.3 12.4 15.5 18.6 21.7 24.8 27.9 Inverted Model 8000 12000 Distance (m) 16000 0.0 3.1 6.2 9.3 12.4 15.5 18.6 21.7 24.8 27.9 31.0 F i g u r e 4.30: (a) T h e correc ted phase pseudosect ion ob ta ined by a three-point decoup l ing m e t h o d , (b) T h e recovered chargeabi l i ty m o d e l ob ta ined after inve r s ion of three-point decoupled da ta . N e x t I compare the results w i t h the s t andard three-point decoupl ing a n d m a n u a l decoup l ing me thods ob ta ined f rom cont rac tor . T h e decoupled da t a us ing a three-point m e t h o d is shown i n F i g u r e 4.30(a). T h e l inear t r e n d is s t rongly ev ident . T h e recovered m o d e l , shown i n F i g u r e 4.30(b) , exh ib i t s h igh chargeabi l i ty be low a d e p t h of 1.2 k m a n d Chapter 4. EM Coupling and a method for its approximate removal 125 at the surface. T h i s is not subs tan t ia ted by the borehole logs. T h e correc ted phases us ing the m a n u a l decoupl ing technique (p ropr ie ta ry to the cont rac tor ) are shown i n F i g u r e 4.31(a). T h e recovered m o d e l , F i g u r e 4.31(b) , indica tes low chargeabih ty near B H 6 w h i c h is not i n accordance w i t h the wel l log da ta . (a) Manual Decoupled Pseudosection BH1 BH9 TO O - . t o 4 / / -:'.^ > \ - . • ) 4000 8000 12000 16000 Distance (m) 0.0 3.1 6.2 9.3 12.4 15.5 18.6 21.7 24.8 27.9 31.0 Inverted Model (b) BH1 BH9 0 500 "g" 1000 ~= 1500 '—i 0 4000 8000 12000 Distance (m) 1 16000 0.0 3.1 6.2 9.3 12.4 15.5 18.6 21.7 24.8 27.9 31.0 F i g u r e 4.31: (a) T h e correc ted phase pseudosect ion ob ta ined by m a n u a l decoup l ing m e t h o d , (b) T h e recovered chargeabihty m o d e l ob ta ined after invers ion of m a n u a l de-coupled da ta . I have shown i n an earl ier sect ion of this C h a p t e r tha t h a v i n g a good es t imate of c o n d u c t i v i t y is c r u c i a l to the success of E M couphng remova l . I n order to examine the i m p o r t a n c e of c o n d u c t i v i t y i n fo rma t ion for the field example , I have considered two differ-ent c o n d u c t i v i t y models . T h e first m o d e l is a homogeneous half-space w i t h c o n d u c t i v i t y Chapter 4. EM Coupling and a method for its approximate removal 126 of 20 mS/m. This is about the average value of the conductivity in Figure 4.26(b). The phase due to E M couphng computed at 0.125 Hz is subtracted from the measured re-sponse to obtain the corrected phases. The recovered model is shown in Figure 4.32(a). The amplitude of the recovered chargeability is 60 mrads and depth range of anomalous chargeable zone is about 1.0-1.5 km. In the second model I decrease the conductivity to 10 mS/m. The corrected phases are then inverted and the recovered chargeability model is shown in Figure 4.32(b). The recovered model indicates the presence of high chargeable zones at depth of 1.5-2.0 km and the structure of the chargeability zone is similar to the model obtained for undecoupled inversion in Figure 4.27(b). Decreasing the conductivity by a factor of two results in a factor of three increase in the recovered phases. This shows that accurate conductivity information is necessary to compute the response due to E M couphng at a desired frequency and reinforces the conclusion made in Section 4.5. E M couphng effects increase with frequency and can swamp the IP signal in the data. Here I apply the method to higher frequency data from the same survey. The phase data collected at 1Hz along this line are shown in Figure 4.33(a). They show a linearly trend and achieve values of 700 mrads for N=6. The inversion in Figure 4.33(b) of undecoupled data shows very high chargeability of 850-950 mrads in the depth range of 1.5-1.7km. The observed data are corrected by computing the E M coupling phase at 1Hz with the conductivity obtained from the 2D DC inversion. The corrected data are shown in Figure 4.34(a) and the recovered model from the inversion is presented in Figure 4.34(b). The maximum amplitude of recovered chargeabil-ity is 70-80 mrads and the structural location of the anomaly is in agreement with the drill-logs and the model obtained in Figure 4.29(b). This example illustrates that the decoupling method presented here is reasonably robust even when E M couphng is severe. Chapter 4. EM Couphng and a method for its approximate removal 127 I n v e r s i o n w i t h CT=0.02 S /m (a) B H 1 B H 9 B H 6 I I , 0 500 -"E" 1000 -f j . 1500 Q 2 0 0 0 -2 5 0 0 --I 1 1 ' I 0 4 0 0 0 8 0 0 0 12000 16000 Distance (m) mrads 0 8 16 24 32 4 0 48 56 64 72 80 I n v e r s i o n w i t h cr=0.01 S /m ( b ) B H 1 B H 9 B H 6 0 4 0 0 0 8 0 0 0 12000 16000 Distance (m) mrads 0 18 36 54 72 90 108 126 144 162 180 F i g u r e 4.32: T h e E M coup l ing phase is c o m p u t e d us ing a homogeneous half-space m o d e l a n d the cor rec ted da t a are inve r t ed us ing the 2 D I P invers ion code, (a) T h e recovered m o d e l after the cor rec t ion is ca r r ied out w i t h the c o n d u c t i v i t y of 0.02 S / m . (b) T h e recovered chargeabi l i ty m o d e l us ing c o n d u c t i v i t y of 0.01 S / m . 4.9.2 Data from Elura Ore Deposit In this e x a m p l e the E M couphng remova l m e t h o d is app l i ed to a da t a set col lec ted at E l u r a ore deposi t near C o b a r area i n N e w S o u t h Wales , A u s t r a l i a . I t is a Z n - P b - A g deposi t s i tua ted 43 K m n o r t h west of C o b a r . T h e ore deposit has been ex tens ive ly s tud ied i n the past b o t h f rom geological and geophys ica l perspect ive . E l u r a is a dense, conduc t ive , mode ra t e ly polar isable ore b o d y w i t h a h igh ly magne t i c core l y i n g under a th i ck conduc t ive layer ( E m e r s o n , 1980). Geo log ica l i n fo rma t ion indica tes tha t the ore Chapter 4. EM Couphng and a method for its approximate removal 128 Undecoupled Pseudosection at 1Hz BH1 B H 9 B H 6 J L 8000 12000 Distance (m) 70 140 210 280 350 420 490 560 630 700 Inverted Model at 1 Hz BH1 B H 9 J L 0 4000 1 ~n 8000 12000 Distance (m) 16000 100 200 300 400 500 600 700 800 900 1000 F i g u r e 4.33: (a) T h e pseudosect ion of the undecoup led phases at 1 H z . (b) T h e recovered m o d e l after inve r s ion of undecoup led d a t a at 1 H z . b o d y is a v e r t i c a l p ipe- l ike body . T h e mine ra l i z a t i on i n the ore b o d y is of three types . T h e outer cover is a siliceous ore, fol lowed by massive p y r i t i c ore and inner core is a mass ive p y r r h o t i t e ore ( A d a m s a n d S c h m i d t , 1980). T h e goal here is to inves t igate the u t i l i t y of the r emova l m e t h o d to the E M coup l ing c o n t a m i n a t e d d a t a tha t were acqu i red i n this area. I consider phase d o m a i n da t a acqu i red along the l ine 50800N f rom the pub l i shed l i t e ra tu re ( S m i t h , 1980). D a t a were co l lec ted us ing a d ipo le -d ipo le array w i t h d ipole l eng th of 100m and N = 1 — 6 us ing the G D P 12 in s t rumen t of Zonge Eng inee r ing . D a t a were acqu i red at four Chapter 4. EM Coupling and a method for its approximate removal 129 Decoupled Pseudosection at 1 Hz ( a ) B H 1 B H 9 B H 6 mrads 0 2 8 5 6 8 4 1 1 2 1 4 0 1 6 8 1 9 6 2 2 4 2 5 2 2 8 0 Inverted Model at 1 Hz ( b ) B H 1 B H 9 B H 6 Q 2 0 0 0 -2 5 0 0 0 4 0 0 0 8 0 0 0 1 2 0 0 0 1 6 0 0 0 Distance (m) mrads F i g u r e 4.34: (a) T h e pseudosect ion of the decoupled phases at 1 H z us ing our m e t h o d , (b) T h e recovered chargeabi l i ty m o d e l ob ta ined after inve r s ion o f decoup led d a t a at 1 H z . different frequencies (0.125, 0.25, 0.5 and 1.0 H z ) . T h e apparent r e s i s t iv i ty d a t a co l lec ted at 0.125 H z i n F i g u r e 4.35(a) were inve r t ed us ing a 2 D D C res i s t iv i ty inve r s ion a l g o r i t h m . T h e recovered c o n d u c t i v i t y m o d e l i n F i g u r e 4.35(b) indica tes a conduc t ive ove rbu rden o f 5-10 o h m - m . T h e r e is also an i n d i c a t i o n of a conduc t ive a n o m a l y between x = 6 5 0 m and 800m. T h i s a n o m a l y coincides w i t h the l oca t ion of the E l u r a ore deposi t . T h e undecoup led phase d a t a at 0.125, 0.25, 0.5 a n d 1.0 H z shown i n F i g u r e 4.36(a)-(d) were i nve r t ed for a 2 D chargeabi l i ty s t ruc ture . T h e recovered models at four frequencies are shown i n F i g u r e 4.37(a)-(d). A l l the models ind ica te a chargeable a n o m a l y be tween Chapter 4. EM Coupling and a method for its approximate removal 130 Pseudo Section (a) 400 800 Distance (m) 1200 1600 -1.7 -1.6 -1.5 -1.4 -1.3 -1.2 -1.1 -1.0 -0.9 -0.8 -0.7 Log1 0CT a (S/m) Conductivity Section (b) -2.5 -2.3 -2.1 -1.9 -1.7 -1.5 -1.3 -1.1 -0.9 -0.7 -0.5 Log 1 0o- (S/m) F i g u r e 4.35: (a) T h e pseudosect ion of the apparent c o n d u c t i v i t y (b) T h e recovered con-d u c t i v i t y m o d e l ob ta ined after invers ion . x = 6 5 0 a n d 800m. However there are several differences be tween t h e m . T h e m a g n i t u d e of recovered chargeabih ty increases w i t h frequency, w h i c h is an i n d i c a t i o n of E M couphng c o n t a m i n a t i o n i n the da ta . T h e m a x i m u m recovered a n o m a l y at 0 .125Hz is 120 mrads c o m p a r e d to 800 mrads at 1 H z . T h e dep th of the recovered chargeable zone is deeper for the h igher frequencies ( 450m for 1Hz data) c o m p a r e d to lower frequencies (250m for 0.25 H z da ta ) . These character is t ics of h igh chargeabi l i ty and deeper anomal ies were also observed w h e n i nve r t i ng undecoup led da t a for the A c r o p o h s example i n F i g u r e 4.27(b) . N e x t I remove E M coup l ing f rom the da ta . T h e decoupled phases at four frequencies Chapter 4. EM Coupling and a method for its approximate removal 131 are presented in Figure 4.38(a)-(d). Comparison of the decoupled phases with the un-decoupled phases in Figure 4.36(a)-(d), suggests that the shapes of the anomaly pattern are similar but the magnitudes of the anomaly are reduced. The decoupled phases in Figure 4.38 are inverted for 2D chargeabihty structure. The models shown in Figure 4.39(a)-(d) indicates that both lateral and vertical depth extents of the chargeable zones are consistent at all four frequencies. The magnitude of the recovered anomaly at 0.125, 0.25 and 0.5 Hz are in the range of 75-85 mrads and 160 mrads for the 1Hz data. The chargeabihty recovered in Figure 4.37(c) and (d) obtained from undecoupled phases at 0.5 Hz and 1Hz is 5 times the magnitude recovered in Figure 4.39(c) and (d). This illus-trates that significant amount E M coupling has been removed from the data at higher frequencies. The recovered anomalies are in good agreement with the location of the ore deposit. 4 . 1 0 C o n c l u s i o n s E M couphng has been a long standing problem in IP data interpretation because the conventional forward modeling to generate IP responses does not take into account the inductive effects of the ground. There can be two approaches to solve the problem: (a) Use a forward modeling that models both IP and the E M coupling effects or (b) remove E M couphng responses from the data and use the traditional DC resistivity formulation to model the IP effects. The first approach requires the solution of Maxwell's equations for electric fields due to finite-length grounded wires; this is computationally intensive to solve especially in two- and three-dimensions. To use the second approach based upon the DC resistivity equation requires that E M couphng first be removed from the IP data. In this Chapter I have developed a practical algorithm for removing E M couphng responses from IP data in the low frequency regime. The methodology is general and Chapter 4. EM Couphng and a method for its approximate removal 132 c a n b e a p p l i e d t o I D , 2 D o r 3 D d a t a s e t s . T h e f o u n d a t i o n o f t h e r e m o v a l m e t h o d i s b a s e d u p o n t h e o r e t i c a l d e r i v a t i o n s w i t h t h e e l e c t r i c f i e l d g e n e r a t e d f r o m a h o r i z o n t a l e l e c t r i c d i p o l e i n a c o m p l e x c o n d u c t i v i t y m e d i u m . I s h o w t h a t t h e o b s e r v e d e l e c t r i c f i e l d c a n b e e x p r e s s e d as a p r o d u c t o f t w o f r e q u e n c y d e p e n d e n t r e s p o n s e s . T h e f i r s t i s a n I P r e s p o n s e f u n c t i o n w h i c h d e p e n d s o n t h e c h a r g e a b i l i t y a n d c o n d u c t i v i t y o f t h e g r o u n d . T h e s e c o n d r e s p o n s e i s a n e l e c t r i c f i e l d d u e t o i n d u c t i v e e f f e c t s w h i c h d e p e n d s o n t h e c o n d u c t i v i t y o f t h e g r o u n d . R e c o g n i z i n g t h a t b o t h a m p l i t u d e a n d p h a s e o f t h e I P r e s p o n s e f u n c t i o n a r e i n d i c a t o r s o f c h a r g e a b i l i t y a l l o w s m e t o d e v e l o p r e m o v a l m e t h o d o l o g i e s f o r t h e P F E a n d p h a s e d a t a . T o c o r r e c t t h e P F E r e s p o n s e s , I n o r m a l i z e t h e m e a s u r e d a m p l i t u d e s w i t h t h e E M r e s p o n s e s . F o r c o r r e c t i o n o f p h a s e d a t a I c o m p u t e t h e p h a s e d u e t o E M c o u p h n g a n d s u b t r a c t i t f r o m t h e m e a s u r e d r e s p o n s e s . I p r e s e n t I D s y n t h e t i c e x a m p l e s t o i l l u s t r a t e E M c o u p h n g r e m o v a l . S y n t h e t i c e x a m p l e s s h o w t h a t i t i s i m p o r t a n t t o k n o w t h e c o n d u c t i v i t y s t r u c t u r e o f t h e g r o u n d f o r c o m p u t i n g E M c o u p l i n g r e s p o n s e s . I n p r a c t i c e t h e t r u e c o n d u c t i v i t y o f t h e g r o u n d i s n o t k n o w n a p r i o r i . T h e b e s t o p t i o n i s t o i n v e r t t h e a m p l i t u d e d a t a a t t h e l o w e s t a v a i l a b l e f r e q u e n c y u s i n g a D C i n v e r s i o n a l g o r i t h m . T h e h o p e i s t h a t t h i s m o d e l i s s u f f i c i e n t l y c l o s e t o t h e t r u e c o n d u c t i v i t y t h a t t h e p r o c e d u r e o u t l i n e d h e r e w i l l w o r k . T h e e x a m p l e s p r e s e n t e d p r o v i d e o p t i m i s m t h a t t h i s m a y b e v a l i d . T o a p p l y t h i s m e t h o d o l o g y t o 2 D p r o b l e m s I i d e a l l y n e e d a 2 D E M f o r w a r d m o d e l i n g . T h i s i s n o t a v a i l a b l e , so I h a v e d e s i g n e d a n a p p r o x i m a t e m e t h o d w h i c h u s e s a l o c a l i z e d a v e r a g e d I D c o n d u c t i v i t y m o d e l , t h a t h a s b e e n c o n s t r u c t e d f r o m t h e 2 D c o n d u c t i v i t y o b t a i n e d f r o m t h e D C r e s i s t i v i t y i n v e r s i o n a l g o r i t h m . E a c h t r a n s m i t t e r p o s i t i o n i s a s s o c i -a t e d w i t h i t s o w n I D c o n d u c t i v i t y . T h e a d v a n t a g e o f u s i n g a l o c a l a v e r a g e d c o n d u c t i v i t y m o d e l , c o m p a r e d t o a s i m p l e h a l f - s p a c e o r a s i n g l e l a y e r e d m o d e l i s t h a t t h e l a t e r a l c o n -d u c t i v i t y v a r i a t i o n a l o n g t h e s u r v e y l e n g t h i s a p p r o x i m a t e l y a c c o u n t e d f o r . H o w e v e r a Chapter 4. EM Couphng and a method for its approximate removal 133 f u l l 2 D E M f o r w a r d m o d e l i n g w o u l d b e i d e a l f o r c o m p u t i n g t h e p h a s e s d u e t o E M c o u p -h n g e s p e c i a l l y w h e n t h e r e i s t o p o g r a p h y a n d s i g n i f i c a n t c o n d u c t i v i t y v a r i a t i o n s l a t e r a l l y . T h e a p p h c a t i o n o f o u r m e t h o d t o t h e field e x a m p l e h a s s h o w n s i g n i f i c a n t i m p r o v e m e n t i n t h e r e c o v e r e d m o d e l c o m p a r e d t o t h e e x i s t i n g m e t h o d s l i k e t h r e e - p o i n t d e c o u p l i n g a n d m a n u a l d e c o u p l i n g t e c h n i q u e s . O v e r a l l t h e i m p r o v e m e n t i n s i g n a l t o n o i s e r a t i o i n a l l o f t h e e x a m p l e s p r e s e n t e d i n t h i s w o r k i s s u b s t a n t i a l a n d t h i s l e a d s t o m u c h g r e a t e r a b i l i t y t o i n t e r p r e t t h e I P d a t a . Chapter 4. EM Couphng and a method for its approximate removal 134 U n d e c o u p l e d P s e u d o S e c t i o n 1 O 2 y 3 TO O - 4 0.125 Hz , (a) 800 1200 1600 ^ • • • ^ m r a d s 0.0 5.4 10.8 16.2 21.6 27.0 32.4 37.8 43.2 48.6 54.0 1 ~r CT> 2 -C " O 3 SS O - 4 H co ^ 5 H 0.25 Hz (b) | 1 1 1 1 I l i r i i 0.0 7.1 14.2 21.3 28.4 35.5 42.6 49.7 56.8 63.9 71.0 1 0 2 2 cr " O 3 co Q_ 4 OO ^ - 5 H 0.5 Hz (c) 400 0.0 1 1 .6 23.2 34.8 46.4 58.0 69.6 81.2 92.8 104.0 116.0 0.0 20.5 41.0 61.5 82.0 102.0 123.0 144.0 164.0 184.0 205.0 F i g u r e 4.36: T h e p s e u d o s e c t i o n o f t h e u n d e c o u p l e d p h a s e s a t ( a ) 0.125 H z , ( b ) 0.25 H z , ( c ) 0.5 H z a n d ( d ) 1 H z . Chapter 4. EM Couphng and a method for its approximate removal 135 F i g u r e 4.37: T h e recovered chargeabihty m o d e l ob ta ined after invers ion of undecoup led da t a at (a) 0.125 H z , (b) 0.25 H z , (c) 0.5 H z and (d) 1 H z . Chapter 4. EM Couphng and a method for its approximate removal 136 F i g u r e 4.38: T h e p s e u d o s e c t i o n o f t h e d e c o u p l e d p h a s e s a t ( a ) 0.125 H z , ( b ) 0.25 H z , ( c ) 0.5 H z a n d ( d ) 1 H z . Chapter 4. EM Couphng and a method for its approximate removal 137 F i g u r e 4.39: T h e recovered chargeabi l i ty m o d e l ob ta ined after invers ion of decoupled da t a at (a) 0.125 H z , (b) 0.25 H z , (c) 0.5 H z and (d) 1 H z . C h a p t e r 5 C o m p l e x c o n d u c t i v i t y i n v e r s i o n i n I D w i t h o u t E M c o u p l i n g r e m o v a l T r a d i t i o n a l p rac t i ce i n I P da t a in t e rp re ta t ion is to remove the E M couphng c o n t a m i n a t i o n f rom the d a t a and use an I P invers ion a l g o r i t h m to o b t a i n a chargeab i l i ty d i s t r i b u t i o n . T h i s is a two-step process. F i r s t the noise due to E M couphng is r emoved f r o m the r aw da ta . T h e second step is to inver t the correc ted I P d a t a w i t h sensi t iv i t ies c o m p u t e d f rom an e s t ima ted c o n d u c t i v i t y s t ruc ture . T h e r emova l step is necessary because the fo rward m o d e l i n g used i n the I P invers ion a l g o r i t h m does not take i n to cons ide ra t ion the c o n t r i b u t i o n due to E M couphng effects. I n the previous chapter I presented a me thodo logy to remove E M couphng f r o m the I P da ta . In this chapter I fo rmula te the p r o b l e m i n a different se t t ing by ask ing the fo l lowing quest ion: Is i t possible to recover chargeab ih ty i n f o r m a t i o n w i thou t r e m o v i n g E M couphng f rom the I P data? If this is possible t hen the ques t ion of r emov ing E M coup l ing does not arise and there w o u l d be no requ i rement to o b t a i n a c o n d u c t i v i t y es t imate to compu te the E M couphng responses. T o answer the ques t ion, I can formula te the p r o b l e m as one p h y s i c a l process i n w h i c h b o t h the I P effect and the E M couphng c o n t r i b u t i o n are s imul t aneous ly cons idered . T o achieve th is goal , the first step is to have some ab i l i t y to do fo rward m o d e l i n g w i t h the fu l l M a x w e l l ' s equa t ion for the e lectr ic f ie ld. T h e second step is to fo rmula te the inverse p r o b l e m i n te rms of a c o m p l e x c o n d u c t i v i t y such tha t the effects due to chargeab ih ty a n d c o n d u c t i v i t y can be ob ta ined at a pa r t i cu l a r f requency as shown i n F i g u r e 5.1. C o m p l e x c o n d u c t i v i t y is a way to represent b o t h c o n d u c t i v i t y and chargeabih ty of the g r o u n d as a f unc t i on of f requency and is a general f ramework to s tudy the f requency d i spers ion 138 Chapter 5. ID complex conductivity inversion without EM couphng removal 139 Conductivity Real Conductivity Complex Estimate o Calculate EM responses Use full EM equation for Electric field V x V x E + irou.o" E = - icou Js Estimate o Calculate EM responses Correct the data Invert for complex conductivity o(ico) Correct the data Invert for n using V . ( a ( 1 - n ) V | ) = - I 5 (r - r s ) (Chapter 5 ) Invert for s(ico) using V. ( a (ico) V4»(ia>)) = -1 (ico) 8 (r - r s ) (Chapter 4; (Chapter 6 ) F i g u r e 5.1: T h r e e different f o rmu la t i on to recover the chargeab i l i ty i n f o r m a t i o n f r o m the da ta . T h e h igh l igh ted sect ion indicates the fo rmu la t i on to be inves t iga ted i n this C h a p t e r . Chapter 5. ID complex conductivity inversion without EM couphng removal 140 of the medium. The goal of the inverse problem is to extract information about the complex conductivity of the earth from IP data that are contaminated by E M couphng. The proposed formulation is general and there is no assumption about any particular complex conductivity model ( like Cole-Cole, Warburg or constant phase model) to take into account the frequency dependence of the conductivity. The data for the inverse problem can be either impedances or apparent resistivities measured at a single frequency and for several dipole separations. It is important to note that only single frequency data can be handled using this method, and thus the data at different frequencies need to be inverted separately. The choice of the frequency should be such that there is an IP effect in the data. 5.1 F o r w a r d P r o b l e m The physical experiment in a frequency domain IP survey shown in Figure 5.1 is described in section (1.2). The datum is considered to be the measured impedance {Z{iu>)) which is the ratio of the complex voltage to the harmonic current at a particular frequency w. The x-component of the electric field due to an x-directed horizontal electric dipole over an isotropic layered complex conductive medium is given by Ex(x,y,z = 0,u) = I(iw)ds d2 47T da: 2 Jo -uo h (1 - rTM) — (1 + rTE) VJJCO u0 Jo (Ar ) i(jjfi0I(iu>)ds f00 e l>co -u0h / (1 + rTE)\J0{\r)d\ (5.1) Jo uo Air where rTE and rTu are the T E and T M reflection coefficients and are functions of the complex conductivity of the medium. The impedance is obtained by linear superposition Chapter 5. ID complex conductivity inversion without EM coupling removal 141 ^ / ^ (ico) Z=Z1 ^ / Cy>(i(0) Z = Z 2 , / Oj (ico) Z= Z i / F i g u r e 5.2: P h y s i c a l s i tua t ion of the c o m p l e x conduc t -i v i t y p r o b l e m i n I D of the vol tage due to s m a l l bu t f ini te d ipole sources and n o r m a l i z i n g by the i n p u t cur ren t . T h i s can represented by 1 / Z(iw) = jrr-r V ) ASj / fi(iu)dr (5.2) where dr is the e lementa l l eng th of receiver, ASJ is the l eng th of the j t h e l emen ta l t rans-m i t t e r d ipole a n d Ex(iu>) = IAsf(iu>) is the e lectr ic field due to a h o r i z o n t a l e lec t r ic d ipole of l eng th A s . A l t h o u g h the measured d a t u m is impedance , i n p rac t i ce the d a t a are presented as c o m p l e x apparent res i s t iv i ty g iven by Pa(iu) - G * Z(iw) (5.3) where G is the geometr ic factor for a pa r t i cu l a r a r ray geometry. Fo r example , for d ipole-d ipo le geomet ry shown i n F i g u r e ( ? ? ) , 1/G = (1/rAM ~ l / ^ i v ) - ( 1 / ^ B M - X/^BN), Chapter 5. ID complex conductivity inversion without EM couphng removal 142 w h e r e A, B a r e t h e p o s i t i o n s o f t h e c u r r e n t e l e c t r o d e s a n d M , N a r e p o s i t i o n s o f t h e p o t e n t i a l e l e c t r o d e s . T h e r e f o r e t h e d a t a t o b e i n v e r t e d a r e t h e r e a l a n d i m a g i n a r y p a r t s o f c o m p l e x a p p a r e n t r e s i s t i v i t y , o r a m p l i t u d e a n d p h a s e o f pa(iijn) t o o b t a i n s u b s u r f a c e c o m p l e x c o n d u c t i v i t y . I n t h e n e x t s e c t i o n I d i s c u s s t h e i n v e r s i o n m e t h o d o l o g y t o r e c o v e r c o m p l e x c o n d u c t i v i t y a(iuj) f r o m c o m p l e x a p p a r e n t r e s i s t i v i t y d a t a pa(iuj). 5.2 I n v e r s i o n M e t h o d o l o g y T h e o b j e c t i v e h e r e i s t o i n v e r t f o r t h e c o m p l e x c o n d u c t i v i t y f r o m t h e c o m p l e x a p p a r e n t r e s i s t i v i t y d a t a . S i n c e t h e d a t a a n d m o d e l p a r a m e t e r s a r e c o m p l e x q u a n t i t i e s , I s t a r t b y f o r m u l a t i n g a g e n e r a l n o n l i n e a r i n v e r s e p r o b l e m i n t h e c o m p l e x d o m a i n s i m i l a r t o t h e l i n e a r p r o b l e m d i s c u s s e d i n A p p e n d i x - D . T h e i n v e r s e p r o b l e m c a n b e m a t h e m a t i c a l l y s t a t e d as, minimize <bm = \\Wm (m — m r e / ) | | 2 (5.4) subject to<f>d = \\Wd (d°bs -F{m))\\2 w h e r e m i s t h e c o m p l e x m o d e l , dob° i s t h e o b s e r v e d c o m p l e x d a t a v e c t o r , T(m) i s t h e p r e d i c t e d d a t a v e c t o r a n d T i s t h e n o n l i n e a r f o r w a r d m o d e l i n g o p e r a t o r . T h e a m p l i t u d e o f c o n d u c t i v i t y c a n v a r y o v e r s e v e r a l o r d e r s o f m a g n i t u d e t h e r e f o r e I c h o s e t h e m o d e l f o r t h e i n v e r s e p r o b l e m m = log(tr(to>)). T h i s i m p l i e s t h a t t h e r e a l p a r t o f t h e m o d e l i s e q u a l t o a m p l i t u d e o f t h e c o m p l e x c o n d u c t i v i t y log(|(j ( ia ; ) | ) a n d t h e i m a g i n a r y p a r t e q u a l t o t h e p h a s e o f t h e c o m p l e x c o n d u c t i v i t y g i v e n b y t a n ~ 1 [ i m a g ( c r ( i a ; ) ) / r e a l ( c T ( i a ; ) ) ] . Wd a n d Wm a r e t h e d a t a a n d m o d e l w e i g h t i n g m a t r i c e s . T o s o l v e t h e i n v e r s e p r o b l e m i n eqn (5 .4) a g l o b a l o b j e c t i v e f u n c t i o n <j)(m) = <j>d(m) + 8(j)m{m) i s m i n i m i z e d . T h e m i n i m i z a t i o n i s c a r r i e d o u t b y d i f f e r e n t i a t i n g t h e m o d e l o b j e c t i v e f u n c t i o n w i t h r e s p e c t t o t h e r e a l a n d i m a g i n a r y p a r t s o f t h e m o d e l p e r t u r b a t i o n a n d e q u a t i n g t h e r e s u l t t o z e r o . L e t m — 7] + i£ w h e r e n i s t h e r e a l p a r t a n d £ i s t h e i m a g i n a r y p a r t o f t h e m o d e l , a n d l e t 8m Chapter 5. ID complex conductivity inversion without EM couphng removal 143 be a p e r t u r b a t i o n . T h e g loba l ob jec t ive func t ion can be w r i t t e n as 4>{m + 8m) = (d°b* - F(m) - J8m)H Wj Wd{d°hs - T{m) (5.5) - J8m) + f3(m + 8m)H Wm(m + 8m) where 8m = 8rj + i8£ and J = dd/dm is the sens i t iv i ty w i t h respect to m o d e l m . T h e s y m b o l H s tands for the conjugate t ranspose i.e. i f z = x + iy t h e n zH = xT — iyT. T h e rea l a n d i m a g i n a r y par t of the sens i t iv i ty J can be ob ta ined by where dR — rea l (d) , c i 7 = imag(d ) and J — JR + i J 7 . E q u a t i o n s (5.6) and (5.7) are the f ami l i a r C a u c h y - R i e m a n n condi t ions . M i n i m i z a t i o n of the p e r t u r b e d ob jec t ive f u n c t i o n i n (5.5) w i t h respect to rea l (8n) and i m a g i n a r y (<!>£) m o d e l p e r t u r b a t i o n i m p l i e s VSri(f)(m + 8m) = 0 VSi<p(m + 8m) = 0 T h u s the m i n i m i z a t i o n w i t h respect to the rea l par t gives v , ^ = [rTwJwdj + jTwJwdr) sv + i (j*TwJwdj - jTwJwdj*) 8t - JTWjWd8d* - J*TWjWd8d + 20W*Wm (8V + v) = 0 (5.8) (5.9) or . R e (jHWjWdj) 8V + I m (dHWjWdj) 8( - R e (jHWjWd8d) + /3W^Wm(8V + V) = 0 T h e m i n i m i z a t i o n w i t h respect to the i m a g i n a r y par t gives, (5.10) Chapter 5. ID complex conductivity inversion without EM couphng removal 144 = (rTwJwdj + jTwJwdr) st - % (rTwJwdj - jTwJwdr) sv - UTWjWd8d* + *J*TWjWd8d + 20WlWm (Si + 0 = 0 (5.11) or, (5.12) Sn (5.13) Re (jHWjWdJ) Si - Im (jHWjWdj) Sr, - Im (JHWjWdSd) + 0W^Wm(8i + O = O The equations can be solved for the real quantities Sn and 8^ given by, JpR + JjJj + BWlWm) - [JlJj - JJJn) (jp! - JJJR) [jpR + JjJj + 0WlWm) Jl8dR + Jj8d! - 0W^WmV JlS^-JJSdn-BWlW^ where JR = WdJR and Jj = WdJi, or they can be combined into a complex system as follows. The objective function (f> is a real quantity, and therefore the V'sv<f> and V$£(/> are also real quantities. Since they are each equal to zero then the linear combination V^ , , ^ + iVst4> = 0. Therefore combining equation ( 5.9) and ( 5.11) I obtain, (jHWjWdJ + BWlWm) 8m = JHWjWd8d - 0WlWmm (5.14) Either the real system (5.13) or the complex system (5.14) can be solved for the perturb-ation. I have chosen to work with (5.13) and solve it using QR decomposition (Golub and Vanloan, 1996). The inverse problem is solved in a similar way as the C S A M T problem in Chapter-2. 5.2.1 S e n s i t i v i t y In order to solve the inverse problem using the linearized Gauss-Newton method, the sensitivity dd(x = Xk,u;)/dmj has to be computed. The data are complex appar-ent resistivities for a single frequency at N receiver electrode locations denoted by Chapter 5. ID complex conductivity inversion without EM couphng removal 145 x,,x2,. .. ,xN. T h u s t h e d a t a a r e d = (pa(x1, u>), pa(x2, u>), • • • , pa(xN, v))T. T h e s e n s i t -i v i t i e s dd(xi,u>)/dmj = o-j{iuj)dd(xi,ui)/d<Tj{ioj) s i n c e m = log(<r(zu;)). T o c o m p u t e t h e s e n s i t i v i t y o f t h e c o m p l e x a p p a r e n t r e s i s t i v i t y w i t h r e s p e c t t o t h e c o m p l e x c o n d u c t i v i t y (dd(xi,u>)/d<Tj(iu>)) t h r e e s t e p s a r e i n v o l v e d , ( a ) F i r s t t h e s e n s i t i v i t y f o r t h e e l e c t r i c field i s c o m p u t e d f r o m t h e s e n s i t i v i t y o f t h e v e c t o r p o t e n t i a l s . T h e s e n s i t i v i t y o f t h e v e c t o r p o t e n t i a l s a r e c o m p u t e d i n s i m i l a r w a y as s h o w n i n C h a p t e r - 2 , e x c e p t t h e c o n d u c t i v i t y i s n o w c o m p l e x i n s t e a d b e i n g r e a l a s w a s t h e c a s e i n t h e C S A M T p r o b l e m , ( b ) I n t h e s e c o n d s t e p t h e s e n s i t i v i t y w i t h r e s p e c t t o t h e e l e c t r i c field i s i n t e g r a t e d o v e r t h e r e -c e i v e r a n d t r a n s m i t t e r d i p o l e l e n g t h s t o o b t a i n s e n s i t i v i t y w i t h r e s p e c t t o t h e i m p e d a n c e dZ(iu})/da. ( c ) I n t h e t h i r d s t e p t h e s e n s i t i v i t y dZ(iu>)/do~ i s m u l t i p l i e d w i t h t h e a p p r o -p r i a t e g e o m e t r i c f a c t o r f o r a g i v e n a r r a y g e o m e t r y t o o b t a i n t h e s e n s i t i v i t y w i t h r e s p e c t t o t h e c o m p l e x a p p a r e n t r e s i s t i v i t y i.e. dpa(iv)/da. 5.2.2 S y n t h e t i c E x a m p l e I n t h i s s e c t i o n I c o n s i d e r a s y n t h e t i c e x a m p l e t o t e s t t h e i n v e r s i o n m e t h o d o l o g y . T h e e a r t h m o d e l i s a t h r e e - l a y e r c o m p l e x c o n d u c t i v i t y m o d e l s h o w n i n F i g u r e 5.3(c) a n d ( d ) . T h e c o m p l e x c o n d u c t i v i t y m o d e l i s p r o d u c e d u s i n g t h e C o l e - C o l e m o d e l . T h e l a y e r p a r a m e t e r s o f t h e t h r e e l a y e r s a r e s h o w n i n T a b l e ( 5.1). T h e c o m p l e x a p p a r e n t r e s i s t i v i t y d a t a a r e g e n e r a t e d a t a f r e q u e n c y o f 1 H z w i t h t h e m o d e l i n F i g u r e 5.3. T h e f r e q u e n c y b f 1 H z i s c h o s e n s o t h a t t h e r e i s a n I P e f f e c t as w e l l as E M c o u p h n g r e s p o n s e . T h e C o l e - C o l e p l o t f o r t h e c h a r g e a b l e l a y e r i s s h o w n i n F i g u r e 5.8. A l t h o u g h t h e e a r t h m o d e l w a s g e n e r a t e d u s i n g t h e C o l e - C o l e f o r m u l a t i o n , t h e i n v e r -s i o n m e t h o d o l o g y i s a p p l i c a b l e t o a n y I D d i s t r i b u t i o n o f c o m p l e x c o n d u c t i v i t y f o r m u -l a t i o n . T h e i n v e r s e p r o b l e m h e r e i s s o l v e d f o r e a c h f r e q u e n c y s e p a r a t e l y . T h e d a t a a r e g e n e r a t e d w i t h a d i p o l e l e n g t h o f 5 0 m a n d a t 8 r e c e i v e r p o s i t i o n s ( i . e . N - s p a c i n g = l , - • • ,8). T h e c o m p l e x a p p a r e n t r e s i s t i v i t y d a t a s h o w n a s r e a l a n d i m a g i n a r y p a r t s i n F i g u r e 5.3(a) Chapter 5. ID complex conductivity inversion without EM couphng removal 146 l a y e r o-oo ( S / m ) C T 0 ( S / m ) V r c h 1 0.0101 0.01 0.01 1.0 0.35 20 2 0.2353 0.2 0.15 10.0 0.45 50 3 0.001 0.001 0.0 1.0 0.35 T a b l e 5.1: M o d e l u s e d t o g e n e r a t e t h e d a t a a n d ( b ) h a s c o n t r i b u t i o n s f r o m b o t h I P e f f e c t s a r i s i n g f r o m t h e c h a r g e a b l e m e d i u m a n d f r o m a n E M c o u p h n g c o n t r i b u t i o n . T h e r e a l p a r t o f t h e d a t a s h o w n i n F i g u r e 5.3(a) a r e c o n t a m i n a t e d w i t h G a u s s i a n r a n d o m n o i s e w i t h s t a n d a r d d e v i a t i o n o f 5 % o f t h e d a t u m v a l u e . T h e i m a g i n a r y p a r t i n F i g u r e 5.3(b) a r e c o n t a m i n a t e d w i t h G a u s s i a n r a n -d o m n o i s e w i t h 5 % s t a n d a r d d e v i a t i o n p l u s a c o n s t a n t s t a n d a r d d e v i a t i o n o f 0.001. T h e c h o i c e o f 5 % i s a r e a s o n a b l e e s t i m a t e g i v e n t h e p r e s e n t d a y t e c h n o l o g y o f t h e a c q u i s i -t i o n e q u i p m e n t . T h e d a t a w e i g h t i n g m a t r i x Wd i s a d i a g o n a l m a t r i x w i t h t h e e l e m e n t s 1 / v ^ e f j + e2-) w h e r e e# a n d e j a r e t h e s t a n d a r d d e v i a t i o n o f t h e r e a l a n d i m a g i n a r y p a r t o f t h e d a t a r e s p e c t i v e l y . A l t h o u g h t h e d a t a v e c t o r i s a c o m p l e x q u a n t i t y t h e d a t a w e i g h t i n g m a t r i x Wd i s a r e a l m a t r i x f o r u n c o r r e l a t e d d a t a s i m i l a r t o t h e l i n e a r p r o b l e m s h o w n i n A p p e n d i x - D . T h e s e d a t a a r e i n v e r t e d b y m i n i m i z i n g a m o d e l o b j e c t i v e f u n c t i o n i n w h i c h t h e c o m p o -n e n t p e n a l i z i n g t h e m o d e l r o u g h n e s s w i t h d e p t h i s d o m i n a n t ( a s = 0.001 a n d az — 1.0). T h e e a r t h i s d i v i d e d i n t o 8 0 l a y e r s w i t h l a y e r t h i c k n e s s i n c r e a s i n g w i t h d e p t h . T h e l a y e r t h i c k n e s s e s a r e k e p t f i x e d t h r o u g h o u t t h e i n v e r s i o n . T h e r e f o r e t h e r e a r e 80 c o m p l e x c o n -d u c t i v i t i e s t o b e d e t e r m i n e d f r o m 8 c o m p l e x r e s i s t i v i t y d a t a . S i n c e t h e e r r o r s a s s i g n e d a r e G a u s s i a n , t h e e x p e c t e d % 2 = 8, t h e n u m b e r o f d a t a . T h e s t a r t i n g m o d e l f o r t h e i n -v e r s i o n i s a h o m o g e n e o u s h a l f - s p a c e w i t h t h e r e a l p a r t o f t h e c o n d u c t i v i t y <TR = O.OlS/m a n d i m a g i n a r y p a r t crj = 0. T h i s i m p l i e s t h a t t h e s t a r t i n g m o d e l i s a n o n - c h a r g e a b l e m e d i u m . T h e r e s u l t i n g r e a l a n d i m a g i n a r y p a r t s o f t h e m o d e l f r o m t h e i n v e r s i o n a r e Chapter 5. ID complex conductivity inversion without EM couphng removal 147 shown in Figure 5.3(c) and (d) respectively. The achieved %2 = 7.99 after 14 iterations. The real part of the recovered model in Figure 5.3(c) indicates that both location and amplitude of the anomaly is well recovered. The imaginary part shown in Figure 5.3(d) also indicates a good agreement with the true model. Figure 5.3 (a) and (b) shows that the predicted data are in good agreement with the observations. Before leaving this example I examine the inversion of couphng contaminated complex resistivity data generated at four other frequencies. The complex conductivity plot for the chargeable layer is shown in Figure 5.8. The real part of the conductivity Re(cr(m>)) shown in Figure 5.8(a) increases from c r 0 to t r ^ , where <7o = t r ^ l — rj) and rj is the chargeability. The imaginary part of the model Im(tr(za;)) in Figure 5.8(b) indicates a peak between 0.01-0.1 Hz. First I generate the complex resistivity data at four different frequencies / = 0.001,0.01,0.1,10Hz shown in Figures 5.4, 5.5, 5.6 and 5.7 (a) and (b) respectively. As expected the amplitude of the imaginary part of the data increases with frequency since E M couphng contamination increases with frequency. Comparing Figure 5.7 (b) with 5.6 (b) indicates that the magnitude of the data at 10Hz is approximately three times greater than at 0.1Hz. The complex resistivity data at each frequency are inverted separately for a complex conductivity model. The recovered model from the inversion at / = 0.001, 0.01, 0.1,10Hz is shown with true model in Figures 5.4, 5.5, 5.6 and 5.7 (c) and (d) respectively. The results from the inversions at these four frequencies indicate that both the real and imaginary part are in good agreement with the true models. The real and imaginary parts of the estimated complex conductivity values obtained at a depth of z=45m (corresponding to the center of the chargeable layer) for the five frequencies ( / = 0.001, 0.01,0.1,1,10, Hz) is plotted in Figure 5.8(a) and (b) respectively. The estimated values shown in circles are in reasonable agreement with the true values at that depth. Chapter 5. ID complex conductivity inversion without EM coupling removal 148 5.3 C o n c l u s i o n s In this chapter I have tried to illustrate that by solving the full Maxwell's equation with complex conductivity, there is no need to remove E M couphng from IP data. This is because the forward modeling takes into consideration the effects due to chargeabihty and E M couphng contribution. To extract the complex conductivity values from apparent complex resistivity data the inversion algorithm is formulated in the complex domain and is solved by minimizing a model objective function subject to adequately fitting the data. The inverse problem is solved by using a standard Gauss-Newton technique. The synthetic example presented in this chapter shows the validity of the approach. The method is general since it directly solves for a(iu>) instead of solving for model dependent quantities that are specific to a particular model. The inversions at different frequencies indicate that the method is applicable to a wide range of frequencies, however the choice of frequency should be such that there is an IP effect in the data. Since this is a single frequency inversion, the imaginary part of the complex conductivity is the indicator of chargeable material in the ground. Thus the choice of frequency should be such that there is phase IP response in the signal, so that the imaginary part of the complex conductivity is non-zero. In Chapter-4 I developed a method to remove E M couphng from the data and then invert the data using a forward mapping which is based on DC resistivity formulation. Thus knowledge of the conductivity structure was required to be able to do E M coup-hng corrections. The formulation presented in this chapter bypasses the need to obtain this conductivity information and does not suffer from any ambiguity of correcting the responses. However at each iteration of the inversion algorithm the full E M equation needs to be solved many times. This can be computationally expensive especially if this method is extended to 3D. In the two-stage inversion, the full Maxwell's equation need Chapter 5. ID complex conductivity inversion without EM couphng removal 149 to be solved on ly once w h e n c o m p u t i n g the E M couphng responses. I n the nex t chapter I discuss the two-stage invers ion process and recover c o m p l e x c o n d u c t i v i t y b y i n v e r t i n g the cor rec ted da ta . Chapter 5. ID complex conductivity inversion without EM couphng removal 150 10° 1 0 1 1 0 2 1 0 3 10° 1 0 ' 1 0 2 1 0 3 Z(m) Z(m) F i g u r e 5.3: I n v e r s i o n o f c o m p l e x a p p a r e n t r e s i s t i v i t y (pa(iu>)) d a t a g e n e r a t e d a t 1 H z . ( a ) R e a l p a r t o f pa{iw) ( i n Q, — m ) c o n t a m i n a t e d w i t h 5 % G a u s s i a n r a n d o m n o i s e s h o w n i n e r r o r b a r s . T h e p r e d i c t e d d a t a f r o m t h e i n v e r s i o n i s s h o w n b y t h e s o l i d l i n e , ( b ) I m a g i n a r y p a r t o f pa(iw) ( i n 0, — m ) c o n t a m i n a t e d w i t h 5 % + 0.001 ( b a s e l e v e l ) G a u s s i a n r a n d o m n o i s e s h o w n i n e r r o r b a r s . T h e p r e d i c t e d d a t a f r o m t h e i n v e r s i o n i s s h o w n b y t h e s o l i d l i n e , ( c ) T h e r e a l p a r t o f <r(iu>) u s e d t o g e n e r a t e t h e s y n t h e t i c d a t a . T h e b l o c k y m o d e l i s t h e t r u e m o d e l a n d t h e s m o o t h m o d e l i s t h e m o d e l f r o m i n v e r s i o n , ( d ) T h e i m a g i n a r y p a r t o f cr(iw) w i t h t h e t r u e b l o c k y m o d e l a n d t h e s m o o t h m o d e l f r o m t h e i n v e r s i o n . Chapter 5. ID complex conductivity inversion without EM couphng removal 151 (c) E C O ^3 e "to CD cr 10" 10 " : 10" ; 10° 1 0 1 1 0 2 Z(m) 1 0 3 -0.004 -0.006 F i g u r e 5.4: I n v e r s i o n o f c o m p l e x a p p a r e n t r e s i s t i v i t y (pa(ioj)) d a t a g e n e r a t e d a t 0 . 0 0 1 H z . ( a ) R e a l p a r t o f pa(iu) ( i n 0 — m ) c o n t a m i n a t e d w i t h 5 % G a u s s i a n r a n d o m n o i s e s h o w n i n e r r o r b a r s . T h e p r e d i c t e d d a t a f r o m t h e i n v e r s i o n i s s h o w n b y t h e s o h d l i n e , ( b ) I m a g i n a r y p a r t o f pa(iu) ( i n Q, — m ) c o n t a m i n a t e d w i t h 5 % + 0.001 ( b a s e l e v e l ) G a u s s i a n r a n d o m n o i s e s h o w n i n e r r o r b a r s . T h e p r e d i c t e d d a t a f r o m t h e i n v e r s i o n i s s h o w n b y t h e s o h d l i n e , ( c ) T h e r e a l p a r t o f a(iu>) u s e d t o g e n e r a t e t h e s y n t h e t i c d a t a . T h e b l o c k y m o d e l i s t h e t r u e m o d e l a n d t h e s m o o t h m o d e l i s t h e m o d e l f r o m i n v e r s i o n , ( d ) T h e i m a g i n a r y p a r t o f <r(iui) w i t h t h e t r u e b l o c k y m o d e l a n d t h e s m o o t h m o d e l f r o m i n v e r s i o n . Chapter 5. ID complex conductivity inversion without EM coupling removal 152 10° 1 0 1 1 0 2 1 0 3 10° 1 0 1 1 0 2 1 0 3 Z(m) Z(m) F i g u r e 5.5: I n v e r s i o n o f c o m p l e x a p p a r e n t r e s i s t i v i t y (pa(iu>)) d a t a g e n e r a t e d a t 0.01Hz. ( a ) R e a l p a r t o f pa(iw) ( i n Cl — m ) c o n t a m i n a t e d w i t h 5 % G a u s s i a n r a n d o m n o i s e s h o w n i n e r r o r b a r s . T h e p r e d i c t e d d a t a f r o m t h e i n v e r s i o n i s s h o w n b y t h e s o l i d l i n e , ( b ) I m a g i n a r y p a r t of pa(iu>) ( i n fi — m ) c o n t a m i n a t e d w i t h 5 % + 0.001 ( b a s e l e v e l ) G a u s s i a n r a n d o m n o i s e s h o w n i n e r r o r b a r s . T h e p r e d i c t e d d a t a f r o m t h e i n v e r s i o n i s s h o w n b y t h e s o l i d l i n e , ( c ) T h e r e a l p a r t o f a(ioj) u s e d t o g e n e r a t e t h e s y n t h e t i c d a t a . T h e b l o c k y m o d e l i s t h e t r u e m o d e l a n d t h e s m o o t h m o d e l i s t h e m o d e l f r o m i n v e r s i o n , ( d ) T h e i m a g i n a r y p a r t o f a{iuj) w i t h t h e t r u e b l o c k y m o d e l a n d t h e s m o o t h m o d e l f r o m t h e i n v e r s i o n . Chapter 5. ID complex conductivity inversion without EM couphng removal 153 (c) E c/5 CO CD cr 1 0 -1 0 " : 1 0 -i r 1 — i 1 0 ° 1 0 1 1 0 2 Z(m) 1 0 3 F i g u r e 5.6: I n v e r s i o n o f c o m p l e x a p p a r e n t r e s i s t i v i t y ( p a ( i t v ) ) d a t a g e n e r a t e d a t 0.1Hz. ( a ) R e a l p a r t o f pa{iu) ( i n Q — m ) c o n t a m i n a t e d w i t h 5 % G a u s s i a n r a n d o m n o i s e s h o w n i n e r r o r b a r s . T h e p r e d i c t e d d a t a f r o m t h e i n v e r s i o n i s s h o w n b y t h e s o h d h n e . ( b ) I m a g i n a r y p a r t oi pa(iu>) ( i n f i — m ) c o n t a m i n a t e d w i t h 5 % + 0.001 ( b a s e l e v e l ) G a u s s i a n r a n d o m n o i s e s h o w n i n e r r o r b a r s . T h e p r e d i c t e d d a t a f r o m t h e i n v e r s i o n i s s h o w n b y t h e s o h d l i n e , ( c ) T h e r e a l p a r t o f cr(iw) u s e d t o g e n e r a t e t h e s y n t h e t i c d a t a . T h e b l o c k y m o d e l i s t h e t r u e m o d e l a n d t h e s m o o t h m o d e l i s t h e m o d e l f r o m i n v e r s i o n , ( d ) T h e i m a g i n a r y p a r t o f a(iu>) w i t h t h e t r u e b l o c k y m o d e l a n d t h e s m o o t h m o d e l f r o m t h e i n v e r s i o n . Chapter 5. ID complex conductivity inversion without EM couphng removal 154 (a) (b) Z(m) Z(m) F i g u r e 5.7: I n v e r s i o n o f c o m p l e x a p p a r e n t r e s i s t i v i t y (pa(iu))) d a t a g e n e r a t e d a t 1 0 H z . ( a ) R e a l p a r t o f pa(iu>) ( i n Q, — m ) c o n t a m i n a t e d w i t h 5 % G a u s s i a n r a n d o m n o i s e s h o w n i n e r r o r b a r s . T h e p r e d i c t e d d a t a f r o m t h e i n v e r s i o n i s s h o w n b y t h e s o l i d l i n e , ( b ) I m a g i n a r y p a r t o f pa(iw) ( i n fl — m ) c o n t a m i n a t e d w i t h 5 % + 0.001 ( b a s e l e v e l ) G a u s s i a n r a n d o m n o i s e s h o w n i n e r r o r b a r s . T h e p r e d i c t e d d a t a f r o m t h e i n v e r s i o n i s s h o w n b y t h e s o l i d l i n e , ( c ) T h e r e a l p a r t o f a{iuj) u s e d t o g e n e r a t e t h e s y n t h e t i c d a t a . T h e b l o c k y m o d e l i s t h e t r u e m o d e l a n d t h e s m o o t h m o d e l i s t h e m o d e l f r o m i n v e r s i o n , ( d ) T h e i m a g i n a r y p a r t o f <r(i<jj) w i t h t h e t r u e b l o c k y m o d e l a n d t h e s m o o t h m o d e l f r o m t h e i n v e r s i o n . Chapter 5. ID complex conductivity inversion without EM couphng removed 155 (a) T I 'I ' "I I "I >—"-TT Frequency (Hz) F i g u r e 5.8: C o m p l e x c o n d u c t i v i t y m o d e l as a f u n c t i o n o f f r e q u e n c y , ( a ) R e a l p a r t a n d ( b ) i m a g i n a r y p a r t o f <r(iu>). T h e s o h d l i n e i s t r u e m o d e l a t t h e d e p t h o f z = 4 5 m . T h e d o t s r e p r e s e n t s t h e e s t i m a t e d v a l u e s o b t a i n e d f r o m t h e i n v e r s i o n o f c o m p l e x r e s i s t i v i t y d a t a a t f i v e f r e q u e n c i e s (0.001, 0.01, 0.1, 1.0, 10.0 H z ) . Chapter 6 Complex conductivity inversion in I D after E M Coupling is removed 6.1 Introduction In the previous chapter I presented the inversion of complex conductivity from apparent resistivity data at a particular frequency. In that formulation there was no requirement to remove E M couphng responses from the data, since the forward modeling used in the inversion algorithm was the solution of full E M equation. In this chapter I remove E M couphng from the data and invert for complex conductivity as shown in Figure 6.1. The forward modehng that is used in the inversion algorithm is V.(cr(z'a;)Vc/>) = — I(iu>)8(r — rs). This paraUels the work in Chapter-4 but works with complex conductivity instead of inverting for chargeabihty. Most IP inversion algorithms are based on the DC resistivity formulation (Oldenburg and L i , 1994; LaBrecque, 1991). Therefore in order to recover chargeable information about the earth using such forward modehng the IP data is assumed to be free of E M couphng contamination. In these situations any E M couphng contamination is considered as noise in the data. The objective is to invert for complex conductivity from complex resistivity data after the noise due to E M couphng have been removed. The goal in this chapter is three fold: (a) invert the true IP data that is not contaminated with E M couphng. Ideally this is the most desirable situation, since the forward modelling does not account for the E M couphng effect. This will establish the model against which other models can be compared, (b) Invert the E M couphng contaminated data with the same 156 Chapter 6. ID Complex conductivity inversion after EM Couphng removal 157 Field Data Conductivity Real Estimate a Calculate EM responses Conductivity Complex Use full EM equation for Electric field V x V x E + icopa E = - icou Js Estimate o Calculate EM responses Correct the data Invert for n using V.(a(1-n) V<|>) = - I5(r-r s) (Chapter 4 ) Invert for complex conductivity o-(ico) (Chapter 5) Correct the data Invert for o(ico) using V. (a(ico) V<j>(ico)) = -1 (ico) 8 (r - rs ) (Chapter 6 F i g u r e 6.1: T h r e e d i f f e r e n t f o r m u l a t i o n t o r e c o v e r t h e c h a r g e a b i l i t y i n f o r m a t i o n f r o m t h e d a t a . T h e h i g h l i g h t e d s e c t i o n i n d i c a t e s t h e f o r m u l a t i o n t o b e i n v e s t i g a t e d i n t h i s Chapter 6. ID Complex conductivity inversion after EM Couphng removal 158 forward modelling. This will establish the artifacts that are generated in the inverted models due to the EM coupling contamination. From an interpretation point of view it is important to know what kind of artifacts are produced in the inverted results if the data are contaminated with EM coupling. These artifacts can be used as a diagnostic check to determine if the data are contaminated by EM couphng. (c) Invert the data after they are corrected for the EM couphng contamination. It was discussed in Chapter-4 that the knowledge of conductivity is important to compute the responses due to EM couphng. This wiU provide an indication of how well the data are corrected for EM couphng contamination. Here two examples will be illustrated, one with the true conductivity and the second with the conductivity obtained from a D.C resistivity inversion. If the EM couphng removal is successful then the model obtained from inversion should be similar to the inverted model in (a). 6.2 F o r w a r d P r o b l e m In a frequency domain IP survey, if the inductive effects are neglected then the governing equation which connects the complex potentials to the current is given by V. (<r(iu>)VV(iw)) = -I(iu>)6(r - r.) (6.1) where cr(iuj) is the complex conductivity of the medium, r is the receiver position and rs is the source position. The derivation for the eqn(6.1) from the original Maxwell's equations is shown in Chapter-1. If the earth is discretized into layers, then the boundary condition for the potential at any interface are given by Vj-i = V5 (6.2) 1 dVj-x _ 1 dVj pj-\(iui) dz Pji}v) dz (6.3) Chapter 6. ID Complex conductivity inversion after EM Couphng removal 159 where tr J-_1(iu;) = 1/pj-\(iuj) and <Tj(iu>) = l/pj(iw) are the conductivities of the (j — l ) t h and jth layer respectively. The solution to the forward problem is similar to the case when the conductivity is real. Here the forward problem is solved by a recursion method (Sunde, 1967). The complex potential measured at the surface is given by V(iu>, r,z = 0)= /(7-) , f°° K ^ M{\ u)Jo(Xr)d\ (6.4) where the J 0 is the Bessel function of zeroth order, <Ti(iu>) is the conductivity of the first layer and the kernel Z ^ I , 2 , . . . , M ( A ) is obtained by recursion which is given by 1 - 6 ,2 M e - 2 A f e l ^ I , 2 , . . . , M ( A , W ) = — — — 1 + ?i,2,...,iwe where, !,3 M The kernel for the intermediate layer j is given by K m 1 ~ J(j-i)j,...,Me~2Xhi~1 where, M " ( i \ , K and the kernel for the bottom half-space is given by K l\ \ 1 - ^ ( M - i ) , M e - 2 A ^ - ' - t t (M-i) ,MlA<* , j — t . t ^TT-— V ^ 1 ~l" £ ( A f - l ) , A f e _ . .M ) , . . . , M (6.5) (6.7) (6.8) (6.9) Chapter 6. ID Complex conductivity inversion after EM Couphng removal 160 layer (Too (S/m) CT 0 (S/m) V r c h 1 0.0101 0.01 0.01 1.0 0.35 20m 2 0.2353 0.2 0.15 10.0 0.45 50m 3 0.001 0.001 0.0 1.0 0.35 '! Table 6.1: Model used to generate the data The measured potential in the field can be represented in terms of a complex apparent resistivity which is given by G V(iu) Pa(iv) (6.11) 2TT where G is the geometric factor which depends on the array configuration used. The data can be represented by the real and imaginary parts of pa(iu>) or by the amphtude and phase of pa(iu>). To model the complex conductivity of the earth, the Cole-Cole formulation is used. Figure (6.2) illustrates the responses due to a three-layer complex conductivity model generated at 1Hz. The layer parameters are given in Table(6.1). The data are generated using eqn(6.1) for a dipole-dipole array geometry with dipole spacing of 50m. The responses, plotted as function of N-spacing, are shown in Figure(6.2). The amphtude of pa(iu) shown in Figure 6.2(a) and real part in Figure 6.2(c) are very similar indicating that the imaginary part of the data is much smaller in magnitude than the real part. The decrease in the amphtude of the resistivity is indicative of the middle conductive layer. Since the second layer is chargeable, the phase values increase with N-spacing. This is shown in Figure 6.2(b). The imaginary part of data presented in Figure 6.2(d) shows an increase in amphtude with N-spacing. This is also indicative of the chargeable layer. Chapter 6. ID Complex conductivity inversion after EM Couphng removal 161 E i E _sz o ^ 101 - i CD " O " Q . E < 10 c (a) J I ' i -1 1 1 1 1 1 1— 0 1 2 3 4 5 6 7 8 N-sp 15 CO 10 5 H CD CO CO sz CL J I I I 1_ o o o o N-sp (b) I I I 1 1 I 1 0 1 2 3 4 5 6 7 8 N-sp 0.0 -0.1 H s ^ -0.2 -0.3 E -0.4 H -0.5 (d) J I I I I I l _ -1 1 1 1 1 1 1— 0 1 2 3 4 5 6 7 8 N-sp Figure 6.2: Synthetic data generated using a three-layer model at / = 1Hz. (a) Ampli-tude of complex apparent resistivity pa(iu>). (b) Phase of pa(iu>). (c) Real part of pa(iuj). (d) Imaginary part of pa(iu>). Chapter 6. ID Complex conductivity inversion after EM Coupling removal 162 6.3 Removal of E M coupling from complex apparent resistivity data In order for the forward problem described in (6.2) to be valid, the data have to be corrected for E M couphng contribution. This work parallels that in Chapter-4 but here I derive an expression to correct the impedances instead of correcting the P F E or the phase data. The electric field generated by a horizontal electric dipole over a complex conductive half-space is given by eqn(4.1). This is given by E EMIP u, ,\ - ~Ids n n _ L _ L I d s \ 2 x 2 -v2 (6.12) The goal is to obtain the second term in eqn(6.12) which can be modeled using eqn(6.1). I denote the second term of eqn (6.12) as EDCIP(iuj) which is given by E DCIP Ids 27rcr(iu>) 2x7 Ids(l — m(iuj)) 2^0-00(1 - 77) 2x2 - y2 Idstb(iu>) 2^(7^ 2a;2 - y2 (6.13) where ip(iu>) is a complex scalar function that contains information about the chargeabil-ity of the ground. Using the expression for the DC electric field in eqn(4.5), the electric field EDCIP(iu>) in eqn(6.13) can be written as EDCIP(iu) = TP(iuj)E DC (6.14) where EDC is the electric field due to DC effect of the medium. Integrating the electric fields on both sides of the eqn(6.14) along the transmitter and receiver lengths I obtain, TDCIP DC (6.15) The complex function ij)(iui) in eqn(6.15) can be expressed in terms of E M I P and E M impedances shown in Chapter-4, with the assumption that the propagation constants of the medium, in the presence and absence of chargeabihty are approximately equal; that Chapter 6. ID Complex conductivity inversion after EM Couphng removal 163 is kEMIP « kEM. T h i s is g iven by ,.. x (l-m(iu>)) ZEMIP(iw) , s ^ M = h r ^ " ^ M • (6-16) T h e goa l is to correct the impedance ZEMIP(iuj) and o b t a i n an i m p e d a n c e ZDCIP(iuj) g iven i n eqn(6.15) . I subs t i tu te the express ion of i n eqn(6.16) i n to eqn(6.15) a n d denote the cor rec ted impedance by ZCORR(iu>). T h i s is g iven b y 7 C O R R ( - \ A \LU>) ryDC (ct>7\ Z W = " P S H Z • ( 6 - 1 7 ) Equa t i on (6 .17 ) is used to correct the impedances . I f the a p p r o x i m a t i o n kEMIP kEM holds , t h e n ZCORR(iu) « ZDCIP(iw). M u l t i p l y i n g the i m p e d a n c e i n eqn(6.17) w i t h the appropr i a t e geometr ic factor (G) results i n the correc ted apparent res is t iv i t ies g iven b y CORR(- \ _ r,ZEMIP(iu>) D c P° - G Z E M { I U J ) Z • (6-18) I n order to do this cor rec t ion ZEM(iu>) needs to be c o m p u t e d us ing a k n o w n c o n d u c t i v i t y s t ruc tu re a n d the d a t a to be correc ted are the measured impedances ZEMIP(iu>). T h i s impl i e s tha t b o t h rea l and i m a g i n a r y par ts of the i m p e d a n c e or the vol tage need to be acqu i red i n the field. T h i s is different f rom the I P responses i n C h a p t e r - 4 , where the a m p l i t u d e (i.e P F E ) or phase responses were correc ted for the E M couphng c o n t r i b u t i o n . H o w e v e r i n p rac t i ce m o d e r n day c o m p l e x res i s t iv i ty in s t rumen t s acqui re b o t h a m p l i t u d e a n d phase d a t a s imul taneous ly . 6.4 I n v e r s i o n M e t h o d o l o g y M y ob jec t ive here is to inver t for the c o m p l e x c o n d u c t i v i t y f r o m c o m p l e x apparent res-i s t i v i t y d a t a w h e n the da t a are not c o n t a m i n a t e d by E M couphng . T h e inverse p r o b l e m can be s ta ted as minimize <f>m = a3 \\WS (m — m r e / ) | | 2 + az \\WZ ( m ) | | 2 (6.19) subject to(j)d=\\Wd (dobs - F{m))\\2 = <fc Chapter 6. ID Complex conductivity inversion after EM Couphng removal 164 where T is the forward mapping operator given in eqn(6.1), m = log(a(ico)) is the model for the inverse problem, W, and Wz are the model weighting matrices and Wd is the data weighting matrix containing the reciprocal of the standard deviations, and 4>*d ^s * n e target misfit to be achieved after the inversion. The inverse problem is solved in a similar way to that outlined in section(5.2). The data for the inversion are the complex apparent resistivities pa(iuS). In the next section I consider a synthetic example to investigate the effects of E M couphng removal on the recovery of the complex conductivity. 6.5 S y n t h e t i c E x a m p l e s The three layer model given in Table(6.1) is used to generate the synthetic data. Two sets of synthetic data are generated: (1) First I generate apparent resistivity data at / = 1Hz in the absence of E M couphng. This is computed using eqn (6.11) where the potentials V(iw) are obtained from eqn (6.1). The data are generated using a dipole-dipole survey with a dipole spacing of 50m for 8 dipole separations (N — 1,... ,8). (2) Next I generate apparent resistivity data contaminated with E M couphng. This is obtained by solving eqn (4.30) and integrating the electric field over the transmitter and receiver dipole lengths. In order to compare the models obtained from the various test cases, a particu-lar model objective function is chosen. The model objective function is that given in eqn(6.19) with az — 1.0 and as = 0.001 and the reference model is a non-chargeable ho-mogeneous half-space model with Re(cr(z'u;)) = O.OlS/m and lm(a(iw)) = 0. The starting model for the inversion is same as the reference model. Both data sets are contaminated with the same Gaussian random noise. The standard deviation chosen for the real part is 5% of the datum value, and the imaginary part is contaminated with noise having a Chapter 6. ID Complex conductivity inversion after EM Couphng removal 165 standard deviation 5% of the datum value plus a constant standard deviation of 0.001. 6.5.1 Inversion of true complex resistivity data First I consider the data generated in the absence of E M couphng. The real and imaginary parts of the synthetic data contaminated with Gaussian random noise are shown in Figure 6.3(a) and (b) respectively. These data were inverted and the recovered complex conductivity is shown in Figure 6.3(c) and (d). The real part of the recovered a(iw) in Figure 6.3(c) indicates that both amphtude and location of the conductor are well determined. At depths where the model is insensitive to the data, a background value of close to O.OlS/m is obtained. The recovered imaginary part of the <r(iuj) in Figure 6.3(d) is in reasonable agreement with the true model. The real and the imaginary parts of the predicted data from the inversion are shown in Figure 6.3(a) and (b) respectively. They are in good agreement with the observations. 6.5.2 Inversion of complex resistivity data contaminated by E M coupling If we consider E M couphng contaminated data and apply an IP inversion algorithm whose forward mapping is based on eqn(6.1), then artifacts in the recovered model wiU be generated because the wrong forward modelling is used in the inversion. To test this hypothesis I invert the E M couphng contaminated data generated for the same model using eqn (4.9). The noise contaminated data are shown in Figure 6.4. Because of E M couphng, the amphtudes of the imaginary data in Figure 6.4(b) are greater than the data shown in Figure 6.3(b). The data in Figure 6.4 are inverted for a complex conductivity structure. The real part of recovered model in Figure 6.4(c) indicates a good representation of the true model. However the recovered imaginary part of the model is in poor agreement with the true model shown in Figure 6.4(d). It shows that: (a) the recovered amphtudes are higher than the true amphtude and (b) the peak of Chapter 6. ID Complex conductivity inversion after EM Couphng removal 166 the anomaly is structurally misplaced and appears at a greater depth. These artifacts were also encountered for the field example in section (4.9) where the recovered phases from the IP inversion appeared at a greater depth and had high amplitudes values. This prompts the need to remove E M coupling if the data are to be inverted using the forward mapping given by eqn(6.1). In the next section I first correct the data using the methodology described in section (6.3), and then invert for the complex conductivity structure. 6.5.3 Inversion of complex resistivity data after E M coupling removal The E M coupling contaminated data shown in Figure 6.4(a) and (b) are corrected for the E M couphng contribution by eqn (6.18). To compute the impedance ZEM in eqn (6.18), the true conductivity structure t r ^ shown in Table 6.1 is used. The real and imaginary part of the corrected data are shown in Figure 6.5 (a) and (b) respectively along with the true data indicated by circles. The corrected data shown in Figure 6.5 (a) and (b) indicate a close agreement with the true data. The real and imaginary parts of the recovered model obtained by inverting the corrected data shown in Figure 6.5 (c) and (d) are in good agreement with the true model and with the model obtained in Figure 6.3 (indicated by dashed line). In the previous example the true conductivity was used to compute the E M coup-ling responses and make the corrections. In practice the true conductivity is not known apriori. Therefore I have considered another example in which the conductivity is ob-tained from the DC resistivity inversion of data generated at a lower frequency. First I generated the amplitude data at 0.1 Hz shown in Figure 6.6(a). These data are inverted for a conductivity structure using a ID DC resistivity inversion algorithm by assuming that there is no E M couphng contamination at this frequency. The model obtained from inversion is shown in Figure 6.6(b) and it is in good agreement with the true model. The Chapter 6. ID Complex conductivity inversion after EM Coupling removal 167 conductivity model in 6.6(b) is then used to compute the E M coupling responses at 1.0 Hz. The corrected real and imaginary part of apparent resistivity are shown in Figure 6.7(a) and (b) respectively along with the true data (indicated by circles). In the next step these data are inverted to generate a complex conductivity model using eqn (6.19). The real part of the model is a good representation of the true model shown in Figure 6.7(c). The imaginary part of the model in 6.7(d) shows good agreement with true model except in the anomalous part where it shghtly overshoots the true value. However we see that the amphtude and structural location of the anomaly is correctly recovered and this is a significant improvement over the result in Figure 6.4(d). My next objective is to test the methodology at various frequencies. I consider the data generated at four other frequencies / = 0.001,0.01,0.1,10Hz using the full E M equation. In Chapter-5, I had inverted these same data for complex conductivity struc-ture which did not require the removal of E M couphng from the data. Here I compute the E M coupling responses with the true conductivity structure and remove E M couphng contribution at their respective frequencies. Figure 6.8-6.11 show the corrected responses at / = 0.001, 0.01, 0.1,10Hz respectively after E M coupling is removed. It is observed that the imaginary part of the corrected data peaks at an intermediate frequency and decreases in magnitude as frequency increases. The corrected data are then inverted for a complex conductivity structure similar to the previous example. Figure 6.8-6.11 indicate that both real and imaginary part of the recovered model are in good agreement with the true model. Comparing the results from two-stage process in Figures 6.8-6.11 with the results from a single stage inversion in Figures 5.4-5.7 shows that the recovered real and imaginary model are in good agreement. The real and imaginary parts of the estimated complex conductivity values obtained at a depth of z=45m (corresponding to the center of the chargeable layer) for the five frequencies ( / = 0.001,0.01,0.1,1,10, Hz) is plotted in Figure 6.12(a) and (b) respectively. The estimated values shown in circles Chapter 6. ID Complex conductivity inversion after EM Coupling removal 168 are i n reasonable agreement w i t h the t rue values at tha t dep th . 6.6 C o n c l u s i o n s I n this chapter I a t t empt to correct for the E M coup l ing effects by a p p l y i n g the m e t h o d -ology deve loped i n Chapter-4 and inver t the correc ted d a t a for a c o m p l e x c o n d u c t i v i t y s t ruc ture . T h e fo rward m a p p i n g used i n the invers ion a l g o r i t h m is the c o m p l e x D C res i s t iv i ty equa t ion . If the da t a are c o n t a m i n a t e d by E M couphng t h e n i nve r s ion us-i n g M a x w e l l ' s equa t ion for D C res i s t iv i ty generates two diagnost ic ar t i facts . F i r s t (a) the recovered a m p l i t u d e of I P responses are higher t h a n the t rue ampl i tudes a n d (b) the anomal ies are s t ruc tu ra l l y mi sp l aced at greater depths . T h i s is because the for-w a r d m o d e l l i n g used i n the invers ion is incorrec t and does not take i n to account the E M couphng effects. T h u s w h e n the E M couphng effects are r emoved f r o m the da ta , the recovered anomal ies agrees we l l w i t h the t rue m o d e l . T h i s is i l l u s t r a t e d w i t h the syn the t i c e x a m p l e and also observed for the f ield da t a i n Chapter-4. T o c o m p u t e the E M couphng responses the c o n d u c t i v i t y s t ruc ture of the m e d i u m is r equ i r ed a n d the q u a l i t y of the recovered c o m p l e x c o n d u c t i v i t y depends u p o n this . T h e e x a m p l e presented here ind ica tes tha t the invers ion w i t h t rue c o n d u c t i v i t y gives a m a r g i n a l l y be t te r m o d e l t h a n us ing the D C conduc t i v i t y . C o m p a r i n g the invers ion results f rom this chapter w i t h the single stage inver s ion results i n Chapter-5, i t is observed tha t b o t h the me thods p roduce s imi l a r models tha t are re la t ive ly i n close agreement. T h i s impl i e s tha t the t w o m e t h o d s are equivalent i n some respect . T h e advantage of the two-step process is tha t the fu l l E M equa t ion need to be solved once, where as i n the single stage inve r s ion i t has to be solved m a n y t imes . Chapter 6. ID Complex conductivity inversion after EM Couphng removal 169 (d) 0.008 0.006 -0.004 -0.002 O) 0.000 CO -0.002 -0.004 -0.006 10° 101 102 Z ( m ) 10 = F i g u r e 6.3: I n v e r s i o n o f t r u e I P d a t a g e n e r a t e d u s i n g a t h r e e - l a y e r m o d e l a t / = 1 H z . T h e a m p l i t u d e a n d p h a s e o f t h e c o m p l e x a p p a r e n t r e s i s t i v i t y pa(iu) i s r e s p e c t i v e l y s h o w n i n ( a ) a n d ( b ) . T h e n o i s e c o n t a m i n a t e d d a t a a r e p l o t t e d w i t h e r r o r b a r s . T h e s o l i d l i n e i n d i c a t e s p r e d i c t e d r e s p o n s e s d u e t o t h e m o d e l r e c o v e r e d f r o m t h e i n v e r s i o n . P a n e l ( c ) a n d ( d ) s h o w s t h e t r u e a n d r e c o v e r e d c o m p l e x c o n d u c t i v i t y . Chapter 6. ID Complex conductivity inversion after EM Couphng removal 170 10° 10 1 10 2 10 3 10° 10 1 10 2 10 3 Z(m) Z(m) F i g u r e 6.4: I n v e r s i o n o f E M c o u p l i n g c o n t a m i n a t e d d a t a u s i n g t h e f o r w a r d m a p p i n g b a s e d o n t h e c o m p l e x D C r e s i s t i v i t y e q u a t i o n . T h e a m p h t u d e a n d p h a s e o f t h e c o m p l e x a p p a r e n t r e s i s t i v i t y pa(iu) a r e r e s p e c t i v e l y s h o w n i n ( a ) a n d ( b ) . T h e n o i s e c o n t a m i n a t e d d a t a a r e p l o t t e d w i t h e r r o r b a r s . T h e s o h d l i n e i n d i c a t e s p r e d i c t e d r e s p o n s e s d u e t o t h e m o d e l r e c o v e r e d f r o m t h e i n v e r s i o n . P a n e l s ( c ) a n d ( d ) s h o w t h e t r u e a n d r e c o v e r e d c o m p l e x c o n d u c t i v i t y . Chapter 6. ID Complex conductivity inversion after EM Couphng removal 171 ( d ) T 101 102 Z ( m ) 10 = F i g u r e 6.5: I n v e r s i o n o f d a t a t h a t h a v e b e e n c o r r e c t e d f o r E M c o u p h n g a t / = 1 H z . H e r e t h e c o r r e c t i o n i s o b t a i n e d b y u s i n g t h e t r u e c o n d u c t i v i t y s t r u c t u r e . T h e a m p l i t u d e a n d p h a s e o f t h e c o m p l e x a p p a r e n t r e s i s t i v i t y pa(iw) i s r e s p e c t i v e l y s h o w n i n ( a ) a n d ( b ) . T h e n o i s e - c o n t a m i n a t e d d a t a a r e p l o t t e d w i t h e r r o r b a r s . T h e d a t a f r o m F i g u r e 6.3 a n d 6.4 a r e i n d i c a t e d b y c i r c l e s a n d d i a m o n d s r e s p e c t i v e l y . T h e s o l i d l i n e i n d i c a t e s p r e d i c t e d r e s p o n s e s d u e t o t h e m o d e l r e c o v e r e d f r o m t h e i n v e r s i o n . P a n e l s ( c ) a n d ( d ) s h o w t h e t r u e a n d r e c o v e r e d c o m p l e x c o n d u c t i v i t y . T h e d a s h e d l i n e i s t h e m o d e l f r o m F i g u r e 6.3. Chapter 6. ID Complex conductivity inversion after EM Couphng removal 172 (a) (b) io° H — i — i — i — i — i — i — h I 1 1 1 1 2 3 4 5 6 7 8 10° 101 102 103 N-sp Z(m) Figure 6.6: Amplitude of apparent resistivity data at f — 0.1Hz is shown in (a). The noise-contaminated data are plotted with error bars. The sohd line indicates the pre-dicted responses from the inversion, (b) The recovered conductivity model from the DC resistivity inversion that will be used to compute the EM couphng responses. Chapter 6. ID Complex conductivity inversion after EM Couphng removal 173 0.008 (d) 0.004 H 0.006 f ^ 10c 101 102 Z(m) 10 = F i g u r e 6.7: I n v e r s i o n o f d a t a t h a t h a v e b e e n c o r r e c t e d f o r E M c o u p h n g a t / = 1 H z . H e r e t h e c o r r e c t i o n i s o b t a i n e d b y u s i n g t h e c o n d u c t i v i t y s t r u c t u r e o b t a i n e d f r o m D C r e s i s t i v i t y i n v e r s i o n i n F i g u r e 6.6(b). T h e a m p l i t u d e a n d p h a s e o f t h e c o m p l e x a p p a r e n t r e s i s t i v i t y pa(iw) a r e r e s p e c t i v e l y s h o w n i n ( a ) a n d ( b ) . T h e n o i s e c o n t a m i n a t e d d a t a a r e p l o t t e d w i t h e r r o r b a r s . T h e d a t a f r o m F i g u r e 6.3 a n d 6.4 a r e i n d i c a t e d b y c i r c l e s a n d d i a m o n d s r e s p e c t i v e l y . T h e s o l i d l i n e i n d i c a t e s p r e d i c t e d r e s p o n s e s f r o m t h e i n v e r s i o n . P a n e l s ( c ) a n d ( d ) s h o w t h e t r u e a n d r e c o v e r e d c o m p l e x c o n d u c t i v i t y . T h e d a s h e d l i n e i s t h e m o d e l f r o m F i g u r e 6.4. Chapter 6. ID Complex conductivity inversion after EM Coupling removal 174 (c) _ 10-1 E CO S 1 0 - 2 CD DC 10-3 H I I 0.008 10° 101 102 Z(m) 10 = -0.004 -0.006 Figure 6.8: Inversion of data that have been corrected for E M couphng at / = 0.001Hz. Here the correction is obtained by using the true conductivity structure. The amphtude and phase of the complex apparent resistivity pa(ioj) are respectively shown in (a) and (b). The noise-contaminated data are plotted with error bars. The uncorrected data are indicated by diamonds. The sohd line indicates predicted responses due to the model recovered from the inversion. Panel (c) and (d) shows the true and recovered complex conductivity. Chapter 6. ID Complex conductivity inversion after EM Couphng removal 175 -0.006 -1 1 1 I 1 1 10° 10 1 10 2 10 3 1 0 ° 10 1 10 2 10 3 Z ( m ) Z ( m ) F i g u r e 6.9: I n v e r s i o n o f d a t a t h a t h a v e b e e n c o r r e c t e d f o r E M c o u p h n g a t / = 0.01Hz. H e r e t h e c o r r e c t i o n i s o b t a i n e d b y u s i n g t h e t r u e c o n d u c t i v i t y s t r u c t u r e . T h e a m p l i t u d e a n d p h a s e o f t h e c o m p l e x a p p a r e n t r e s i s t i v i t y pa(iw) a r e r e s p e c t i v e l y s h o w n i n ( a ) a n d ( b ) . T h e n o i s e - c o n t a m i n a t e d d a t a a r e p l o t t e d w i t h e r r o r b a r s . T h e u n c o r r e c t e d d a t a a r e i n d i c a t e d b y d i a m o n d s . T h e s o l i d l i n e i n d i c a t e s p r e d i c t e d r e s p o n s e s d u e t o t h e m o d e l r e c o v e r e d f r o m t h e i n v e r s i o n . P a n e l s ( c ) a n d ( d ) s h o w t h e t r u e a n d r e c o v e r e d c o m p l e x c o n d u c t i v i t y . Chapter 6. ID Complex conductivity inversion after EM Couphng removal 176 10 1 10 2 Z ( m ) Figure 6.10: Inversion of data that have been corrected for E M couphng at / = 0.1Hz. Here the correction is obtained by using the true conductivity structure. The amphtude and phase of the complex apparent resistivity pa(iu) are respectively shown in (a) and (b). The noise-contaminated data are plotted with error bars. The uncorrected data are indicated by diamonds. The sohd line indicates predicted responses due to the model recovered from the inversion. Panels (c) and (d) show the true and recovered complex conductivity. Chapter 6. ID Complex conductivity inversion after EM Couphng removal 177 0 . 0 0 8 0 . 0 0 6 -" ~Q 0 . 0 0 4 -J f . 0 . 0 0 2 ^ 0 . 0 0 0 -0 .002 -0 .004 -0 .006 1 0 3 1 0 ° F i g u r e 6.11: I n v e r s i o n o f d a t a t h a t h a v e b e e n c o r r e c t e d f o r E M c o u p h n g a t / = 1 0 H z . H e r e t h e c o r r e c t i o n i s o b t a i n e d b y u s i n g t h e t r u e c o n d u c t i v i t y s t r u c t u r e . T h e a m p l i t u d e a n d p h a s e o f t h e c o m p l e x a p p a r e n t r e s i s t i v i t y pa(iu>) a r e r e s p e c t i v e l y s h o w n i n ( a ) a n d ( b ) . T h e n o i s e - c o n t a m i n a t e d d a t a a r e p l o t t e d w i t h e r r o r b a r s . T h e u n c o r r e c t e d d a t a a r e i n d i c a t e d b y d i a m o n d s . T h e s o l i d l i n e i n d i c a t e s p r e d i c t e d r e s p o n s e s d u e t o t h e m o d e l r e c o v e r e d f r o m t h e i n v e r s i o n . P a n e l s ( c ) a n d ( d ) s h o w t h e t r u e a n d r e c o v e r e d c o m p l e x c o n d u c t i v i t y . Chapter 6. ID Complex conductivity inversion after EM Couphng removal 178 10 10 10 10 ' 10" 10' Frequency (Hz) 10 10 10 Frequency (Hz) F i g u r e 6.12: C o m p l e x c o n d u c t i v i t y m o d e l as a f u n c t i o n o f f r e q u e n c y , ( a ) R e a l p a r t a n d ( b ) i m a g i n a r y p a r t o f a(iu>). T h e s o h d l i n e i s t r u e m o d e l a t t h e d e p t h o f z = 4 5 m . T h e d o t s r e p r e s e n t s t h e e s t i m a t e d v a l u e s o b t a i n e d f r o m t h e i n v e r s i o n o f c o m p l e x r e s i s t i v i t y d a t a a t five f r e q u e n c i e s (0.001, 0.01, 0.1, 1.0, 10.0 H z ) . Chapter 7 Two-dimensional Inversion of Spectral Parameters from Frequency Domain Data 7.1 Introduction I P d a t a i n the f ie ld are the end result of the various con t r ibu t ions m a d e b y different I P sources. T h e t r a d i t i o n a l ob jec t ive i n I P da t a in t e rp re t a t ion is to o b t a i n i n f o r m a t i o n about the chargeab i l i ty of the subsurface. C u r r e n t me thodo logy is to inver t the I P d a t a (ei ther P F E or phases i n f requency d o m a i n , or m i l l i v o l t / v o l t i n t i m e d o m a i n ) and recover a m o d e l tha t is a m a p of chargeabi l i ty d i s t r i b u t i o n ( O l d e n b u r g and L i , 1994). H i s t o r i c a l l y the I P m e t h o d was p r i m a r i l y used i n base m e t a l exp lo ra t i on to detect su lph ide ore bodies especia l ly p o r p h y r y deposits i n the subsurface. A l t h o u g h most su lph ide minera l s are chargeable , so too are graphi tes and clays. T h e current inve r s ion a lgor i thms tha t w o r k w i t h the conven t iona l d a t a types l ike phases, P F E or m i l l i v o l t / v o l t , p rov ide o n l y a m a p of cha rgeab i l i ty a n d this does not help to d i s t ingu i sh be tween the different ma te r i a l s tha t can p o t e n t i a l l y cause the I P effects i n the g round . T h i s raises the ques t ion: C a n we de te rmine w h i c h minera l s are caus ing the I P effects i n the ground? C a n the detai ls about the I P response be used to d i sc r imina t e between different rock types or minera l s ( l ike su lphides , graphi tes , clays etc) caus ing the I P effects? T o answer these quest ions , geophys ica l i n s t r u m e n t a t i o n was developed that were capable of acqu i r ing de ta i led i n f o r m a t i o n of I P da ta . T h i s l ed to I P measurements at m u l t i p l e frequencies i n f requency d o m a i n a n d at m u l t i p l e t i m e channels i n the t i m e d o m a i n . T h e m u l t i p l e f requency d a t a are also ca l l ed 179 Chapter 7. 2D Inversion for Cole-Cole Parameters 180 spectral IP or complex resistivity measurements. The variation in the amphtude and phase of the measured impedance (voltage/current) with frequency provided the motivation to study the problem. It has been suggested that IP phenomena are essentially a low frequency dispersion in the conductivity of the earth materials (Wait,1959a,b; Madden and Marshall, 1959; Van Voohris et al., 1973; Shuey and Johnson, 1973; Angoran and Madden, 1977; Pelton et al., 1978a; Wong, 1979). Therefore to model this frequency dispersion simplified circuit models were developed that could explain or at least represent the frequency dependence of the data in fre-quency domain or the decay characteristic of the transient curve in time domain. This lead to the emergence of various simplified frequency dispersion models for the earth materials to represent the IP effect observed in field and laboratory data. It was hoped that the parameters of these models could then be used as diagnostic tools to answer the question of mineral discrimination. Inital attempts to investigate the mineral discrimi-nation problem using the spectral IP measurements first appeared in the work of Pelton et al. (1978a). In the spectral IP data interpretation we are interested in the information about the complex conductivity of the earth. To extract information about the complex conductiv-ity there can be various options to investigate the problem. Here I outline three options which are possible routes to approach the inverse problem. These are: (a) Data at each frequency u>k can be inverted separately to obtain a conductivity model a(x, z,u>k)- Therefore for each ijth cell in the 2D model there are conductivity values cr(xi, Zj, u>k) for k = 1,..., NF where NF is the number of frequencies used in the survey. The next step is to make inferences from the complex conductivity values at each cell obtained at different frequencies. One possible approach to obtain such information is by fitting an appropriate frequency dispersion model to the complex conductivity values for each cell. The parameters of the frequency dispersion model can then be used to make Chapter 7. 2D Inversion for Cole-Cole Parameters 181 some inferences about the mineral discrimination problem. For example, if a single Cole-Cole model is fitted to the complex conductivity values obtained at different frequencies, then inferences are based on the 2D distribution of the Cole-Cole parameters. A n analogous procedure described above for multiple frequency domain data can be applied in the time-domain for multiple time channel measurements. Previous attempts to recover Cole-Cole parameters in 2D have been carried out by Yuval and Oldenburg (1997) using time-domain IP data. Their procedure was to invert data at individual time channels and ultimately obtain a value of 77, r and c in each cell of the parameterized model. The procedure was as follows: First the data at individual time channels (tk) are inverted to obtain a chargeability value r,(tk) in each cell. Thus the inversion of multiple time channel data resulted in a decay curve for chargeability in each cell. The next step was to use very fast simulated annealing to carry out parametric inversion of the decay curve in each cell to recover the parameters of the Cole-Cole model. In their approach the inversion for an individual time channel was carried out using a normalization cri-teria such that results obtained from separate time channel inversions could be properly combined. Because there is a question of compatibility of the models obtained from dif-ferent time channel inversions the result will depend on the normalization criteria used, which can be questionable. The difficulty in their approach is that the distribution of the recovered parameter is noise sensitive and therefore a smooth distribution is not easily achieved. This is because parameters of each cell are inverted independently and there is no regularization imposed that would generate smooth distribution of the spectral parameters. (b) In the second approach to obtain spectral information about the polarization, data obtained at multiple frequencies can be inverted jointly to obtain an estimate of the complex conductivity. This method is yet to be implemented but the possibility exists to proceed along this direction. To do so the model for the inverse problem will Chapter 7. 2D Inversion for Cole-Cole Parameters 182 be three-dimensional in nature given by <T(X,Z,OJ), where the first two dimensions are spatial and the third dimension is frequency. So a generic model objective function in this case will have components from the two spatial terms and one frequency term. And the model objective function should have some ability to take into account the realistic changes in conductivity with frequency. Thus the goal will be to extract the real and imaginary part of complex conductivity (Tr{x, z,u>) and <7i(x, z,u>) from the multiple frequency data d = d(x = xi, ui — u>k) for I = 1,..., N and k = 1,. . . , NF, where N is the number of electrode positions on the surface and NF is the number of frequencies. The advantage of this method hes in its general approach and the ability to handle all of the data together. The method does not assume any particular form of frequency dispersion model. However there may be some difficulty in formulating the model objective function that incorporates the behaviour of the real and imaginary conductivity with frequency. Also the 3D nature of the model will result in large system of equations to be solved. (c) In the third approach, a particular frequency dispersion model is assumed for the complex conductivity of the medium (for example the Cole-Cole model) and the parameters of this model are then obtained by inverting all of the data at multiple frequencies. Various possibilities exist in chosing the frequency dispersion model. Among the several models that have been tested, the Cole-Cole model is the most commonly used among the workers in the field. Pelton et al. (1978a) first suggested the Cole-Cole model to study the complex res-istivity behaviour of the earth. It should be noted that the Cole-Cole model, as strictly defined refers to the response of complex dielectrics, however the same form of frequency dispersion has been applied to complex resistivity responses. The three parameter model with n, T and c has been used by Pelton et al. (1978a) in an attempt to discriminate between the IP sources. Although the Cole-Cole model was developed for frequency do-main responses, but it is also used to interpret time-domain responses (Johnson; 1984; Chapter 7. 2D Inversion for Cole-Cole Parameters 183 Seigel et a l . , 1997). Inves t iga t ion w i t h the C o l e - C o l e m o d e l have i n d i c a t e d tha t , the ra te of decay, a n d the shape of the t ransient decay, can p rov ide i n f o r m a t i o n about the chargeable b o d y (Seigel et a l . , 1997). It has been r epor t ed tha t the ra te of decay of voltages is i n d i c a t i v e of the m e a n size of me ta l l i c grains. T h i s w o u l d m e a n tha t the t i m e constant r is suggestive of average g ra in size d i s t r i b u t i o n of the I P targets . V a n h a l a a n d P e l t o n i e m i (1992) r epor t ed that r varies as the square of the average g r a i n size. T h e r e l a x a t i o n constant c, is suggestive of the d i s t r i b u t i o n p a t t e r n of the par t ic les . Seigel et al . (1997) suggests tha t a h igh value of c indica tes a more u n i f o r m d i s t r i b u t i o n t h a n low In this w o r k I use the t h i r d approach and de te rmine the C o l e - C o l e parameters f r o m m u l t i p l e f requency c o m p l e x res i s t iv i ty da ta . T h e I P da t a con ta in d o m i n a n t i n f o r m a t i o n about chargeab i l i ty c o m p a r e d to r and c. So e x t r a c t i n g i n f o r m a t i o n about ( r , c) is a difficult ob jec t ive c o m p a r e d to recover ing on ly i n f o r m a t i o n about rj. T h e ob jec t ive here is to fo rmula te the p r o b l e m as a nonl inear inverse p r o b l e m and o b t a i n a s m o o t h d i s t r i b u t i o n of the three parameters (77 , r , c) . 7.2 F o r w a r d P r o b l e m I n a c o m p l e x res is t iv i ty , or spec t ra l I P survey, a h a r m o n i c current is t r a n s m i t t e d i n to the g r o u n d a n d a c o m p l e x vol tage is measured at the receiver. T h e ra t io of the measu red vol tage to the i npu t current gives the measured impedance . Idea l ly i f the g r o u n d is chargeable, and the effects due to E M couphng are absent, t h e n the govern ing equa t ion for the measured c o m p l e x vol tage can be represented as (Wel le r et a l . , 1996), A l t h o u g h the quant i t ies are c o m p l e x i n eqn (7.1), the f o r m of the equa t ion is the same as the u sua l D C res i s t iv i ty equa t ion . T h e c o m p l e x res i s t iv i ty for the C o l e - C o l e m o d e l for values of c. (7.1) Chapter 7. 2D Inversion for Cole-Cole Parameters 184 t h e jth c e l l a t f r e q u e n c y u)k i s g i v e n b y w h e r e p0j i s t h e D C r e s i s t i v i t y , m t h e c h a r g e a b i H t y , Tj t h e t i m e c o n s t a n t a n d Cj i s t h e r e l a x a t i o n c o n s t a n t i n t h e jth c e l l o f t h e d i s c r e t i z e d m e d i u m . L e t t h e r e s p o n s e d u e t o t h e D C r e s i s t i v i t y o f t h e m e d i u m b e d e n o t e d b y J~^c(po), a n d t h a t d u e t o t h e c o m p l e x r e s i s t i v i t y b y Tdc(p(iu>)). E q u a t i o n ( 7.2) c a n b e w r i t t e n as p(iu>) — p0(l — m(iu>)), w h e r e m(iu>) i s t h e c o m p l e x c h a r g e a b i l i t y g i v e n b y L i n e a r i z i n g t h e r e s p o n s e d u e t o c o m p l e x r e s i s t i v i t y u s i n g T a y l o r ' s e x p a n s i o n , a n d n e g -l e c t i n g t h e h i g h e r o r d e r t e r m s , I o b t a i n , • ^ d c ^ - ^ d c W ^ ) ] I , X /• X — S - f l = ~ 22 Jimfak) (7.4) ^ d c ^ ° J j=l w h e r e Jij = —dJ^c/dhi(pj) w h e r e / d e n o t e s p a r t i c u l a r e l e c t r o d e p o s i t i o n . T h e d a t a r)la(wk) c a n b e o b t a i n e d i f t h e i m p e d a n c e s a r e a v a i l a b l e a t e a c h f r e q u e n c y . F o r e x a m p l e i f t h e i m p e d a n c e m e a s u r e d a t f r e q u e n c y u>k i s d e n o t e d b y Z(iujk), a n d t h e m e a s u r e d D C i m p e d a n c e i s d e n o t e d b y Zdc, t h e n t h e c o m p l e x a p p a r e n t c h a r g e a b i l i t y rjla(uJk) a t t h e Ith r e c e i v e r l o c a t i o n c a n b e w r i t t e n as V a M = 1 =j • (7.5) T h i s t y p e o f d a t a c a n b e g e n e r a t e d s i n c e i m p e d a n c e s a t m u l t i p l e f r e q u e n c i e s a r e c o l l e c t e d i n a c o m p l e x r e s i s t i v i t y s u r v e y . nla(u>k) i s a c o m p l e x q u a n t i t y a n d t h e r e f o r e t h e d a t a c a n b e e i t h e r r e a l a n d i m a g i n a r y , o r a m p l i t u d e a n d p h a s e . H e r e t h e d a t a a r e c o n s i d e r e d t o b e t h e r e a l a n d i m a g i n a r y p a r t o f r/la(u)k)-Chapter 7. 2D Inversion for Cole-Cole Parameters 185 7.3 S e n s i t i v i t y In order to solve the inverse p r o b l e m we need to c o m p u t e the sens i t iv i ty w i t h respect to three parameters n, r a n d c. T h e in t r in s i c chargeabih ty n a n d the r e l a x a t i o n cons tant c varies be tween 0 and 1. T h e t i m e constant can va ry over order of magn i tudes ( P e l t o n et a l . , 1978; Seigel et a l . , 1997). Therefore the parameters for the inve r s ion are chosen to be n,log{r) a n d c. T h e d a t u m for the observa t ion poin t xi a n d at f requency u>k c a n be represented as M dik = Va(xi,ujk) = ^ 2 J^VjR(Tji c,-,w f c) (7.6) 3 = 1 where , i i ( r ) , c ^ t ) = ( l - 1 + ( ^ T j ) , ) (7 .7 ) T h e sens i t iv i ty w i t h respect to r]j, log(rj) and Cj can be ob ta ined by dif ferent ia t ion of R. T h i s is g iven b y ddth JljR(Tj,Cj,Llk) (7.8) ddiu dR(Tj,Cj,ujk) = JUVi a > . / X ( 7 - 9 ) dlogin) dlogiTj) ddik dRjr^c^Uk) , . -de- = J l j V j ^ ( 7 ' 1 0 ) T h e sens i t iv i ty w i t h respect to rjj can be easi ly ob ta ined since R(Tj,Cj,u>k) is a l ready k n o w n . T h e sens i t iv i ty for log( r j ) and Cj can be c o m p u t e d ana ly t i caUy b y di f ferent ia t ing the express ion of R. A f t e r different iat ion the fo l lowing expressions are ob t a ined , dR{Tj,Cj,u;k) ((iojkTj)C3 log ( tW) 8Cj \ [1 + (iutTtf'f (7.11) (7.12) Chapter 7. 2D Inversion for Cole-Cole Parameters 186 7.4 I n v e r s e P r o b l e m The objective here is to formulate a robust inversion algorithm to recover the spectral parameters from multi-frequency data. The model is parameterized into rectangular cells and in each cell there are three parameters 7?,f,c to be recovered where r = log(r). A generic model objective function for the three parameters can be written as (7.13) where a s , Q j and az are the control parameters which control the relative importance of the different components of the model norms. For each parameter there are three components: the smallest model norm which minimizes the closeness of the model with a reference model and two flattest model norm which penalizes the derivative in x- and z-directions. T7 0 , fo and CQ are the reference models for intrinsic chargeability, time constant and relaxation constant. an,a? and ac are the control parameters which regulate the amount of model norm of each of the parameters in the model objective function. The general inverse problem in the discretized form can be represented mathematically as min 4>m = av\\Wv(v - 7 ? 0 ) | | 2 + a*\\W*(T - f 0 ) | | 2 +ac\\Wc(c-c0)\\2 subject to 4>d=\\Wd{d°hs - F(T7,f,c))112 = ^ Chapter 7. 2D Inversion for Cole-Cole Parameters 187 where (j>m is the m o d e l ob jec t ive func t ion , <f>d is the da t a cons t ra in t and 4>*d is the target misf i t to be achieved i n the invers ion . Wn, Wf and Wc are the we igh t ing ma t r i ces for 77, f a n d c respect ively . E a c h of the weigh t ing mat r ices (Wv, W?,WC) is composed of a smaUest m o d e l we igh t ing m a t r i x W„, and two flattest m o d e l we igh t ing mat r i ces Wx and Wz. F o r example WTWV = ct.WfW, + axWTWx + azWTWz. Wd is the d a t a we igh t ing m a t r i x , w h i c h is a d iagona l m a t r i x h a v i n g the r ec ip roca l of the s t a n d a r d d e v i a t i o n for each d a t u m . T h e m o d e l ob jec t ive func t ion <j>m can also be w r i t t e n as <t>m = \\Wv(n - Vo)\\2 + ( 0 , / a O ( f - r0)\\2 +(ac/cXrl)\\Wc(c - c0)\\2 T h e inverse p r o b l e m is solved by m i n i m i z i n g a g loba l ob jec t ive func t i on <f> = (f>d + (3(f>m where 0 is the regular i sa t ion parameter . M i n i m i z a t i o n of the g loba l ob jec t ive func t i on leads to a sys t em of equat ions g iven by (JTJ + 0kWlWm) 8m = JT8d - 0kWlWm{mk - m0) (7.16) where J = [Jv,Jf,Jc), W^Wm = a.WfWr, + afWTWr + aeW?Wc, m = ( 7 / , f , c ) r and 8d = dobs — F(rjh,rk,ck). A t each i t e r a t ion (k) we choose a r egu la r i za t ion pa rame te r 0k and solve for the m o d e l p e r t u r b a t i o n 8m. T h e u p d a t e d m o d e l for the next i t e r a t i o n is g iven by mk+1 = mk -f £ * 8m, where £ £ [0,1] is the step l eng th pa ramete r . ( is chosen such tha t the g loba l ob jec t ive func t ion decreases i n successive i t e ra t ions . T h i s is 0k<t>m(™k+l) + Mmk+l) < BkMmk) + fa{mk) (7.17) T h e r egu la r i za t i on paramete r 0k is changed i n successive i te ra t ions us ing a coo l ing sched-ule g iven b y 0k+1 = \ * 0k where A € [0,1) (7.18) Chapter 7. 2D Inversion for Cole-Cole Parameters 188 For this problem I have chosen A = 0.5. I start with a high value of 3° and use the cooling schedule to decrease it. However as the data misfit approaches the target misfit 4>d, A —> 1. Therefore the choice of A can be written as A = max (Mmk) - ti) 0.5,1 (7.19) ti In situations where the predicted data misfit undershoots the target misfit (<j>d(mk+1) < ti) then the regularization 3k is increased (since A > 1), so that we avoid over fitting the data. 7 . 5 S y n t h e t i c E x a m p l e In this section a synthetic example is considered to test the recovery of chargeabihty and spectral parameters of the Cole-Cole model. The model is a two-block prism in a ho-mogeneous half-space shown in Figure 7.1. Figure 7.1(a) indicates the true chargeabihty model. The left prism has chargeabihty r,i = 0.12 and the right prism has 7?2 = 0.1. The background 77 = 0.01. The time constant for the left prism T i = 100sec and that for the right prism r 2 = lOsec shown in Figure 7.1(b). The background r = lsec. The relaxation constants for the left and right prisms are c a = 0.55 and c 2 = 0.35 respectively. The background relaxation constant c = 0.25. The medium is discretized into 528 cells. The data are generated using eqn(7.4) for a dipole-dipole geometry with dipole length of 50m. The conductivity model used to compute the sensitivity J in (7.6) is a homo-geneous half-space of 0.015/ra. The data are collected using 14 transmitters with 6 n-spacings, and at three frequencies logarithmically sampled between 0.125 Hz and 10 Hz. The total number of data to be inverted are 414. The data are contaminated by Gaussian random noise with standard deviation of 5%. For displaying purposes, I present the data in percentage for both the real and imaginary part, i.e., Re(?7a(ct>fc)) * 100 and Im(?7a(u;fc)) * 100. Figure 7.2(a), (c) and (e) display the real part of the data at 0.125 Chapter 7. 2D Inversion for Cole-Cole Parameters 189 S Y N T H E T I C M O D E L "EL 200 g 250 0.000 0.012 0.024 0.036 0.048 0.060 0.072 0.084 0.096 0.108 0.120 (b) Q _ 200 S 250 300 350 0 100 200 300 400 500 600 700 800 D i s t ance (m) I o g 1 0 ( x ) Q_ 200 S 250 100 200 300 400 500 D i s t a n c e (m) 600 700 800 0.25 0.28 0.31 0.34 0.37 0.40 0.43 0.46 0.49 0.52 0.55 Figure 7.1: Synthetic model used to generate the data, (a) Chargeability ( 7 7 ) (b) time constant (r) and (c) relaxation constant (c). Chapter 7. 2D Inversion for Cole-Cole Parameters 190 H z , 1.0 H z a n d 10.0 H z respect ively. F i g u r e 7.3(a), (c) and (e) are the i m a g i n a r y par t s of the d a t a at 0.125 H z , 1.0 H z and 10.0 H z respect ively . T h e m a g n i t u d e of the rea l pa r t of the d a t a increases m a r g i n a l l y w i t h the frequency shown i n F i g u r e 7.2. T h i s is expec t ed because Re(ra( iu ; ) ) —» r? as u> —> oo. T h e i m a g i n a r y par t of the d a t a i n F i g u r e 7.3 indica tes tha t the magn i tude decreases w i t h frequency. T h e inve r s ion is ca r r ied out by m i n i m i z i n g the h o r i z o n t a l a n d v e r t i c a l der iva t ives of the m o d e l g iven i n eqn (7.13) subject to f i t t ing the da ta . O n e of the i m p o r t a n t issues i n the inve r s ion a l g o r i t h m is the choice of the re la t ive weights a+/av a n d ac/av i n eqn (7.15). A l t h o u g h there m a y be various possibi l i t ies to choose the re la t ive weights , I choose a+ av av F o r th is example af/av = 0.0006 and a c l a v = 0.2. T h e s ta r t ing value of the regu-l a r i z a t i o n pa ramete r is set equa l to the s ta r t ing misfi t value (8° = <f>d{m°)). T h e start-i n g r egu la r i za t i on paramete r 8° shou ld be large enough so tha t the s ingular values of gvyV^Wm is greater t h a n the s ingular values of JTJ i n eqn (7.16). T h e s t a r t i ng m o d e l for the inve r s ion is the same as the reference m o d e l w i t h w0 = 0, rQ — 5sec a n d c = 0.25. T h e recovered m o d e l for r,, r and c f rom the invers ion is shown i n F i g u r e 7.4(a),(b) a n d (c) respect ively . T h e models f rom the invers ion ind ica te a smoother represen ta t ion of the t rue m o d e l shown i n F i g u r e 7.1. T h i s is because the m o d e l ob jec t ive f u n c t i o n has a d o m i n a n t flatness componen t . B o t h the a m p l i t u d e a n d the l o c a t i o n of the anomal ies of the three parameters i n F i g u r e 7.4 are i n good agreement w i t h the t rue m o d e l . T h e in t r i n s i c chargeab ih ty (n) a n d the re l axa t ion constant (c) ob t a ined f rom the inve r s ion are pos i t ive , a l t hough no p o s i t i v i t y const ra int was i m p o s e d i n the inver s ion a l g o r i t h m . T h e \w*( true |2 - Vo)\\2 ||T^ c( ct,Ue _ C o ) | | 2 (7.20) (7.21) Chapter 7. 2D Inversion for Cole-Cole Parameters Distance (m) 0.050 0.545 1.040 1.530 2.030 2.520 3.020 3.510 4.010 4.510 5.000 F i g u r e 7.2: R e a l p a r t o f t h e o b s e r v e d d a t a i n p e r c e n t a g e a t ( a ) 0.125 H z ( c ) 1.0 H z a n ( e ) 10 H z . T h e p r e d i c t e d d a t a f r o m i n v e r s i o n a t ( b ) 0.125 H z ( d ) 1.0 H z a n d ( f ) 10 H z Chapter 7. 2D Inversion for Cole-Cole Parameters 192 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 F i g u r e 7.3: I m a g i n a r y p a r t o f t h e o b s e r v e d d a t a i n p e r c e n t a g e a t ( a ) 0.1 H z ( c ) 1.0 H z a n d ( e ) 10 H z . T h e p r e d i c t e d d a t a f r o m i n v e r s i o n a t ( b ) 0.125 H z ( d ) 1.0 H z a n d ( f ) 10 H z . Chapter 7. 2D Inversion for Cole-Cole Parameters 193 p r e d i c t e d rea l and i m a g i n a r y da t a f rom the invers ion , shown i n F i g u r e 7.2 a n d F i g u r e 7.3 respec t ive ly are i n good agreement w i t h the observed da ta . I n p rac t i ce we are in teres ted i n the spec t ra l i n f o r m a t i o n i n the region where there is h i g h chargeabihty . In the region where the chargeabih ty is s m a l l the c o n t r i b u t i o n to the d a t a f r o m the spec t ra l parameters is also s m a l l because d(xi,<x>k) = JiorloP-{TiicjiU}k)-I n o ther words as Vj —•> 0 there is no effect of Tj and Cj is h igher i n regions where there is m o r e chargeabihty . T h e in t e rp re ta t ion of F i g u r e 7.4(b) and (c) shou ld therefore be done i n con junc t ion w i t h F i g u r e 7.4(a). T h e p r i m a r y goal of this inverse p r o b l e m is have some ab i l i t y to del ineate regions of differing r and c. C o n v e n t i o n a l I P inve r s ion a lgor i thms w h i c h w o r k w i t h single f requency d a t a m a y p roduce models s i m i l a r to F i g u r e 7.4(a). T h u s i f we in terpre t on ly F i g u r e 7.4(a), t hen b o t h the chargeable zones w o u l d be equa l ly i m p o r t a n t . Therefore some a d d i t i o n a l i n f o r m a t i o n is r equ i red to d i s t i ngu i sh the two anomal ies . F igures 7.4(b) and (c) show tha t regions h a v i n g different r a n d c values are recovered f r o m the invers ion . T h i s a d d i t i o n a l i n f o r m a t i o n of the chargeable zones o b t a i n e d o n the basis of different r and c values, helps to d i s t i ngu i sh the two I P targets . T o compare the recovered C o l e - C o l e parameters w i t h the t rue m o d e l , I m a s k the regions o f l ow chargeab ih ty i .e. , (77 < 0.02), and replot the recovered m o d e l i n F i g u r e 7.5 a long w i t h the t rue m o d e l . T h e a m p l i t u d e and loca t ion of the recovered 77, r a n d c shown i n F i g u r e 7.5(a), (b) and (c) respec t ive ly are i n good agreement w i t h the t rue m o d e l . 7.6 F i e l d E x a m p l e I n th is sec t ion I consider a f ield example i n w h i c h m u l t i p l e f requency c o m p l e x res is t iv-i t y d a t a are co l lec ted over a copper p o r p h y r y deposi t i n P i m a county, A r i z o n a , U S A ( C o u r t e s y of K e n Zonge) . T h e ore deposi t is ca l led the N o r t h S i lver B e l l deposi t a n d i t is l oca t ed 35 miles nor thwest of Tucson . P o r p h y r y copper m i n e r a l i z a t i o n i n th is area Chapter 7. 2D Inversion for Cole-Cole Parameters 194 Q _ 200 H S 250 0 100 I 200 i i 300 400 500 D i s t a n c e (m) i 600 700 800 1 — • 0.000 0.012 0.024 0.036 0.048 0.060 0.072 0.084 0.096 0.108 0.120 (b) Q _ 200 S 250 300 350 0 100 200 300 400 500 D i s t ance (m) 600 700 800 I og10 ( x ) I— I— I — o H 50 100 -150 Q _ 200 -CD Q 250 -300 -350 -300 400 500 D i s t a n c e (m) 700 800 c 0.25 0.28 0.31 0.34 0.37 0.40 0.43 0.46 0.49 0.52 0.55 F i g u r e 7.4: Recove red m o d e l f rom the invers ion . P a n e l (a) is chargeabi l i ty (rj), (b) t i m e constant ( r ) a n d (c) r e l axa t ion constant (c). Chapter 7. 2D Inversion for Cole-Cole Parameters 195 (a) o_ CD Q o 50 I 100 | 150 200 250 300 350 1 1 1 1 1 1 1 0 100 200 300 400 500 600 700 800 Distance (m) I i i i I i i 0.000 0.024 0.048 0.072 0.096 0.120 ~i 1 1 1 1 r 0 100 200 300 400 500 600 700 800 Distance (m) o.o 50 -100 -sz 150 -Q - 200 -CD 250 -Q 300 -350 -0 100 200 300 400 500 600 700 800 Distance (m) 0.25 0.31 0.37 0.43 0.49 0.55 100 200 300 400 500 600 700 800 Distance (m) 0.000 0.024 0.048 0.072 0.096 0.120 (e) ~i 1 r 0 100 200 300 400 500 600 700 800 Distance (m) ~\ 1 1 i i r 0 100 200 300 400 500 600 700 800 Distance (m) 0.25 0.31 0.37 0.43 0.49 0.55 Figure 7.5: The recovered Cole-Cole parameters in the region of high chargeability. A threshold value of 77 > 0.02 is used to trim the original model in Figure 7.4. Panel (a) is chargeability (77), (b) time constant (r) and (c) relaxation constant (c) from the inversion. This is compared with the true model for (d) 77 (e) t and (f) c. Chapter 7. 2D Inversion for Cole-Cole Parameters 196 lies along the southwest flank of the mountains in hydrothermally altered igneous rocks. The complex resistivity method involves the measurement of impedance, both in-phase and quadrature components, over a frequency range of 0.1 Hz - 100 Hz. The data were acquired using dipole-dipole survey with dipole length of 76.2m (250ft) for 8 N-spacings. 26 transmitter location were used in the survey. The inversion of complex resistivity data for the spectral parameters involves three steps. In the first step, the amphtudes of apparent resistivity data at the lowest frequency are inverted for a 2D conductivity structure a(x,z). This conductivity structure is used to compute the sensitivity matrix J for the forward problem given in eqn (7.6). In the second step the complex apparent chargeabihty data nak are generated from the measured impedance data using eqn (7.5). For generating these data the impedance measured at the lowest frequency is assumed to be ZDC — Z(u> = ijoiouj). In the third step the complex apparent chargeability data rjak are inverted for the Cole-Cole parameters ( T J , T , c). The observed apparent resistivity data obtained at the lowest frequency (0.125 Hz) are shown in Figure 7.6(a). The 2D conductivity structure recovered from the inversion is shown in Figure 7.7. The conductivity model indicates a shallower conductive zone at depth of 100m near x= 1000m and a deeper conductive region between x= 1200m and 1500m. The predicted apparent resistivity data shown in Figure 7.6(b) indicates a good agreement with the observations in Figure 7.6(a). The conductivity model shown in Figure 7.7 is used to compute the sensitivity matrix J needed in the forward modehng computation for spectral IP responses (7.4). In the next step I generate the spectral IP responses (nak) from the measured im-pedances at seven frequencies f=[0.625, 0.875, 1.125, 3.0, 5.0, 7.0, 9.0] Hz. The complex impedances at these seven frequencies are normalized with the impedance measured at 0.125 Hz to produce the apparent complex chargeabihty data given in eqn (7.5). For display purposes I present the real and imaginary part of the complex chargeabihty data Chapter 7. 2D Inversion for Cole-Cole Parameters 197 1 2 g> 3 § i & 5 1 2 g> 3 2.51 Conductivity Pseudosection 3.80 •TV Vs5-K< '•1 500 1000 1500 Distance (m) I | - Obs 0 500 1000 1500 2000 ' ! / P r e d 2000 CTa(mS/m) 5.75 8.71 13.20 20.00 F i g u r e 7.6: (a) O b s e r v e d apparent res i s t iv i ty da t a ob ta ined at the lowest frequency, (b) P r e d i c t e d apparent res i s t iv i ty da t a f rom the invers ion . a 0 Q 200 500 1000 1500 Distance (m) 2000 a (mS/m) 1.78 2.54 3.63 5.19 7.41 10.60 15.10 21.60 30.90 44.20 63.10 F i g u r e 7.7: 2 D C o n d u c t i v i t y m o d e l ob ta ined f rom the invers ion . Chapter 7. 2D Inversion for Cole-Cole Parameters R e a l P a r t E 2 C O b ^ 8 E 2 ° - 6 obs 0.625Hz C O 8 pred X X X X X \ 'X x X \ " - " X N — " / / 1 \ \ \ X x X \ * f i X X \ X 0.625Hz j r 0 5 0 0 1 0 0 0 1 5 0 0 2 0 0 0 E 2 "8 4 CO b ± 8 pred j u \ : 0 ^A'lM /<)% **/ \ "V- ~ i \ 0.875Hz ¥) \ v J \ 0 5 0 0 1 0 0 0 1 5 0 0 2 0 0 0 cr> "o CO o_ co I C D Q . co obs I o; /-> X ^ - X - X X - .. • 1.125Hz | 0 5 0 0 1 0 0 0 1 5 0 0 Dis tance (m) I I | i— pred \j ) ~ l k J X \ ^ R X ( 1.125Hz 2 0 0 0 0 . 0 1 . 5 3 . 0 4 . 5 6 . 0 7 . 5 9 . 0 1 0 . 5 1 2 . 0 1 3 . 5 1 5 . 0 F i g u r e 7.8: R e a l p a r t o f t h e d a t a Chapter 7. 2D Inversion for Cole-Cole Parameters 199 at s ix frequencies. F i g u r e 7.8 (a),(c) and (e) shows the rea l par t of the d a t a at 0.625, 0.875 a n d 1.125 H z respect ively. F i g u r e 7.9 (a),(c) a n d (e) shows the rea l par t of the d a t a at 5.0, 7.0 a n d 9.0 H z respect ively. T h e rea l par t of d a t a shown i n F i g u r e 7.9 a n d 7.10 i nd i ca t e tha t the a m p l i t u d e of the d a t a increases w i t h the frequency suggest ing tha t there is subs tan t i a l f requency d ispers ion i n the da ta . T h u s the d a t a are su i tab le for spec t r a l I P analysis us ing frequency d ispers ion m o d e l l ike C o l e - C o l e . T h e i m a g i n a r y par t of the c o m p l e x chargeabih ty da t a at 0.625, 0.875 and 1.125 H z is shown i n F i g u r e 7.10(a), (b) and (c) respect ively . T h e da t a at 5.0, 7.0 and 9.0 H z are presented i n F i g u r e 7.11(a), (b) a n d (c) respect ively . In the next step I inver t the apparent c o m p l e x chargeabih ty d a t a for the C o l e - C o l e parameters . T h e m o d e l ob jec t ive func t ion for the inve r s ion is a c o m b i n a t i o n of smallest and flattest m o d e l no rms w i t h a dominan t flattest componen t i n eqn(7.13) . T h e re la t ive weights for the m o d e l ob jec t ive func t ion i n (7.15) were chosen to be c t ^ / a^ = 0.0002 a n d otf/ajf = 0.2. T h e choice was made by r u n n i n g a n u m b e r of invers ions for different values of the re la t ive weights . F i g u r e 7.12(a), (b) and (c) shows the recovered m o d e l for T/, T a n d c o b t a i n e d f rom the invers ion . T h e models have been p l o t t e d to depths greater t h a n the d e p t h of inves t iga t ion ( i n this case a p p r o x i m a t e l y 350m) to show the general anomalous features. T h e in t r i n s i c chargeabihty m o d e l indica tes two regions of h i g h chargeab ih ty i n F i g u r e 7.12(a) w h i c h has cor responding h igh values of c i n F i g u r e 7.12(c). T h e t i m e constant r shown i n F i g u r e 7.12(b) indica tes tha t i n the top 100m of the m o d e l there is a region of l ow t i m e constant ( r < 0.01) o n the left par t of the m o d e l (be tween x = 3 0 0 m a n d x = 1 2 0 0 m ) c o m p a r e d to the r ight side (where r > 0.05). T h e fit to d a t a are shown i n con junc t i on w i t h the observed responses. B o t h rea l a n d i m a g i n a r y d a t a i nd i ca t e g o o d agreement w i t h the observat ions. T o m a k e a geological i n t e rp re t a t ion I have rep lo t t ed the spec t ra l I P mode ls to a d e p t h of 274m and super imposed on t h e m some avai lable geological i n f o r m a t i o n k n o w n Chapter 7. 2D Inversion for Cole-Cole Parameters 2 0 0 Chapter 7. 2D Inversion for Cole-Cole Parameters I m a g i n a r y P a r t i> 2 "° 4 ca Q . t o °- 6 8 obs 1 . 1 2 5 H z r-1^ nHflraSfRSHHl W 2 CO B ^ 8 p r e d 1 . 1 2 5 H z o 500 1000 1500 Distance (m) 2000 0.0 0.8 1.6 2.4 3.2 4.0 4.8 5.6 6.4 7.2 8.0 F i g u r e 7.10: I m a g i n a r y p a r t o f t h e d a t a . Chapter 7. 2D Inversion for Cole-Cole Parameters 202 I m a g i n a r y P a r t 500 1000 1500 2000 0 500 1000 1500 2000 0 500 1000 1500 2000 D i s t a n c e (m) 0.0 0.8 1.6 2.4 3.2 4.0 4.8 5.6 6.4 7.2 8.0 F i g u r e 7.11: I m a g i n a r y p a r t o f t h e d a t a . Chapter 7. 2D Inversion for Cole-Cole Parameters 203 from the nearby drill hole and surface geological mapping (Forman, 1994). The two main geologic units are intrusive quartz monzonite porphyry (denoted by Mz) and dacite porphyry (denoted by Dp). Drillhole information extends downward from the surface to a maximum depth of 250 ft (76.2m). The mapped location of the chalcocite (Cu2S) enrichment blanket, shown in white (Figure 7.13), is based on a logging criterion of greater than 0.4% total copper. The intrinsic chargeabihty model in Figure 7.13(a) indicates that the high chargeabihty region with n = 0.2 — 0.25 between x=1200 and x=1500m correlates with the zone of dacite porphyry that has undergone phyllic alteration (Forman, 1994). It has been suggested by Forman (1994) that the phyllic alteration assemblage is one of the cause of strong IP responses in this region that has overprinted the IP response from the enrichment blanket. In general, the right side of the model has high chargeabihty values compared to left side. The intrinsic chargeabihty recovered in the region of enrichment blanket is in the range 0.1-0.15. The time constant r of the enrichment zone is marked by low values compared to the surrounding region. Unfortunately there is no ground truth to verify the values of spectral Cole-Cole parameters obtained from the inversion. 7.7 C o n c l u s i o n s The goal of this chapter is to develop an inversion algorithm to obtain spectral IP in-formation from multi-frequency IP data. As a first step I assume that there is no E M couphng contamination or that it has been removed from the data. I consider that the frequency dispersion of conductivity (complex conductivity) for earth materials can be adequately modeled by a Cole-Cole model which has been proposed by number of work-ers in this field. The forward modehng for the spectral IP data is then carried out using a linearised mapping between the intrinsic complex chargeabihty and the data. This is valid when the chargeabihty values are small enough so that higher order contributions Chapter 7. 2D Inversion for Cole-Cole P a r a m e t e r s 204 Time constant (x) §- 400 Q 500 1 0 300 i 600 l 900 i 1200 I 1500 l I 1800 2100 Iog10(x) i | pm^ms^ | | | p p -2.000-1.880 -1.750 -1.620-1.500-1.380 -1.250 -1.120-1.000 -0.875 -0.750 Relaxation constant (c) 900 1200 1500 Distance (m) 1800 2100 0.30 0.34 0.38 0.42 0.46 0.50 0.54 0.58 0.62 0.66 0.70 Figure 7.12: Recovered mode l obtained from inversion of field data from the N o r t h Silver B e l l deposit. Pane l (a) intr insic chargeabihty (b) t ime constant and (c) relaxation constant Chapter 7. 2D Inversion for Cole-Cole Parameters 205 Intrinsic Chargeability (r\) (a) 0.000 0.026 0 .052 0.078 0.104 0.130 0.156 0.182 0.208 0.234 0.260 Time constant (x) (b) - 2 . 0 0 0 - 1 . 8 8 0 -1.750 - 1 . 6 2 0 - 1 . 5 0 0 - 1 . 3 8 0 -1 .250 - 1 . 1 2 0 - 1 . 0 0 0 -0.875 -0 .750 Relaxation constant (c) (c) 0 3 0 0 6 0 0 9 0 0 1200 1500 1800 2 1 0 0 Distance (m) 0.30 0.34 0 .38 0.42 0.46 0 .50 0.54 0.58 0.62 0.66 0.70 Figure 7.13: Geological information overlayed on the inversion model . T h e white region is the zone of supergene enrichment. T h e symbols D p : dacite porphyry and M z : quartz monzonite. Panel : (a) intr insic chargeability (b) t ime constant (sec) and (c) relaxation constant Chapter 7. 2D Inversion for Cole-Cole Parameters 2 0 6 c a n b e n e g l e c t e d . T h e n e x t s t e p w a s t o d e v e l o p a n i n v e r s i o n a l g o r i t h m w h i c h e x t r a c t s t h e s p e c t r a l C o l e - C o l e p a r a m e t e r s f r o m m u l t i - f r e q u e n c y c o m p l e x r e s i s t i v i t y d a t a . T h e i n v e r s i o n i s f o r m u l a t e d b y m i n i m i z i n g a m o d e l o b j e c t i v e f u n c t i o n s u b j e c t t o a d e q u a t e l y f i t t i n g t h e d a t a . T h e i n v e r s i o n i s s o l v e d i t e r a t i v e l y u s i n g G a u s s - N e w t o n m e t h o d . T h e o b j e c t i v e f u n c t i o n i s m i n i m i z e d b y f i x i n g t h e r e g u l a r i z a t i o n p a r a m e t e r a t e a c h i t e r a t i o n . A n a p p r o p r i a t e s t e p l e n g t h i s c h o s e n so t h a t t h e m a g n i t u d e o f t h e o b j e c t i v e f u n c t i o n r e d u c e s f r o m o n e i t e r a t i o n t o t h e n e x t . I s t a r t w i t h l a r g e r e g u l a r i z a t i o n p a r a m e t e r a n d d e c r e a s e i t . A t t h e f i n a l i t e r a t i o n w h e n t h e d a t a m i s f i t i s c l o s e t o t h e d e s i r e d m i s f i t , t h e r e g u l a r i z a t i o n p a r a m e t e r i s d e c r e a s e d s l o w l y . T h i s e n s u r e s c o n v e r g e n c e o f t h e a l g o r i t h m w i t h o u t b u i l d i n g e x t r a s t r u c t u r e i n t h e m o d e l . T h e i n v e r s i o n a l g o r i t h m i s t e s t e d o n b o t h s y n t h e t i c a n d f i e l d d a t a e x a m p l e . T h e s y n t h e t i c e x a m p l e s h o w s t h a t t h e r e g i o n s w i t h d i f f e r e n t v a l u e s o f s p e c t r a l C o l e - C o l e p a r a m e t e r s c a n b e r e c o v e r e d f r o m m u l t i - f r e q u e n c y d a t a . T h i s h a s i m p o r t a n t i m p l i c a t i o n s , i f m i n e r a l d i s c r i m i n a t i o n c a n b e a c c o m p l i s h e d o n t h e b a s i s o f t h e i r C o l e - C o l e s p e c t r a l p a r a m e t e r s . T h e i n v e r s i o n m e t h o d d e v e l o p e d i n t h i s c h a p t e r i s a p p l i e d t o a f i e l d d a t a o b t a i n e d f r o m a m i n e r a l e x p l o r a t i o n s u r v e y . D u r i n g t h e t i m e o f t h e r e s e a r c h t h i s w a s t h e o n l y s p e c t r a l I P d a t a s e t a v a i l a b l e w h i c h h a d v e r y l i t t l e E M c o u p h n g c o n t a m i n a t i o n . T h e f o r w a r d m o d e l i n g d o e s n o t t a k e i n t o a c c o u n t E M c o u p l i n g e f f e c t s , s o a n y d a t a t h a t h a v e E M c o u p h n g e f f e c t s n e e d t o b e c o r r e c t e d b e f o r e t h i s i n v e r s i o n a l g o r i t h m c a n b e a p p l i e d . T h e r e s u l t s f r o m t h e f i e l d d a t a r e c o v e r e d z o n e s o f h i g h c h a r g e a b i h t y w h i c h c o r r e s p o n d e d t o t h e z o n e s o f p h y l l i c a l t e r a t i o n s . I n t h e r e g i o n w h e r e t h e m i n e r a l i z e d b o d y o c c u r s t h e r e c o v e r e d i n t r i n s i c c h a r g e a b i l i t i e s a r e l o w t o m o d e r a t e v a l u e s (0.1-0.12). I t i s s u g g e s t i v e t h a t t h e I P r e s p o n s e d u e t o t h e m i n e r a l i z e d z o n e i s m a s k e d b y t h e s t r o n g r e s p o n s e f r o m t h e a l t e r a t i o n z o n e w h i c h i n a c c o r d a n c e w i t h t h e rj v a l u e s i n t h i s r e g i o n . T h e m i n e r a l i s e d z o n e i s m a r k e d b y a l o w t i m e c o n s t a n t v a l u e s c o m p a r e d t o t h e o t h e r r e g i o n s . U n f o r t u n a t e l y , i n t r i n s i c v a l u e s o f t h e t i m e c o n s t a n t s a r e n o t a v a i l a b l e t o v e r i f y Chapter 7. 2D Inversion for Cole-Cole Parameters 207 the values obtained from inversion, but overall, the results are encouraging given the available geological information in the area. The strength of this approach is the ability to handle multi-frequency data and obtain maps of spectral parameters of the Cole-Cole model. The inversion is formulated such that all three parameters can be obtained in a single inversion. However there is some further research required to address some of the issues which are not covered in this work. They are: (a) What is a better way to choose the relative weights of the model objective function. It may be possible to obtain an estimate of the relative weights from the apparent values of the spectral parameters, if they are available, (b) The spectral parameters of the Cole-Cole model are generally bounded with 0 < r, < 1,10 - 4 < r < 103 and 0 < c < 1. These hard bounds can be incorporated in the inversion algorithm using interior-point methods (Li and Oldenburg, 1999). The inversion algorithm presented in this work did not recover any unrealistic values of the parameters, but having this feature in the algorithm will guard against any unrealistic values that are likely to arise when working with other data, (c) A thorough understanding of the relation between the spectral parameters and the mineral type is essential for this method to be successful. Good data sets are required to investigate the problem further. In summary the ability to extract additional information from IP data with inversion methodology is a challenging problem and work presented here provide optimism in that direction. Chapter 8 Conclusions and Future scope Induced polarization surveys have been established as a powerful mineral exploration tool and have also found their application in environmental problems. If we can invert the surface or borehole IP data to delineate the distribution of chargeabihty then we may directly image the region of interest (i.e. chargeable zones). The region of interest can be the mineralized and/or alteration zones for the mineral exploration problem or the zone of contaminants in case of an environmental problem. The survey involves inputing current into the ground using two electrodes and meas-uring the electric fields away from the current source. For a frequency domain survey the source can be a sinusoidal current and for a time-domain surveys the current is a step turn off. The measured electric field from the survey has contributions from both galvanic and inductive response. The IP signal of the chargeable material is buried in the galvanic component. The recorded electric fields, in principle, demand that the full Maxwell's equations for the electric fields be solved. However, when the frequen-cies are small (and the conductivities and lengths of the grounded wires are small) the inductive component can be neglected. The governing DC resistivity equation is: V.(a{iu})VV{iu))) = — I(iu>)8(r — rs). In the frequency domain, either the phase of the impedance V(iu>)/I(iu>), or the amphtude of the impedance at two different frequencies, can be used as IP response. The difficulty arises when there are inductive effects in the data because then the DC resistivity equation is no longer valid. There are two options: (a) either make the data conform to the DC resistivity equation or (b) abandon the D C 2 0 8 Chapter 8. Conclusions and Future scope 2 0 9 r e s i s t i v i t y e q u a t i o n a n d r e p l a c e i t w i t h t h e f u l l M a x w e l l ' s e q u a t i o n s f o r t h e e l e c t r i c f i e l d . A m a j o r p a r t o f t h i s t h e s i s i s f o c u s s e d u p o n t h e f i r s t o p t i o n , t h a t i s , h o w t o r e m o v e E M c o u p h n g c o m p o n e n t f r o m t h e d a t a a n d r e c o v e r c h a r g e a b i h t y d i s t r i b u t i o n o f t h e s u b s u r -f a c e . T h e m a j o r s t e p s a r e o u t l i n e d i n t h e l e f t h a n d c o l u m n o f t h e flow c h a r t i n F i g u r e 8.1. Correct the data Invert for r\ using v • ( o (1 - r,) v § ) = - l 8 ( r-rs) (Chapter 4; Estimate a Estimate a I J Calculate EM responses Calculate EM responses Use full EM equation for Electric field v » v » E + ico|aoE =-icouJ Correct the data Invert for complex conductivity o(ico) Invert for o(ico) using v . (o(ico) v 4>(ico)) = -l(ico) 8( r-rs) (Chapter 5 ) (Chapter 6; F i g u r e 8.1: T h r e e d i f f e r e n t f o r m u l a t i o n s t o r e c o v e r t h e c h a r g e a b i h t y i n f o r m a t i o n f r o m E M c o u p h n g c o n t a m i n a t e d f i e l d d a t a . T h e p h y s i c a l n a t u r e o f t h e I P a n d t h e E M c o u p h n g r e s p o n s e s i s i n v e s t i g a t e d i n C h a p t e r - 4 w i t h f o u r d i f f e r e n t e l e c t r i c fields g e n e r a t e d f r o m a h o r i z o n t a l e l e c t r i c d i p o l e Chapter 8. Conclusions and Future scope 2 1 0 o v e r a c o m p l e x c o n d u c t i v e e a r t h . T h e s e e l e c t r i c fields a r e d e f i n e d , b a s e d o n t h e f r e q u e n c y o f t h e t r a n s m i t t e r c u r r e n t a n d c h a r g e a b i h t y o f t h e g r o u n d . T h e r e m o v a l m e t h o d o l o g y t o s e p a r a t e t h e E M c o u p h n g r e s p o n s e s f r o m t h e I P s i g n a l i s d e r i v e d u s i n g t h e s e e l e c t r i c fields. T h e o b s e r v e d e l e c t r i c field i s e x p r e s s e d as a p r o d u c t o f a n I P r e s p o n s e f u n c t i o n , w h i c h d e p e n d s o n t h e c h a r g e a b i h t y a n d c o n d u c t i v i t y o f t h e g r o u n d , a n d a n e l e c t r i c f i e l d t h a t a r i s e s f r o m i n d u c t i v e a n d D C r e s i s t i v i t y e f f e c t s o f t h e g r o u n d . T h i s r e p r e s e n t a t i o n r e q u i r e s a s s u m p t i o n t h a t c h a r g e a b l e m a t e r i a l s d o n o t g r e a t l y a l t e r t h e p r o p a g a t i o n c o n -s t a n t o f t h e d i f f u s i v e E M w a v e s . T h i s f u n d a m e n t a l r e l a t i o n s h i p b e t w e e n t h e fields w i t h , a n d w i t h o u t , c h a r g e a b i h t y a l l o w s m e t o s e p a r a t e E M c o u p h n g f r o m t h e d a t a . T h e a m p -h t u d e a n d p h a s e o f t h e I P r e s p o n s e f u n c t i o n a r e b o t h i n d i c a t o r s o f c h a r g e a b i h t y . T h i s e n a b l e s m e t o d e v e l o p r e m o v a l m e t h o d s f o r t h e a m p h t u d e a n d p h a s e d a t a . A m p h t u d e d a t a m e a s u r e d i n t h e f i e l d ( a s a p p a r e n t r e s i s t i v i t i e s o r i m p e d a n c e s ) a r e n o t t h e a b s o l u t e m e a s u r e o f c h a r g e a b i h t y . R a t h e r , t o d e t e c t a n I P a n o m a l y , m e a s u r e m e n t s o f a m p h t u d e s a t t w o d i f f e r e n t f r e q u e n c i e s a r e r e q u i r e d . T h e d e v i a t i o n o f t h e r a t i o o f t h e a m p h t u d e s f r o m u n i t y i s a n i n d i c a t i o n o f c h a r g e a b i h t y ; c o m m o n l y k n o w n as P F E d a t a . T h u s t o c o r r e c t t h e P F E r e s p o n s e s , f i r s t t h e E M r e s p o n s e s a r e c o m p u t e d a t t h e t w o f r e q u e n c i e s a n d t h e n t h e y a r e u s e d t o n o r m a l i z e t h e m e a s u r e d a m p h t u d e s a t t h e t w o f r e q u e n c i e s . U n h k e a n a m p h t u d e r e s p o n s e , p h a s e i s a n a b s o l u t e m e a s u r e o f c h a r g e a b i h t y a n d t h e r e f o r e i s a s i n g l e f r e q u e n c y m e a s u r e m e n t . P h a s e d a t a a r e c o r r e c t e d b y s u b t r a c t i n g t h e p h a s e d u e t o E M c o u p h n g f r o m t h e m e a s u r e d p h a s e s . S y n t h e t i c e x a m p l e s i n C h a p t e r -4 s h o w t h a t t h e c o r r e c t e d r e s p o n s e s f o r t h e P F E a n d p h a s e d a t a h a v e a c l o s e m a t c h w i t h t h e t r u e I P r e s p o n s e s , i n d i c a t i n g t h a t s i g n i f i c a n t E M c o u p h n g r e s p o n s e h a s b e e n r e m o v e d . T o c o m p u t e t h e E M c o u p h n g r e s p o n s e s , c o n d u c t i v i t y i n f o r m a t i o n o f t h e g r o u n d i s n e c e s s a r y . T h e I D a n d 3 D s y n t h e t i c e x a m p l e s i l l u s t r a t e t h a t a g o o d a p p r o x i m a t i o n t o t h e t r u e c o n d u c t i v i t y s t r u c t u r e i s r e q u i r e d t o r e m o v e E M c o u p h n g f r o m t h e d a t a . I f w r o n g c o n d u c t i v i t y i n f o r m a t i o n i s s u p p l i e d t h e n i t c a n l e a d t o o v e r - o r u n d e r - c o r r e c t i o n Chapter 8. Conclusions and Future scope 211 of the data. In practice the true conductivity information is not generally available, so a DC conductivity model obtained by inverting the amplitude data at the lowest frequency can be used as an approximation to the true conductivity. Synthetic ID and 3D examples indicate the usefulness of the DC conductivity to compute the E M couphng responses. Typically IP data collected in the field are along profiles, so a 2D interpretation is most appropriate. To handle 2D data sets, an approximate removal methodology is developed in Chapter-4 and is integrated with the 2D IP inversion algorithm to recover 2D chargeability distribution. The method is applied to field data sets from the Acropolis area and the Elura ore deposit in Australia. The results from the Acropolis example indicate that a significant amount of E M couphng has been removed from the data, and the recovered chargeabihty model from the inversion agrees qualitatively with the geological information of the boreholes. The results from this method show significant improvements compared to the three-point and manual decoupling techniques. The data from the Elura ore deposit indicate that the E M couphng is more dominant at higher frequencies. Application of the approximate 2D removal methodology indicate that the model obtained by inverting the decoupled data are in good agreement with the location of the ore deposit. The synthetic and field examples presented in Chapter-4 provide optimism about the removal method developed in the thesis. However, there are several questions that need to be addressed in the future that are important for successful E M couphng removal from the IP data, (a) The approximation kEMIP & kEM should be valid for a given conductivity, frequency and for a particular geometry. The question is whether a quantification of this approximation can be made when the chargeabihty of the medium is unknown. If it can, then the analysis can help provide error bounds on the decoupled data or perhaps be a basis for a further correction of the data, (b) In practice, we have an approximation to the true conductivity ( C T ^ ) to do E M couphng correction. The ID example presented in Chapter 8. Conclusions and Future scope 2 1 2 Chapter-4 iUustrates that a smooth representation of the model might be adequate. This model which fits the low frequency data, is certainly superior to the half-space models. A quantification of how closely a^ needs to be approximated would be useful, (c) Inverting the lowest frequency amphtude data generates a DC conductivity model a0 = c r ^ l — 7 7 ) . Since n(x,y,z) is generally less than 0.3 and since rj(x,y,z) is non-zero only over small portions of the domain, then a0 is expected to be reasonable approximation to a^. If 77 is very large, then the a^ is not well approximated. In such cases it might be necessary to iteratively refine the conductivity used for E M coupling removal. As a possibility, first the data at the lowest frequency are inverted to obtain a conductivity model ao The corrected data with ao are then inverted for a chargeabihty distribution 77. The recovered chargeabihty 77 can used to obtain a new estimate of the conductivity structure a = < T o / ( l — 7 7 ) , that can be used to correct the responses. The common practice is to remove the E M coupling contamination from the data and use an IP inversion algorithm to obtain a chargeability distribution. This is a two-step process. In Chapter-4, I presented this methodology and inverted for P F E and phase responses. In Chapter-6, I followed same procedure but formulated the problem in terms of complex conductivity shown in second column of Figure 8.1. The data were the real and imaginary parts of the complex apparent resistivity data. I show that if the data are contaminated by E M couphng then inversion using conventional DC resistivity formulation generates two diagnostic artifacts: (a) the recovered amphtude of IP responses are higher than the true amphtudes and (b) the anomahes are displaced to depths greater than true depth. In contrast, after the E M couphng is removed the recovered model agrees with the true model. A n alternative method to obtain chargeabihty information, is to formulate the prob-lem as one physical process in which both the IP effect and the E M couphng contribution are simultaneously considered. To do this, the first step is to forward model with the Chapter 8. Conclusions and Future scope 213 full Maxwell's equation for the electric field. The second step is to formulate the inverse problem in terms of a complex conductivity such that the effects due to chargeabihty and conductivity can be obtained at a particular frequency shown in the third column of Figure 8.1. This is discussed in Chapter-5. It is shown that complex conductivity can be recovered at different frequencies by inverting the real and imaginary parts of the apparent complex resistivity data. The growth path for this line of attack can be advant-ageous since there is no E M couphng correction involved and also both conductivity and chargeabihty information can be obtained in a single inversion. However there are various questions and computational difficulties that will have to be overcome if this method is to be successfully extended to 3D: (a) The formulation presented in Chapter-5 works with a single frequency. However, in practice, field data are collected at many frequencies,1 since the frequency at which the phase peak of the chargeable material occurs, is not known a priori. We note that this is related to the imaginary part of the complex conductivities which is sensitive to the chargeable material, and that the imaginary part peaks at an intermediate frequency. Therefore the question is: Can the formulation in Chapter-5 be extended to invert multi-frequency data jointly to recover complex conductivities at dif-ferent frequencies. In such a case the model objective function should have some ability to take into account the realistic changes in conductivity with frequency, (b) In a typical IP survey there are many transmitters, so the forward problem needs to be solved many times, and this prompts the need for an efficient forward modeling algorithm, (c) At each iteration the sensitivity needs to be computed, so there is a question if approximate sensitivities can be used to expedite the inversion process. A l l of these issues fall into the category of 3D computational complexities, and the road ahead to solve this inverse problem seems challenging. The final part of the thesis revisits a long-standing problem in IP interpretation and attempts to extract even more geological information from the data. Chargeabihty can Chapter 8. Conclusions and Future scope 214 be caused by various minerals like sulphides, graphites, clays etc. Current inversion methodologies that recover a map of chargeabihty do not differentiate between these different IP targets. In Chapter-7, I investigate the possibility of formulating an inverse problem, to recover Cole-Cole parameters from multi-frequency IP data. It is my hope that these parameters can ultimately be used for discriminating different minerals in the subsurface. The synthetic and field example presented in Chapter-7 suggests that differing values of the spectral parameters can be recovered from multi-frequency data. This has important implications, if mineral discrimination can be carried out on the basis of spectral parameters from a frequency dispersion model. Although there are several case histories from field tests and laboratory experiments that relate the spectral values to mineral properties (such as grain size and distribution), a thorough understanding of what these parameters signify is lacking. It is my recommendation that more research is required to establish a link between the different values of these parameters with different mineral types. This involves fundamental research in electrical rock physics to understand the IP responses and develop phenomenological models whose parameters would be diagnostic of different minerals. Appendix A Derivation for the Schelkunoff's potential A and F for a horizontally layered earth Following Ward and Hohmann (1988) here I wiU outline the derivation of the vector potential A and F using propagator matrix method. The solution is developed for a horizontal electric dipole source over a layered earth. Expression for the potential at the surface are used to compute the electric field and magnetic field at the surface. And the expressions for the potential inside the medium are used for computing the sensitivity using adjoint formulation. In a source free region the vector potential A is due to only electric sources ( J 5 ) and F due to only magnetic sources (Ms)- This is given by H = V x A (A . l ) E = - V x F (A.2) The solution for the E M field due to finite sources in presence of homogeneous or layered earth can be represented as superposition of plane wave solutions. The problem can be simplified if the solution is developed in terms of T E and T M modes, where transverse means normal to vertical(z) direction. For the I D situation T E (F) and T M (A) vector potentials can be written as F = Fzz A = AJ. (A.3) Thus only scalar potential Az and Fz need to be evaluated. For notational simplicity the subscripts z will not be presented. In each layer of constant conductivity the scalar 215 Appendix A. Derivation for the vector potentials A and F for ID earth 216 potentials A and F satisfy V2A + k2A = 0 V2F + k2F = 0 (AA) Using 2D Fourier transform the two ODE's can be written as d2A dz2 d2F -u'A = 0 (A.5) -u2F = 0 (A.6) dz2 where the 2D Fourier transform pair is given by /O O / " O O / F(x,yJz)e-i^x+kyyUxdy (A.7) • o o J —oo /C O i » C © / F(kw,ky,z)eiV"x+k*yUkxdky (A.8) - O O J— oo where u — -\Jk2 + k2 — k2 and k2 = u>2fi0eo — iu}(j,0cr. Solutions for eqn (A.6) and eqn (A.8) can be written in terms of upward and downward propagating coefficients. This is given by A(kx,ky,z,u>) = DA(kx,ky,uj)e u z + UA(kx,ky,uj)e% (A.9) F(kx,ky,z,u>) = DF(kx,ky,co)e u z + UF(kx,ky,u>)ev (A.10) where D(kx,ky,to) and U(kx,ky,u>) are the downward and upward propagating coeffici-ents respectively. Appendix A. Derivation for the vector potentials A and F for ID earth 217 A.0.1 S o l u t i o n f o r Az p o t e n t i a l The solution of Aj(kx, ky, z,u>) in the jth layer is given by Ai{kx.,ky,z,u) = Df(kx,ky,u;)e-U^z-Z^ + Uf(kx, ky,<jj)eu^z~Zj-^ ( A . l l ) The interface condition at the top of the jth layer(z = - Z j - i ) is given by: Aj(kx,ky,z = Zj-uw) - Aj-1{kx,ky,z = z,-_i,u;) (A.12) dAi 1 Oj + iivej dz <Tj_i + iu/ej-! dz Applying interface conditions to eqn ( A . l l ) at z = Zj-i yields, (A.13) Df + Uf = Df_1e~ui-lhi-1 + Uf_xeUi-lhi~l (A.14) i - (-UjDf + UjUf) = - i - (-Uj^Df^e-"'-**-* + « i _iL7/_ 1 e" i -^>->) (A.15) t r , - _ i where hj-i—Zj — Zj-\. Rewriting eqn (A.14) and (A.15) in matrix form yields, 1 1 Df Uf e-uj-ihj-i <7j_l ° J - 1 ^ 1 ^ - 1 (A.16) Inverting the 2 x 2 matrix in eqn (A.16) analytically, the upward and downward coeffici-ents of (j — l)th layer can be expressed in terms of bottom layer (jth) coefficients. This is given by °t-i ( 1 + 32=L 2 V Uj^l<Tj (A.17) Appendix A. Derivation for the vector potentials A and F for ID earth 218 This can be written as Df Uf (A.18) where, 2 V u i - i ' r j J . - f l — 2iZi=L) \ 2 V U3-l<rj J 2 ^ Ui-lTj J 1 + -2u (A.19) • i - l " i - l The relation given in eqn (A.18) can be used to propagate the components through a stack of layers from the bottom half-space to surface. To compute the electric and magnetic field at the surface the coefficients DA and U A need to be determined and it can be obtained by propagating the components upward. This is given by DA UA N-l = exP(J2uihi)Uf=1T: i=l A [ D N JjA UN (A.20) In the above expression DA is downward decaying solution above the air-earth interface, which is zero in absence of any source. And in presence of a source DA is expressed in terms of source function. For example in presence of horizontal electric dipole DA is given by (Ward and Hohmann, 1988) DA I ds ik 2 kl + kl x e - u 0 h (A.21) And the solution for A0 is given by A°z(kx,ky,z,u;) = DAe-u°z + U*e"°' (A.22) At the bottom half-space the downward decaying solution exists and upward decaying solution UM is zero. To compute the upward decaying component U A I consider eqn Appendix A. Derivation for the vector potentials A and F for ID earth 219 ( A . 2 0 ) a n d rewr i te i t as * = e*rtI>M ( P°11 ] [ ( A . 2 3 ) where , p o p o 1 1 1 2 1 = nf_ x rf po pO ^21 "^ 22 T h i s al lows to express the u p w a r d decay ing component UA c an be expressed i n t e rms of d o w n w a r d decay ing DA. T h i s is g iven by U0A = ett^FZD* = ^ D A . ( A . 2 4 ) "i i W e note tha t UA do not depend on pos i t ive exponent ia ls . U s i n g eqn ( A . 2 3 ) the d o w n w a r d decay ing coefficient D^ i n te rms of DA is g iven by DAr = e - ^ l 1 ^ - L D A ( A . 2 5 ) M l T a k i n g 2 D inverse Four ie r t r ans fo rm of eqn ( A . 2 2 ) y ie lds , -I fl oo poo A° = — / / (DAe-U0Z + i t f e " 0 2 ) ei{-k*x+k«y)dkxdky. ( A . 2 6 ) 4TT2 S u b s t i t u t i n g and £7^ i n to the above equa t ion we get ITT2 7 y l^'7-' 1 "° 1 ' a,KxaKy ikxI ds _Uo(z+h) P2\ —ikxIds 4TT2 J_„ J_„ \2(k>x Akl)& + P « 2{kl + kl)' ( A . 2 7 ) s i m p l i f y i n g the above equa t ion by no t ing that d / dx[el(k*x+kyy)} = ikxe^kxX+kyy^ y i e lds , L L +1'-"-*') w*^ (A-28) Appendix A. Derivation for the vector potentials A and F for ID earth 220 T a k i n g the Four i e r H a n k e l t r ans form of eqn ( A . 2 8 ) y ie lds , -Ids d A° = J°° ^ e-uo{z+h) + PkeM>-^ JJMdX. ( A . 2 9 ) 47T dx N o t e tha t the ra t io P 2 0 1 ( A , a ; ) / P 1 0 1 ( A , a ; ) is same as rxM as used i n W a r d a n d H o h m a n n (1988). T h e next ob jec t ive is to derive an exp l i c i t express ion for the p o t e n t i a l ins ide the m e d i u m . Cons ide r tha t we are in teres ted i n j t h layer i n the m e d i u m . T h e genera l so lu t ion for A p o t e n t i a l i n the j t h layer is g iven b y -i /too * o o A{ = — / ( 2 ^ e — i ( * - * i - 0 + ufe^'-'i-1)) e^+^dKdky. ( A . 3 0 ) 4^ J—OO v — O O T o c o m p u t e the above in tegra l we need to k n o w the Df and Uf w h i c h can be expressed i n te rms of k n o w n quan t i t y DQ. SO the so lu t ion are p ropaga ted f rom the b o t t o m h a l f space to the top of j t h layer . T h i s is g iven by 3 = exp(Y Uihi)V1 N (AM) where , P J = ( ^ ^ ) = n f - + i r f • ( A 3 2 ) T h i s al lows to express the components Df a n d Uf i n te rms of D^. T h i s is g iven b y Df = PiMe^=tUkhk Uf = P^Dffe^T UKHK. S u b s t i t u t i n g the express ion of Dfj f rom ( A . 2 5 ) y ie lds , ik, e - ^ E i : ; u t k t 2 kl+k'y i» ^ Appendix A. Derivation for the vector potentials A and F for ID earth 221 A -Ids ikx e ^hP^ Y^Ukhk ( , 2 kl + kl J * { A ' 6 ^ Substituting Df and Uf into eqn (A.30) yields, -Ids d f°° f°° e-\u°h+^3k='lUkhk\ Ai = /O O fl c • C O J — c STT2 dxJ^J^ kl + Wy (A.35) pi pi \ _ i i e - u i ( ^ - z j - i ) _ l ? ! u,-(z-z,-_i) I J.{Kx+kyy)ji dk po po / B M l M l / Taking the Fourier Hankel transform of the above equation gives the expression for A potential in the jth layer. This is given by — 7V/« f) f°° «> - [ U o ' l +£i=i ukhk] ( pi pi \ (A.36) A.0.2 S o l u t i o n f o r Fz p o t e n t i a l The solution for the Fj(kx, ky, z,u>) potential in the jth layer is given by F^ky.z^) = Df(kx,ky,u)e-U^-Z>-^ + Uf(kx,ky,u;)eu^-Z^ (A.37) The interface conditions at the top of the jth layer (z = -Zj-i) is given by (Ward and Hohmann, 1988) Fj(kx,ky,z = ZJ-UU)) = Fj-i(kx,ky,z = Zj-Uu) (A.38) L°A = ±.°A± (A.39) /tj dz fij-i dz Applying the interface conditions given in eqn (A.38) and (A.39) to eqn (A.37) at z = Z j _ i yields, Df + Uf = Df_ 1e- u ' ' - l f c>- 1 + Uf_xeui-lhi-1 (A.40) Appendix A. Derivation for the vector potentials A and F for ID earth 222 E q u a t i o n ( A . 4 0 ) a n d ( A . 4 1 ) can be expressed i n a m a t r i x f o r m g iven by ( A . 4 1 ) ( — p u i - i h i - i i + g —2uj_i hj — i 1 ( 1 — uJH-i-2 I " j - i M i 2 V u i - l M j / ( A . 4 2 ) T h i s can be w r i t t e n as where the propaga tor m a t r i x is g iven by Uf ( A . 4 3 ) r f = ( A . 4 4 ) 2 ^ ' Uj-iHj J 2 \_ Uj-iHj J I i / j _ Uj/ij-l A -2v.j-i.hj-x I f l J- " J ^ J - 1 N \ g - 2 « i _ i ^ _ 1 \ 2 \ 2 V uy-iA*i / / P r o p a g a t i n g the so lu t ion t h r o u g h the stack of layers f rom b o t t o m half-space to the surface yie lds the coefficients at the surface. T h i s is g iven by UF N-l nF ( A . 4 5 ) 0 F o r a h o r i z o n t a l e lec t r ic d ipole DF is g iven by ( W a r d a n d H o h m a n n , 1988) -zolds iky h DF 2«o kl + k2y T h e comple t e so lu t ion for Fz po t en t i a l at a i r -ear th interface is g iven b y F° = ~ / / (DFe-U0Z + U£euaz) e^kxX+kyy)dkxdky J-co J_oo ( A . 4 6 ) ( A . 4 7 ) T h e u p w a r d a n d d o w n w a r d decay ing coefficient at the surface can be c o m p u t e d us ing the same m e t h o d o l o g y used for A po ten t i a l . T o c o m p u t e these coefficients I r ewr i t e ( A . 4 5 ) Appendix A. Derivation for the vector potentials A and F for ID earth 223 as fol lows, D ° ) = ^ ( f > A ) ( ^ Q")(DFn I ( A . 4 8 ) where , ( A . 4 9 ) Q21 Q22 / S u b s t i t u t i n g DF and t/jf in to the above equa t ion y ie lds , 0 _ iixJ/x0Ids r°° r00 -iky ' ~ J-00 J-00 «o(*2 + ^) f-u0(z+h) + 9^eu0(z-h) \ ei{^'+^v)dkxdky. \ Qn J T a k i n g the Four i e r H a n k e l t r ans form of eqn ( A . 4 9 ) y ie lds , 4TT dyJo V Q11 ) ^oA V 7 N o t e tha t the ra t io Q^{\, w ) / Q ° 1 ( A , u) is same as r^E as used i n W a r d a n d H o h m a n n (1988). N e x t I der ive an exp l i c i t express ion for the Fz p o t e n t i a l ins ide the m e d i u m . T h e genera l so lu t ion for F po t en t i a l i n the j t h layer is g iven by 1 AOO poo Fi = ~ / (Dfe-^'-'t-1) + Uf e^'-''-^) ei{k*x+k*v) dkxdky. ( A . 5 1 ) 4^r J—00 J—00 U s i n g the same analogy as i n previous sect ion the coefficients for the j t h l ayer is g iven by D F \ N-i / D F \ j =eXp(J2uihi)UljTf[ N ( A . 5 2 ) where Df a n d Uf is g iven by F -iupolds iky e-^hQ3n yJ-\Ukhk ( A 5 3 1 2u0 kl + k2y Q°u { • ) Appendix A. Derivation for the vector potentials A and F for ID earth 2 2 4 F _ -iunolds iky e-»°hQj _yi-_]u^ U' - 2u0 kl + k2y % ( A - M ) S u b s t i t u t i n g DF a n d Uf i n eqn ( A . 5 1 ) y ie lds , -iwnoids 8 r°° f°° e - l u o h + ^ u k h k ] F? = STT2 OyJ^J^ uQ(kx + kl) ^ l i e - u i ( 2 - 2 i - i ) 4_ Q2±euj(z-zi-i)\ e^kxX+kyV^dkxdky ( A . 5 5 ) a n d t a k i n g the Four i e r H a n k e l t r ans fo rm of the above equa t ion gives the express ion for F p o t e n t i a l i n the j t h layer . T h i s is g iven by -iu>u.0Ids d f°° e ~ l U o h + ^ U k h k ] Fl = F Jo . * * 9 y J ° . U ° X , ( A . 5 6 ) ^n e-«i(*-*i-i) + !hke"i{*-*i-i)\ j0(\r)dx Appendix B Derivation for the smallest (Ws) and flattest (Wz) model norm matrices I n a C S A M T s u r v e y d a t a a r e a c q u i r e d a t f r e q u e n c i e s w h i c h s p a n s e v e r a l d e c a d e s (e.g. 0.5-4096 H z ) . T h e c o r r e s p o n d i n g s k i n d e p t h s a l s o s p a n s e v e r a l d e c a d e s a n d s o a l o g -a r i t h m i c d e p t h s p a c i n g i s a p p r o p r i a t e t o d i s c r e t i z e t h e m e d i u m f o r I D e a r t h . I a d o p t E q u a t i o n (2.61) as a s u i t a b l e m o d e l o b j e c t i v e f u n c t i o n t o b e m i n i m i z e d . H e r e I d e r i v e t h e d i s c r e t i z e d f o r m o f t h e m o d e l o b j e c t i v e f u n c t i o n i n (2.61) f o r t h e s m a l l e s t a n d t h e f l a t t e s t m o d e l c o m p o n e n t . T h e c o m p o n e n t f o r t h e s m a l l e s t m o d e l n o r m i n E q u a t i o n (2.61) i s g i v e n b y 0 m = J (m(z) ~ mref(z))2 dmz = j (^j (m(z) - mref(z))2 dz ( B . l ) w h e r e m = l n ( c r ) i s t h e m o d e l . W e n o t e t h a t t h e a d d i t i o n a l w e i g h t i n g f u n c t i o n ws(z) d e f i n e d i n e q n ( 2 . 1 6 ) i s e q u a l t o (l/z). T o p r e v e n t e f f e c t s o f s i n g u l a r i t y i n t h e w e i g h t i n g I s e t ws(z) = 1/zo w h e n z < ZQ w h e r e z0 i s a n o n - z e r o d e p t h l e s s t h a n t h e t h i c k n e s s o f t h e f i r s t l a y e r . F o r p r a c t i c a l p u r p o s e s t h e u p p e r l i m i t o f t h e i n t e g r a l i s e v a l u a t e d u p t o zmax w h i c h i s c h o s e n t o b e s u f f i c i e n t l y l a r g e . F o r t h e d i s c r e t e I D m o d e l s h o w n i n F i g u r e 2.1 I c h o o s e a d i s c r e t e s e t o f w e i g h t s f o r wz(z). T h i s i s g i v e n b y ws(z) = — f o r 0 < z < Zi = f o r Zj-i < z < zy, j = 2, • • • , M. zi-i D i s c r e t i z i n g t h e i n t e g r a l f o r a n M - l a y e r e d e a r t h y i e l d s , M / 1 \  M / h \ C = E to " K « / ) i ) a 7 - W = E to - to*/);)2 r 2 - ( B . 3 ) 225 ( B . 2 ) Appendix B. Derivation for the model norm matrices 2 2 6 w h e r e AZJ = Zj — Zj-i i s t h e t h i c k n e s s o f t h e jth l a y e r . E q u a t i o n ( B . 3 ) c a n b e r e a r r a n g e d s u c h t h a t i t c a n b e w r i t t e n i n t h e m a t r i x f o r m g i v e n b y M 3=1 Zj-1 (mj - {mre-f)j) = mTWjWsm. ( B . 4 ) T h e r e f o r e Ws = d i a g ( - ^ y ) f o r j = 1, • • • , M. F o r t h e b o t t o m h a l f - s p a c e I c h o o s e t h e t h i c k n e s s t o b e e q u a l t o t h e t h i c k n e s s o f t h e l a y e r a b o v e i t i.e. HM-I = ^ M - W e n o t e t h a t s i n c e t h e l a y e r t h i c k n e s s i n c r e a s e s e x p o n e n t i a l l y w i t h d e p t h , t h e r a t i o o f KM-\IZM-\ i s f i n i t e a n d i n t h e o r d e r o f u n i t y . T h u s a t g r e a t e r d e p t h s w h e r e t h e d a t a a r e i n s e n s i t i v e t o t h e s t r u c t u r e , t h e i n v e r s i o n r e c o v e r s t h e r e f e r e n c e m o d e l . T h e c o m p o n e n t f o r t h e f l a t t e s t m o d e l n o r m f r o m e q n ( 2 . 6 1 ) i s g i v e n b y * = /( dm(z) dlnz dmz dm(z) dz dz ( B . 5 ) W e n o t e t h a t t h e a d d i t i o n a l w e i g h t i n g f u n c t i o n wz(z) i n e q n ( 2 . 1 6 ) i s e q u a l t o z. D i s c r e t -i z i n g t h e i n t e g r a l i n ( B . 5 ) f o r a n M - l a y e r e d m o d e l y i e l d s , M - l Zj + Zj-! m; M - l ^2(mj+1-mj) 3=1 Azj+1 + AZJ ( B . 6 ) 2 | Z3 + Z J - 1 hj+1 + hj T h e E q u a t i o n ( B . 6 ) c a n b e r e a r r a n g e d s u c h t h a t i t c a n b e w r i t t e n i n t h e m a t r i x f o r m g i v e n b y M 3 = 1 Zj + Zj-! l3+l + hj m •3 + 1 m 3) mTWTWzm. ( B . 7 ) T h i s i m p l i e s t h a t t h e f l a t t e s t m o d e l n o r m m a t r i x Wz i s M — 1 x M m a t r i x w i t h t h e e l e m e n t s s h o w n i n e q n ( 2 . 6 3 ) . Appendix C EMIP ^ uEM Validity of the approximation k Being able to write the electric field EEMIP as a product of an electric field EEM, which is not affected by chargeabihty, and i>(iu>), which depends only on chargeability, is crucial to this work. The derivation of eqn(4.14) assumes kEMIP « kEM. In this appendix I investigate the validity of this approximation. In order to quantify this I consider a homogeneous half-space model for a finite length dipole source and receiver and plot the amplitude and phase of £(iu>) which is given by 7EMIP(- \ where £(iu>) depends on t r ^ , 77, r, c, dipole length and dipole spacings. If the approxim-ation holds well, then the amplitude of £(i<x>) should equal unity and the phase will be zero. Figures A-1(a) and (b) show that the discrepancy increases with increasing value of the conductivity (&00)) for a dipole-dipole survey with dipole length of300m. For a conductivity of lS/m the error in the amplitude is less than 2% (Figure C.l(a)) and for phase it is less than 15 mrads (Figure C.l(b)) in the frequency range of 0.01-1000Hz. In most IP surveys the frequencies used, rarely exceed 100Hz, so we are mostly interested in the low frequency regime. The effect of intrinsic chargeability 77 is shown in Figure C.l(c) and (d). We note that for the same frequency range (0.01-1000Hz) and for in-trinsic chargeabihty 77 < 0.3 the error in amplitude is less than 2% (Figure C.l(c)) and the error in phase is less than 15 mrads (Figure C.l(d)). Figure C.2(a) and (b) shows the validity of approximation as function of time constant r . As r decreases the discrepancy increases in both amplitude and phase. For r > O.lsec the error in amplitude is less than 227 Appendix C. Validity of the approximation kEMIP & k 228 3% a n d less t h a n 25 mrads i n phase for the same frequency range. F i g u r e C . 2 ( c ) a n d (d) ind ica tes tha t the d i sc repancy increases w i t h the decrease i n c. Fo r c > 0.2 the error i n a m p h t u d e is less t h a n 2% (F igu re 0 .2 (c ) ) and less t h a n 20 mrads for the phase ( F i g u r e A - 2 ( d ) ) i n the same frequency range. F i g u r e 0 .3(a) and (b) shows tha t the d i sc repancy increases w i t h the d ipole l eng th . Fo r d ipole l eng th less t h a n 300m the error i n a m p l i t u d e is less t h a n 1% a n d error i n phase is less t h a n 10 mrads . F i g u r e 0 .3 (c ) a n d (d) shows tha t the d i sc repancy increases w i t h the d ipole spacings. F o r a d ipo le l eng th of 3 0 0 m i f N < 6 t h e n the error i n a m p h t u d e is less t h a n 2% and error i n phase is less t h a n 20 mrads . Appendix C. Validity of the approximation kEMIP w k 229 1.020 -1.017 -1.014 -1.011 -CD 1.008 -T3 1.005 -Q -E 1.002 -< 0.999 -0.996 -0.993 -0.990 -(7=1 (a) a = 0.1 a = 0.011 (b) a = 0.001 10 •a co « o CD tn cc -5H -io H -15 a = 0,01 102 10"' 10° 101 102 Frequency(Hz) 103 10-2 10° 10' 102 Frequency(Hz) 103 10"' 10° 101 102 Frequency(Hz) io-2 101 10° 101 102 Frequency(Hz) Figure C . l : The validity of the approximation kEMIP ?» kEM is tested by plotting the amplitude and phase of the quantity £(iu>) = / ( ^ o o , Vi Ti c> w i °> N — sp) for a fi-nite length dipole source and receiver, (a) Amplitude and (b) phase of £(iu>) for varying conductivity tr^ with fixed n = 0.2, r = 10, c = 0 .3 ,^ = l , o = 300m. (c) Amplitude and (d) phase of £(iuj) for various intrinsic chargeabilities rj with fixed (r^ = OAS/m, T = 10, c = 0.3, N =l,a = 300m. Appendix C. Validity of the approximation kEMIP « k 2 3 0 0.990 H 1 1 1 1 1- -20 -\ 1 1 1 1 h 10-2 10-' 10° 10' 102 103 10-2 10-1 10° 101 102 103 Frequency(Hz) Frequency(Hz) F i g u r e C.2: ( a ) A m p l i t u d e a n d ( b ) p h a s e o f <f(«t>) f o r v a r y i n g t i m e c o n s t a n t T w i t h fixed cr^ = O.lS/m,/? = 0.2, c = 0.3, N = l , o = 3 0 0 m . ( c ) A m p h t u d e a n d ( d ) p h a s e o f £(«*;) f o r v a r y i n g r e l a x a t i o n c o n s t a n t c w i t h fixed c r ^ = 0AS/m,n = 0 . 2 , r = 10, JV = l , a = 3 0 0 m . Appendix C. Validity of the approximation kEMIP m kEM 2 3 1 (a) 1.01 CD T3 ~ 1.00 Q . E < 0.99 (b) j i i i_ a = 300m a = 50m -10 10-2 10' 10° 10' 102 Frequency(Hz 103 10"2 10' 10° 10' 102 Frequency(Hz) 103 10° 10' 102 Frequency(Hz) 10' 10° 10' 102 Frequency(Hz) F i g u r e C.3: ( a ) A m p l i t u d e a n d ( b ) p h a s e o f £(iu>) f o r v a r y i n g d i p o l e l e n g t h a w i t h fixed (Too = 0.1S/m,r? = 0.2, r = 10, c = 0.3,7V = 1. ( c ) A m p h -t u d e a n d ( d ) p h a s e o f f o r v a r y i n g d i p o l e s p a c i n g s o r N - s p a c i n g s c w i t h fixed (Too = 0.15/m, rj = 0.2, r = 10, c = 0.3, a = 3 0 0 m . A p p e n d i x D L i n e a r i n v e r s i o n w i t h c o m p l e x q u a n t i t i e s D . l I n t r o d u c t i o n In many electromagnetic geophysical surveys, data can have both amplitude and phase and therefore they can be treated as a complex quantities. In a typical frequency domain IP survey data are collected at a suite of frequencies using a particular array configuration. The goal of inversion is to extract information about the conductivity and chargeability of the medium from these data. As outlined in the Chapter-3 both the physical properties i.e. conductivity and chargeabihty can be combined in a complex resistivity model. So a possible objective is to formulate the inversion algorithm to recover complex model from complex data. Also the kernel which connects the data with the model can be a complex quantity. The inverse problem can be solved using complex equation or by breaking up the problem so as to work with real and imaginary components. Either approach should be valid but there was some uncertainty in my mind regarding how the errors should be treated. The objective of this appendix is to investigate the procedure of extracting a complex model from complex data as a linear inverse problem and determine how errors should be assigned to the data when the data are treated as complex quantities. D . 2 L i n e a r I n v e r s e P r o b l e m The theory of linear inverse theory is well established (Oldenburg,1984; Parker, 1994). The framework in which most of the linear inverse problem are set up recovers real model 232 Appendix D. Linear inversion with complex quantities 2 3 3 p a r a m e t e r s f r o m c o m p l e x d a t a o r r e a l d a t a . I n t h i s s e c t i o n t h e h n e a r i n v e r s e p r o b l e m o f e x t r a c t i n g a c o m p l e x m o d e l f r o m c o m p l e x d a t a w i l l b e c o n s i d e r e d . L e t u s c o n s i d e r t h a t t h e d a t a a r e h n e a r l y r e l a t e d t o t h e m o d e l g i v e n b y d = Gm ( D . l ) w h e r e d i s t h e d a t a v e c t o r o f l e n g t h N, m i s t h e m o d e l v e c t o r o f l e n g t h M a n d G i s t h e k e r n e l m a t r i x o f s i z e N x M. I f t h e r e a r e m o r e m o d e l p a r a m e t e r s t h a n d a t a M > N t h e n t h e s o l u t i o n f o r m i s n o n - u n i q u e . O n e o f t h e w a y s t o s o l v e t h e p r o b l e m i s : m i n i m i z e a m o d e l o b j e c t i v e f u n c t i o n s u b j e c t t o fitting t h e d a t a . T h e p r o b l e m c a n b e f o r m u l a t e d i n t w o w a y s : ( a ) o n e i n w h i c h t h e d a t a , m o d e l a n d k e r n e l a r e t r e a t e d as c o m p l e x q u a n t i t y ( b ) a n d i n t h e s e c o n d a p p r o a c h t h e m o d e l , d a t a a n d k e r n e l a r e s e p a r a t e d i n t o r e a l a n d i m a g i n a r y p a r t s . D . 2 . 1 M e t h o d - 1 T h i s i n v e r s e p r o b l e m i n w h i c h t h e m o d e l , d a t a a n d t h e k e r n e l m a t r i x a r e t r e a t e d as c o m p l e x c a n b e s t a t e d as m i n i m i z e 4>m = \\Wm(m - m0)\\2 ^ s u b j e c t t o 4>d = 11Wd(d°bs - Gm)\\2 = <f>*d T h e q u a n t i t i e s i n t h e a b o v e e q u a t i o n a r e c o m p l e x . F o r a n y c o m p l e x v e c t o r v t h e Z 2 n o r m is g i v e n b y ||v||2 = vHv, H d e n o t e s t h e c o m p l e x c o n j u g a t e t r a n s p o s e . Wd i s t h e d a t a w e i g h t i n g m a t r i x f o r t h e c o m p l e x d a t a a n d Wm i s t h e m o d e l w e i g h t i n g m a t r i x w h i c h c a n h a v e b o t h t h e s m a l l e s t a n d f l a t t e s t m o d e l c o m p o n e n t s i.e. W^Wm = asWjW, + ctzWTWz. B o t h t h e d a t a a n d m o d e l w e i g h t i n g m a t r i c e s a r e r e a l m a t r i c e s . T o s o l v e t h e i n v e r s e p r o b l e m a g l o b a l o b j e c t i v e f u n c t i o n <f> = (f>d+S<j>m w h i c h c o n t a i n s b o t h m o d e l n o r m a n d d a t a m i s f i t i s m i n i m i z e d . A l t h o u g h t h e d a t a a n d m o d e l a r e c o m p l e x t h e d e r i v a t i o n Appendix D. Linear inversion with complex quantities 2 3 4 w i l l b e c a r r i e d b y s e p a r a t i n g t h e m o d e l i n t o r e a l a n d i m a g i n a r y p a r t s . F o r s i m p l i c i t y o f t h i s d e r i v a t i o n t h e r e f e r e n c e m o d e l m0 a n d t h e d a t a w e i g h t i n g m a t r i x Wd w i l l b e l e f t o u t o f t h e e q u a t i o n i n D.2. C o n s i d e r t h a t t h e m o d e l m c a n b e w r i t t e n as m = n + i£. T h e o b j e c t i v e f u n c t i o n t o b e m i n i m i z e d c a n b e w r i t t e n as <f> = (G(V + iO - df (G(V + it) -d) + 8(V + %t)BWlWm(r, + it). ( D . 3 ) T o m i n i m i z e e q n ( D . 3 ) w i t h r e s p e c t t o m I c a n d i f f e r e n t i a t e <f> w i t h r e s p e c t t o t h e r e a l p a r t o f m i.e. 77 a n d t h e i m a g i n a r y p a r t i.e. t s e p a r a t e l y a n d e q u a t e t o z e r o . T h e r e f o r e d i f f e r e n t i a t i n g t h e o b j e c t i v e f u n c t i o n w i t h r e s p e c t t o t h e r e a l p a r t rj a n d e q u a t i n g t o z e r o g i v e s , Vv4>= ((G*)TG + GTG*)n + i((G*)TG-GTG*)t )'• ( D - 4 ) - (G*)Td - GTd* + 28WZWmr, = 0 a n d d i f f e r e n t i a t i n g w i t h r e s p e c t t o t h e i m a g i n a r y p a r t £ g i v e s , = ((G*)TG + GTG*) t + i ((G*)TG - GTG*) n + ( D . 5 ) i ({G*fd - GTd*) + 28WlWmt = 0 w h e r e G* i s t h e c o m p l e x c o n j u g a t e o f G. B o t h e q n ( D . 4 ) a n d ( D . 5 ) c a n b e s h o w n t o b e r e a l . S i n c e Vv4> = 0 a n d = 0 t h e n V'v(f> + iV£<f) = 0. T h e r e f o r e c o m b i n i n g t h e r e s u l t s o f e q n ( D . 4 ) a n d ( D . 5 ) s u c h t h a t Vv(j) + i V ^ c / i = 0 y i e l d s , (GHG + 8WlWm) m = GHd ( D . 6 ) I n c l u d i n g t h e d a t a w e i g h t i n g m a t r i x Wd a n d t h e r e f e r e n c e m o d e l m o w i l l r e s u l t i n t h e f o l l o w i n g s y s t e m o f e q u a t i o n , (GHWjWdG + 8WlWm) m = GHWjWdd + 8WlWmm0 ( D . 7 ) Appendix D. Linear inversion with complex quantities 2 3 5 T h i s i s a c o m p l e x s y s t e m o f e q u a t i o n s Am = b w h e r e A = GHWdWdG + 0W^Wm a n d b = GHWjWdd - f OW^Wmmo. T h e s y s t e m i n ( D . 7 ) c a n b e s o l v e d t o o b t a i n m s u c h t h a t (f>d = <j>*d- I n o r d e r t o a s s i g n e r r o r s t o c o m p l e x d a t a l e t u s c o n s i d e r t h e d e f i n i t i o n o f v a r i a n c e i n c o m p l e x p l a n e . V a r i a n c e i s d e f i n e d as t h e n o r m a l i z e d s u m o f t h e s q u a r e d d i s t a n c e o f a n e n s e m b l e o f p o i n t s f r o m t h e m e a n . T h i s d i s t a n c e o r t h e l e n g t h i s a r e a l q u a n t i t y . S u p p o s e t h e r e i s a n e n s e m b l e o f p o i n t s s a y zk = xk + iyk f o r k = 1, • • • , N a n d l e t t h e i r m e a n b e z = x + iy. T h e n t h e s q u a r e d d i s t a n c e o f a n y s i n g l e p o i n t , s a y zk, f r o m t h e m e a n i s g i v e n b y (xk — x)2 + (yk — y)2- T h i s c a n a l s o b e w r i t t e n a s (zk — z)H(zk — z). I f t h e c o v a r i a n c e o f p o i n t s i s z e r o ( i . e . t h e r e i s n o c o r r e l a t i o n b e t w e e n t h e p o i n t s ) t h e n t h e v a r i a n c e o f t h e e n t i r e e n s e m b l e i s g i v e n b y m a t r i x Wd i s a r e a l m a t r i x f o r u n c o r r e c t e d d a t a . T h e r e f o r e t h e d a t a w e i g h t i n g m a t r i x c a n b e w r i t t e n a s T h u s i f t h e m o d e l a n d d a t a a r e t r e a t e d a s c o m p l e x q u a n t i t i e s t h e n t h e c o m p l e x s y s t e m o f e q u a t i o n s g i v e n i n ( D . 7 ) , r e p r e s e n t e d b y Am = b, c a n b e s o l v e d b y s e p a r a t i n g i n t o r e a l a n d i m a g i n a r y p a r t s . ( D . 8 ) T h e r e f o r e t h e s t a n d a r d d e v i a t i o n £ 2 = \/{Var(z)) i s a l s o a r e a l q u a n t i t y . T h i s i m p l i e s t h a t i f t h e r e a l p a r t o f t h e d a t a h a s a s t a n d a r d d e v i a t i o n e q u a l t o eR a n d t h e i m a g i n a r y p a r t h a s s t a n d a r d d e v i a t i o n e 7 , t h e n t h e e l e m e n t s o f t h e d a t a w e i g h t i n g m a t r i x Wd i s l / y / ( ( e H ) 2 + ( e J ) 2 ) - A l t h o u g h t h e d a t a v e c t o r i s a c o m p l e x q u a n t i t y t h e d a t a w e i g h t i n g f o r i = 1,.. . , N. ( D . 9 ) Appendix D. Linear inversion with complex quantities 236 D . 2 . 2 M e t h o d - 2 In this approach the data,model and the kernel are separated into real and imaginary part. The inverse problem can be stated as minimize <f>m = av \ \Wv(r] - T?0)||2 + \ \Wt(i - £0] subject to <f>d WR 0 GR -G1 G1 GR ti-(D.10) (d°b')R (d°bsy where m = n + i£, d = dR + id1, G — GR + iG1, Wf = diag(l/e H) and Wj = diag(l/e J). eR and e1 are the standard deviation of the real and imaginary part of the data. av and ct{ are the control parameters which determine relative contribution of the model objective due to real part of the model and the imaginary part of the model respectively. The inverse problem given in eqn(D.lO) can be solved in a standard way by minimizing the objective function (j) = + 0^m- This leads to the following system of equation given by {jTWjWdJ + 0W*Wm) P - JTWjWdd°b° + 0WlWmVo ( D . l l ) where, G1 GR ) \ 0 Wj J \ 0 y/a{Wi For this method to be equivalent to the method in (D.2.1) the control parameters av = ot£ = 1. This implies that there is equal weighting for the real and the imaginary part of the model. A synthetic numerical example is considered to illustrate the equivalence of the two methods. J = (D.12) Appendix D. Linear inversion with complex quantities 2 3 7 x F i g u r e D . l : P l o t o f k e r n e l as f u n c t i o n o f d e p t h x. ( a ) R e a l p a r t ( b ) I m a g i n a r y p a r t D . 3 S y n t h e t i c E x a m p l e A n u m e r i c a l e x a m p l e i n o n e - d i m e n s i o n i s c o n s i d e r e d t o i U u s t r a t e t h e t w o i n v e r s i o n m e t h -o d s . T h e f o r w a r d p r o b l e m i s g i v e n b y dj= ( G,{x)m(x)dx j = l,...,N ( D . 1 3 ) Jo w h e r e dj i s t h e jth d a t u m , Gj(x) i s t h e k e r n e l f u n c t i o n , N i s t h e n u m b e r o f d a t a , m(x) i s t h e m o d e l . T h e k e r n e l Gj(x) i s g i v e n b y Gj(x) = ( a + ib)e~^cx c o s - 1 ) * ) ( D . 1 4 ) F o r t h i s p a r t i c u l a r e x a m p l e a = 10.0,6 = 1.0, c = 0.05 a n d N = 20. T h e p l o t o f t h e r e a l a n d i m a g i n a r y p a r t o f t h e k e r n e l i s s h o w n i n F i g u r e D . l ( a ) a n d ( b ) r e s p e c t i v e l y . Appendix D. Linear inversion with complex quantities 238 B o t h the rea l and i m a g i n a r y par t of the ke rne l decay w i t h d e p t h a n d each also has a constant dc par t . T h e rea l a n d i m a g i n a r y par ts of the t rue m o d e l used to generate the d a t a are shown as b l o c k y models i n F i g u r e D.2(a ) and (b) respect ively. T h e rea l pa r t has two anomal ies a n d the i m a g i n a r y par t has a single anomaly . T h e rea l and i m a g i n a r y par t of the d a t a are genera ted us ing eqn(D.13) a n d are shown by circles i n F i g u r e D .3 (a ) a n d (b) respect ive ly . B o t h the rea l and i m a g i n a r y par t of da t a are c o n t a m i n a t e d w i t h G a u s s i a n r a n d o m noise w i t h 5% s t anda rd dev ia t i on of the d a t u m value and a base l eve l error of 0.001. T h e region is d i v i d e d in to 100 layers between x = [0,1]. T h u s there are 20 c o m p l e x d a t a a n d 100 c o m p l e x m o d e l parameters to be solved. T h e desired da t a misf i t to be ach ieved i n the inve r s ion is 20 since this is the expec ted var iance. T h e p r o b l e m is u n d e r d e t e r m i n e d , so a m o d e l ob jec t ive func t ion is m i n i m i z e d subject to f i t t ing the d a t a ( eqn (D .2 ) ) . T h e m o d e l ob jec t ive func t ion chosen for the invers ion has a d o m i n a n t flatness componen t w i t h as = 0.0001 a n d az = 1.0. Wd has elements accord ing to eqn ( D . 9 ) . T h e i n v e r t e d m o d e l us ing the me thod-1 is shown F i g u r e D . 2 . T h e recovered m o d e l for b o t h the rea l ( F i g u r e D .2 (a ) ) a n d i m a g i n a r y par t ( F i g u r e D .2 (b ) ) indica tes a s m o o t h represen ta t ion of the t rue m o d e l . T h e s m o o t h representa t ion is the result of the m o d e l ob jec t ive func t i on i n w h i c h smoothness componen t is dominan t . T h e p red ic t ed d a t a p r o d u c e d b y the recovered m o d e l is i n d i c a t e d by star i n F i g u r e D .3 (a ) and (b) indica tes a good agreement w i t h the observat ions . N e x t I consider the same da t a and use method-2 to recover the m o d e l parameters . T h e m o d e l ob jec t ive func t ion i n this case has two components shown i n e q n ( D . l O ) . T h e i n d i v i d u a l m o d e l we igh t ing mat r ices i .e. , Wv and are of the same f o r m as i n the p r e v i -ous case. I n order for this m e t h o d be equivalent to method-1 the re la t ive c o n t r i b u t i o n of the m o d e l ob jec t ive func t ion for the rea l and the i m a g i n a r y par t are cons idered equa l i .e. c t^ /a^ = 1. He re the da ta , m o d e l and the ke rne l are separated in to rea l a n d i m a g i n a r y Appendix D. Linear inversion with complex quantities 239 Real part of Model (a) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Figure D.2: Inversion results with method-1. The true model is shown in blocky sohd line and the smooth model in solid is obtained from inversion, (a) Real part of the model, (b) Imaginary part of the model. parts prior to inversion. Thus there 40 data and 200 model parameters in the problem. The desired data misfit to be achieved in the inversion is %2 = 40. The recovered model is shown in Figure D.5. The real (Figure D.5(a)) and imaginary part (Figure D.5(b)) of the recovered model indicates a smooth representation of the true model. The predicted data produced by the recovered model is indicated by star in Figure D.4(a) and (b) indicates a good agreement with the observations. The recovered model from method-1 in Figure D.2 shows extremely good agreement with the model recovered using method-2, shown in Figure D.5. For this example the model objective function was same for both methods and the only difference was the Appendix D. Linear inversion with complex quantities 240 F i g u r e D . 3 : S y n t h e t i c da t a con t amina t ed w i t h noise are shown b y circles a n d the pre-d i c t e d d a t a f r o m invers ion us ing method-1 are show by stars, (a) R e a l par t of the d a t a a n d (b) i m a g i n a r y par t . choice of d a t a errors. In the method-1 the da t a were c o m p l e x a n d the d a t a we igh t ing m a t r i x was g iven by eqn (D .9 ) where as i n the method-2 the rea l a n d i m a g i n a r y d a t a were i n d i v i d u a l l y n o r m a l i z e d b y d a t a errors. T h i s suggests tha t the er ror ass ignment to the c o m p l e x d a t a us ing eqn (D .9 ) holds i f the inverse p r o b l e m is f o rmu la t ed i n t e rms of c o m p l e x quant i t ies . DA C o n c l u s i o n s In th is chapter I have presented the so lu t ion to l inear inverse p r o b l e m i n w h i c h b o t h d a t a a n d m o d e l are c o m p l e x quant i t ies . T h e so lu t ion is ob ta ined us ing two approaches . Appendix D. Linear inversion with complex quantities 241 Real part of Model 0.6 I 1 1 1 1 1 1 1 1 r I i i i i i i i i i I 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Imaginary Part of Model (b) 0.61 1 1 1 1 1 < 1 1 1 L10 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 F i g u r e D . 4 : S y n t h e t i c da t a con t amina t ed w i t h noise are shown b y circles a n d the pre-d i c t e d d a t a f r o m invers ion us ing method-2 are show by stars, (a) R e a l par t of the d a t a a n d (b) i m a g i n a r y par t . I n the first m e t h o d the inverse p r o b l e m is fo rmula ted b y t r ea t ing the m o d e l , d a t a a n d ke rne l as c o m p l e x quant i t ies . In this case the s t anda rd dev i a t i on of the c o m p l e x d a t a are g iven b y \/e2R + e\ where e# a n d er are the s t andard dev i a t i on for the rea l a n d i m a g i n a r y respect ively . In the second m e t h o d i t is solved by separa t ing the da ta , m o d e l a n d ke rne l i n t o rea l a n d i m a g i n a r y par t . Resu l t s f rom the syn the t ic example shows the equivalence of the two me thods . Appendix D. Linear inversion with complex quantities 2 4 2 10 12 14 16 18 20 (b) 0 2 4 6 8 10 12 14 16 18 20 F i g u r e D.5: I n v e r s i o n r e s u l t s w i t h m e t h o d - 2 . 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