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Tutorial : 2-D and 3-D electrical imaging surveys

By

Dr. M.H.Loke

Copyright (1996-2021)

email : drmhloke@yahoo.com, geotomosoft@gmail.com

Web : www.geotomosoft.com

(All rights reserved)

(Revision date : 27th August 2021)

Copyright and disclaimer notice

The author, M.H.Loke, retains the copyright to this set of notes. Users may print a copy

of the notes, but may not alter the contents in any way. The copyright notices must be

retained. For public distribution, prior approval by the author is required.

It is hoped that the information provided will prove useful for those carrying out 2-D

and 3-D field surveys, but the author will not assume responsibility for any damage or

loss caused by any errors in the information provided. If you find any errors, please

inform me by email and I will make every effort to correct it in the next edition.

You can download the programs mentioned in the text (RES2DMOD, RES2DINV,

RES3DMOD, RES3DINV) from the following web site

www.geotomosoft.com

M.H.Loke

Table of Contents

1 Introduction to resistivity surveys ....................................................................................... 1

1.1 Introduction and basic resistivity theory ...................................................................... 1

1.2 Electrical properties of earth materials ......................................................................... 5

1.3 1-D resistivity surveys and inversions – applications, limitations and pitfalls ............ 6

1.4 Basic Inverse Theory .................................................................................................. 11

1.5 2-D model discretization methods .............................................................................. 15

2 2-D electrical surveys – Data acquisition, presentation and arrays ................................... 18

2.1 Introduction ................................................................................................................ 18

2.2 Field survey method - instrumentation and measurement procedure ......................... 18

2.3 Available field instruments ......................................................................................... 21

2.4 Pseudosection data plotting method ........................................................................... 24

2.5 A comparison of the different electrode arrays .......................................................... 26

2.5.1 The Frechet derivative for a homogeneous half-space ........................................ 26

2.5.2 A 1-D view of the sensitivity function - depth of investigation .......................... 27

2.5.3 A 2-D view of the sensitivity function ................................................................. 30

2.5.4 Wenner array ........................................................................................................ 31

2.5.5 Dipole-dipole array .............................................................................................. 31

2.5.6 Wenner-Schlumberger array ................................................................................ 35

2.5.7 Pole-pole array ..................................................................................................... 37

2.5.8 Pole-dipole array .................................................................................................. 38

2.5.9 Multiple gradient array ........................................................................................ 41

2.5.10 High-resolution electrical surveys with overlapping data levels ...................... 44

2.5.11 Summary of array types .................................................................................... 45

2.5.12 Use of the sensitivity values for multi-channel measurements or streamers .... 46

3 A 2-D forward modeling program .................................................................................... 48

3.1 Finite-difference and finite-element methods ............................................................. 48

3.2 Using the forward modeling program RES2DMOD .................................................. 49

3.3 Forward modeling exercises ....................................................................................... 50

4 A 2-D inversion program .................................................................................................. 52

4.1 Introduction ................................................................................................................ 52

4.2 Pre-inversion and post-inversion methods to remove bad data points ....................... 52

4.3 Selecting the proper inversion settings ....................................................................... 55

4.4 Using the model sensitivity and uncertainty values ................................................ 61

4.5 Methods to handle topography ................................................................................... 66

4.6 Incorporating information from borehole logs and seismic surveys .......................... 68

4.7 Model refinement ....................................................................................................... 72

4.8 Fast inversion of long 2-D survey lines ...................................................................... 76

4.8.1 Preprocessing steps .............................................................................................. 76

4.8.2 Data set and computer system used for tests ....................................................... 76

4.8.3 Finite-element mesh size...................................................................................... 77

4.8.4 Limit calculation of Jacobian matrix values ........................................................ 77

4.8.5 Use sparse inversion techniques .......................................................................... 78

4.8.6 Use wider cells ..................................................................................................... 78

4.8.7 Overview of methods to reduce the calculation time........................................... 79

4.9 Model resolution and automatic array optimization methods .................................... 81

4.9.1 Concept of model resolution ................................................................................ 81

4.9.2 A heuristic explanation of model resolution – through a glass darkly ................ 81

4.9.3 Examples of model resolution with standard arrays ............................................ 82

4.9.4 Array optimization methods ................................................................................ 83

4.9.5 DOI versus model resolution? ............................................................................. 86

4.9.6 The streamer design problem using a model resolution approach. ...................... 87

4.10 Pitfalls in 2-D resistivity surveys and inversion...................................................... 89

4.11 The pole-pole inversion paradox ............................................................................. 94

5 I.P. inversion ..................................................................................................................... 96

5.1 Introduction ................................................................................................................ 96

5.2 The IP effect ............................................................................................................... 96

5.3 IP data types................................................................................................................ 98

5.4 IP mathematical models............................................................................................ 100

5.5 I.P. surveys with multi-electrode systems ................................................................ 101

6 Cross-borehole imaging .................................................................................................. 104

6.1 Introduction .............................................................................................................. 104

6.2 Electrode configurations for cross-borehole surveys ............................................... 104

6.2.1 Two electrodes array – the pole-pole ................................................................. 104

6.2.2 Three electrodes array – the pole-bipole ............................................................ 106

6.2.3 Four electrodes array – the bipole-bipole .......................................................... 108

6.3 Single borehole surveys ............................................................................................ 111

6.4 Cross-borehole optimized arrays .............................................................................. 113

6.5 Optimized arrays with subsurface electrodes ........................................................... 113

7 2-D field examples .......................................................................................................... 117

7.1 Introduction .............................................................................................................. 117

7.2 Landslide - Cangkat Jering, Malaysia ...................................................................... 117

7.3 Old Tar Works - U.K. ............................................................................................... 117

7.4 Holes in clay layer - U.S.A. ...................................................................................... 118

7.5 Time-lapse water infiltration survey - U.K. ............................................................. 120

7.6 Pumping test, U.K. ................................................................................................... 122

7.7 Wenner Gamma array survey - Nigeria .................................................................... 123

7.8 Mobile underwater survey - Belgium ....................................................................... 124

7.9 Floating electrodes survey – U.S.A. ......................................................................... 126

7.10 Oil Sands, Canada ................................................................................................. 126

8 3-D electrical imaging surveys ........................................................................................ 128

8.1 Introduction to 3-D surveys ...................................................................................... 128

8.2 Array types for 3-D surveys ..................................................................................... 128

8.2.1 The pole-pole array ............................................................................................ 128

8.2.2 The pole-dipole array ......................................................................................... 131

8.2.3 The dipole-dipole, Wenner and Schlumberger arrays ....................................... 131

8.2.4 Summary of array types ..................................................................................... 133

8.3 3-D roll-along techniques ......................................................................................... 136

8.4 A 3-D forward modeling program ............................................................................ 138

8.5 3-D inversion algorithms and 3-D data sets ............................................................. 140

8.6 A 3-D inversion program .......................................................................................... 141

8.7 Banding effects in 3-D inversion models ................................................................. 142

8.8 The use of long electrodes in 3-D surveys ............................................................... 145

8.9 Data grid formats and model discretizations ............................................................ 147

8.9.1 Types of surveys and model grids ..................................................................... 147

8.9.2 Model grid optimization for large surveys with arbitrary electrodes positions . 148

8.10 3-D array optimization – grids and perimeters...................................................... 152

8.11 Unstable arrays and the geometric factor relative error ........................................ 159

8.12 Not on firm foundations – inversion with shifting electrodes in 2-D and 3-D ..... 161

8.13 Examples of 3-D field surveys .............................................................................. 166

8.13.1 Birmingham field test survey - U.K. .............................................................. 166

8.13.2 Sludge deposit - Sweden................................................................................. 167

8.13.4 Copper Hill - Australia ................................................................................... 170

8.13.5 Athabasca Basin – Canada ............................................................................. 172

8.13.6 Cripple Creek and Victor Gold Mine, Colorado – USA ................................ 174

8.13.7 Burra copper deposit, South Australia : Model reliability determination ...... 176

8.14 Closing remarks on the 3-D method ..................................................................... 179

Acknowledgments.................................................................................................................. 180

References .............................................................................................................................. 181

Appendix A The smoothness constraint and resolving deeper structures in I.P. surveys 192

A.1 A problem with I.P. inversion with a conductive overburden .................................. 192

A.2 2-D synthetic model test ........................................................................................... 192

A.3 3-D field data set test ................................................................................................ 193

A.4 Software implementation .......................................................................................... 195

Appendix B Modeling long electrodes ............................................................................ 196

B.1 Two methods to model a long electrode ................................................................... 196

Appendix C New I.P. systems, negative apparent resistivity values and vector arrays .. 197

C.1 The distributed receivers I.P. systems and negative apparent resistivity values. ..... 197

C.2 Vector array surveys and data inversion................................................................... 198

Appendix D Arrays and artifacts in areas with large resistivity contrasts. ...................... 200

D.1 The problem of a near-surface low resistivity structure. .......................................... 200

Appendix E Strategies for the inversion of very large data sets...................................... 203

E.1 Challenges of very large data sets and models ......................................................... 203

E.2 Field data set and inverse models ............................................................................. 203

E.3 Summary ................................................................................................................... 205

E.4 Mass storage options ................................................................................................ 205

Appendix F Constrained I.P. inversion ........................................................................... 208

F.1 Methods of constraining inverse model parameters ................................................. 208

F.2 Using the method of transformations for I.P. inversion ........................................... 209

F.3 Using the range bound transformation for I.P. inversion ......................................... 211

Appendix G Using a post-inversion L-curve method for large data sets ......................... 212

G.1 Errors in electrode positions with mobile surveys.................................................... 212

G.2 Using a post-inversion L-curve method ................................................................... 212

G.3 Using the L-curve method to separate structure from noise for mobile surveys ...... 214

List of Figures

Figure 1. The flow of current from a point current source and the resulting potential

distribution. ......................................................................................................................... 3

Figure 2. The potential distribution caused by a pair of current electrodes. The electrodes are

1 m apart with a current of 1 ampere and a homogeneous half-space with resistivity of 1

m. ..................................................................................................................................... 3

Figure 3. A conventional array with four electrodes to measure the subsurface resistivity. .... 4

Figure 4. Common arrays used in resistivity surveys and their geometric factors. Note that

the dipole-dipole, pole-dipole and Wenner-Schlumberger arrays have two parameters, the

dipole length “a” and the dipole separation factor “n”. While the “n” factor is commonly

an integer value, non-integer values can also be used......................................................... 4

Figure 5. The resistivity of rocks, soils and minerals. ............................................................... 6

Figure 6. The three different models used in the interpretation of resistivity measurements. ... 7

Figure 7. A typical 1-D model used in the interpretation of resistivity sounding data for the

Wenner array. ...................................................................................................................... 7

Figure 8. A 2-D two-layer model with a low resistivity prism in the upper layer. The

calculated apparent resistivity pseudosections for the (a) Wenner and (b) Schlumberger

arrays. (c) The 2D model. The mid-point for a conventional sounding survey is also

shown. ................................................................................................................................. 9

Figure 9. Apparent resistivity sounding curves for a 2-D model with a lateral inhomogeneity.

(a) The apparent resistivity curve extracted from the 2D pseudosection for the Wenner

array. The sounding curve for a two-layer model without the low resistivity prism is also

shown by the black line curve. (b) The apparent resistivity curves extracted from the 2-D

pseudosection for the Schlumberger array with a spacing of 1.0 meter (black crosses) and

3.0 meters (red crosses) between the potential electrodes. The sounding curve for a two-

layer model without the low resistivity prism is also shown. ........................................... 10

Figure 10. Coupling between neighboring model cells through the roughness filter in a 2-D

model. (a) In the horizontal and vertical diretcions only, and (b) in diagonal diretcions as

well. ................................................................................................................................... 13

Figure 11. (a) Coupling between corresponding model blocks in two time-lapse models using

a cross-model time-lapse smoothness constraint. (b) Example Jacobian matrix structure

for five time series data sets and models. Each grey rectangle represents the Jacobian

matrix associated with a single set of measurements. ....................................................... 15

Figure 12. The different models for the subsurface used in the interpretation of data from 2-D

electrical imaging surveys. (a) A purely cell based model. (b) A purely boundary based

model. (c) The laterally constrained model. (d) A combined cell based and boundary

based model with rectangular cells, and (e) with boundary conforming trapezoidal cells.

........................................................................................................................................... 16

Figure 13. Example of a cell based model with a variable boundary. (a) The test model. (b)

The apparent resistivity pseudosection. (c) The inversion model with a variable sharp

boundary that is marked by a black line. ........................................................................... 17

Figure 14. The arrangement of electrodes for a 2-D electrical survey and the sequence of

measurements used to build up a pseudosection. .............................................................. 19

Figure 15. The use of the roll-along method to extend the area covered by a 2-D survey. ..... 20

Figure 16. Sketch outline of the ABEM Lund Imaging System. Each mark on the cables

indicates an electrode position (Dahlin, 1996). The cables are placed along a single line

(the sideways shift in the figure is only for clarity). This figure also shows the principle

of moving cables when using the roll-along technique. .................................................... 22

Figure 17. The Aarhus Pulled Array System. The system shown has two current (C)

electrodes and six potential electrodes (Christensen and Sørensen 1998, Bernstone and

Dahlin 1999). ..................................................................................................................... 23

Figure 18. The Geometrics OhmMapper system using capacitive coupled electrodes.

(Courtesy of Geometrics Inc.). .......................................................................................... 23

Figure 19. Schematic diagram of a possible mobile underwater survey system. The cable has

two fixed current electrodes and a number of potential electrodes so that measurements

can be made at different spacings. The above arrangement uses the Wenner-

Schlumberger type of configuration. Other configurations, such as the gradient array, can

also be used. ...................................................................................................................... 24

Figure 20. The apparent resistivity pseudosections from 2-D imaging surveys with different

arrays over a rectangular prism. ........................................................................................ 25

Figure 21. The parameters for the sensitivity function calculation at a point (x,y,z) within a

half-space. A pole-pole array with the current electrode at the origin and the potential

electrode “a” meters away is shown. ................................................................................ 26

Figure 22. A plot of the 1-D sensitivity function. (a) The sensitivity function for the pole-pole

array. Note that the median depth of investigation (red arrow) is more than twice the

depth of maximum sensitivity (blue arrow). (b) The sensitivity function and median

depth of investigation for the Wenner array...................................................................... 28

Figure 23. 2-D sensitivity sections for the Wenner array. The sensitivity sections for the (a)

alpha, (b) beta and (c) gamma configurations. .................................................................. 32

Figure 24. 2-D sensitivity sections for the dipole-dipole array. The sections with (a) n=1, (b)

n=2, (c) n=4 and (d) n=6. ................................................................................................. 33

Figure 25. Two possible different arrangements for a dipole-dipole array measurement. The

two arrangements have the same array length but different “a” and “n” factors resulting

in very different signal strengths. ...................................................................................... 35

Figure 26. 2-D sensitivity sections for the Wenner-Schlumberger array. The sensitivity

sections with (a) n=1, (b) n=2, (c) n=4 and (d) n=6. ........................................................ 36

Figure 27. A comparison of the (i) electrode arrangement and (ii) pseudosection data pattern

for the Wenner and Wenner-Schlumberger arrays. ........................................................... 37

Figure 28. The pole-pole array 2-D sensitivity section. ........................................................... 38

Figure 29. The forward and reverse pole-dipole arrays. .......................................................... 39

Figure 30. The pole-dipole array 2-D sensitivity sections. The sensitivity sections with (a)

n=1, (b) n=2, (c) n=4 and (d) n=6. .................................................................................... 40

Figure 31. Sensitivity sections for the gradient array. The current electrodes are fixed at x=0

and 1.0 meters, and the distance between the potential electrodes is 0.1 m. The position

of the P1-P2 potential electrodes at (a) 0.45 and 0.55 m., (b) 0.55 and 0.65 m., (c) 0.65

and 0.75 m, (d) 0.75 and 0.85 m. and (e) 0.80 and 0.90 m. .............................................. 42

Figure 32. (a) Example of multiple gradient array data set and inversion model. (b) Profile

plot using exact pseudodepths. (c) Profile plot using approximate pseudodepths. ........... 43

Figure 33. The apparent resistivity pseudosection for the dipole–dipole array using

overlapping data levels over a rectangular prism. Values of 1 to 3 meters are used for the

dipole length ‘a’, and the dipole separation factor ‘n’ varies from 1 to 5. Compare this

with Figure 20c for the same model but with ‘a’ fixed at 1 meter, and “n” varying from 1

to 10. In practice, a ‘n’ value greater than 8 would result in very noisy apparent resistivity

values. ................................................................................................................................ 45

Figure 34. Cumulative sensitivity sections for different measurement configurations using (a)

a dipole-dipole sequence, (b) a moving gradient array and (c) an expanding Wenner-

Schlumberger array. .......................................................................................................... 47

Figure 35. The output from the RES2DMOD software for the SINGLE_BLOCK.MOD 2-D

model file. The individual cells in the model are shown in the lower figure, while the

upper figure shows the pseudosection for the Wenner Beta (dipole-dipole with n=1)

array. .................................................................................................................................. 49

Figure 36. An example of a field data set with a few bad data points. The most obvious bad

datum points are located below the 300 meters and 470 meters marks. The apparent

resistivity data in (a) pseudosection form and in (b) profile form. ................................... 53

Figure 37. Selecting the menu option to remove bad data points manually. ........................... 54

Figure 38. Error distribution bar chart from a trial inversion of the Grundfor Line 1 data set

with five bad data points. .................................................................................................. 55

Figure 39. Different options to modify the inversion process. ................................................ 56

Figure 40. Example of inversion results using the l2-norm smooth inversion and l1-norm

blocky inversion model constrains. (a) Apparent resistivity pseudosection (Wenner array)

for a synthetic test model with a faulted block (100 m) in the bottom-left side and a

small rectangular block (2 m) on the right side with a surrounding medium of 10 m.

The inversion models produced by (b) the conventional least-squares smoothness-

constrained or l2-norm inversion method and (c) the robust or l1-norm inversion method.

........................................................................................................................................... 57

Figure 41. Different methods to subdivide the subsurface into rectangular prisms in a 2-D

model. Models obtained with (a) the default algorithm, (b) by allowing the number of

model cells to exceed the number of data points, (c) a model which extends to the edges

of the survey line and (d) using the sensitivity values for a homogeneous earth model. .. 58

Figure 42. The options to change the thickness of the model layers. ...................................... 59

Figure 43. The options under the ‘Change Settings’ menu selection. ..................................... 60

Figure 44. The dialog box to limit the model resistivity values. ............................................. 61

Figure 45. Landfill survey example (Wenner array). (a) Apparent resistivity pseudosection ,

(b) model section, (c) model sensitivity section (d) model uncertainty section, (e)

minimum and maximum resistivity sections. .................................................................... 62

Figure 46. Landfill survey depth of investigation determination. (a) Model section with

extended depths and (b) the normalized DOI index section. ............................................ 65

Figure 47. Beach survey example in Denmark. The figure shows the model section with

extended depths and the normalized DOI index section. .................................................. 65

Figure 48. Different methods to incorporate topography into a 2-D inversion model. (a)

Schematic diagram of a typical 2-D inversion model with no topography. A finite-

element mesh with four nodes in the horizontal direction between adjacent electrodes is

normally used. The near surface layers are also subdivided vertically by several mesh

lines. Models with a distorted grid to match the actual topography where (b) the

subsurface nodes are shifted vertically by the same amount as the surface nodes, (c) the

shift in the subsurface nodes are gradually reduced with depth or (d) rapidly reduced with

depth, and (e) the model obtained with the inverse Schwartz-Christoffel transformation

method. .............................................................................................................................. 67

Figure 49. Fixing the resistivity of rectangular and triangular regions of the inversion model.

........................................................................................................................................... 70

Figure 50. The inversion model cells with fixed regions. The fixed regions are drawn in

purple. Note that the triangular region extends beyond the survey line. ........................... 71

Figure 51. Example of an inversion model with specified sharp boundaries. (a) The

boundaries in the Clifton survey (Scott et al., 2000) data set is shown by the blue lines.

(b) The measured apparent resistivity pseudosection and the inversion model section. ... 71

Figure 52. The effect of cell size on the model misfit for near surface inhomogeneities. (a)

Model with a cell width of one unit electrode spacing. (b) A finer model with a cell width

of half the unit electrode spacing. The near surface inhomogeneities are represented by

coloured ovals. .................................................................................................................. 72

Figure 53. Synthetic model (c) used to generate test apparent resistivity data for the pole-

dipole (a) and Wenner (b) arrays....................................................................................... 73

Figure 54. The effect of cell size on the pole-dipole array inversion model. Pole-dipole array.

(a) The apparent resistivity pseudosection. The inversion models obtained using cells

widths of (b) one, (b) one-half and (c) one-quarter the unit electrode spacing. ................ 74

Figure 55. Example of the use of narrower model cells with the Wenner-Schlumberger array.

(a) The apparent resistivity pseudosection for the PIPESCHL.DAT data set. The

inversion models using (b) cells with a width of 1.0 meter that is the same as the actual

unit electrode, and (c) using narrower cells with a width of 0.5 meter. ............................ 75

Figure 56. Example of reduction of near-surface 'ripples' in inversion model. (a) Apparent

resistivity pseudosection for the BLUERIDGE.DAT data set. (b) Normal inversion using

the robust inversion norm and model refinement. (c) Inversion using higher damping

factor for first layer to reduce the 'ripple' effect in the top layer. ...................................... 75

Figure 57. Inversion models for a long survey line using different settings to reduce the

calculation time. (a) The measured apparent resistivity pseudosection for the Redas

underwater survey for the first 2600 meters. Inversion models using (b) standard

inversion settings with 4 nodes between adjacent electrode positions, (c) standard

inversion settings with 2 nodes between adjacent electrode positions, (d) with calculation

of Jacobian matrix values for selected model cells, (e) with sparse inversion technique,

(f) with 2 meter model cell width and (g) with 3 meter model cell width. ....................... 80

Figure 58. Part of finite-element mesh used to model a survey with submerged electrodes.

The resistivity of mesh cells in the water layer are fixed at the known water resistivity. 80

Figure 59. Part of a finite-element mesh for a long survey line. The electrodes used in the

same array are 3 meters apart, while the data unit electrode spacing is 1 meter. Using 2

nodes between adjacent electrodes in the mesh will actually result in 6 nodes between

electrodes in the same array. ............................................................................................. 81

Figure 60. Model resolution sections for the (a) Wenner, (b) Wenner-Schlumberger and (c)

simple dipole-dipole array (d) dipole-dipole array with overlapping data levels. ............ 83

Figure 61. (a) Model resolution section for comprehensive data set. (b) Model resolution

section for optimized data set generated by the ‘Compare R’ method. ............................ 84

Figure 62. Test inversion model for the different arrays. (a) Apparent resistivity

pseudosection for the simple dipole-dipole array with a dipole length ‘a’ of 1.0 meter

with the dipole separation factor ‘n’ of 1 to 6. (b) Apparent resistivity pseudosection for

the dipole-dipole array with overlapping data levels. (c) The synthetic model with 4

rectangular blocks of 100 .m embedded in a medium of 10 .m. as used by Wilkinson

et al. (2006b). .................................................................................................................... 85

Figure 63. Inversion results with the (a) Wenner array, (b) Wenner-Schlumberger array, (c)

simple dipole-dipole array and (d) dipole-dipole array with overlapping data levels. ..... 86

Figure 64. Inversions models with the (a) optimized data set with 4462 data points and (b) a

truncated optimized data set with 413 data points. ........................................................... 86

Figure 65. Comparison between the (a) model resolution, (b) model resolution index and (c)

DOI index sections for the Landfill data set. .................................................................... 87

Figure 66. The model resolution plots for the three streamer configurations. ......................... 88

Figure 67. An example of 3-D effects on a 2-D survey. (a) Apparent resistivity

pseudosections (Wenner array) along lines at different y-locations over (b) a 3-D

structure shown in the form of horizontal slices. .............................................................. 91

Figure 68. The 2-D sensitivity sections for the pole-dipole array with a dipole length of 1

meter and with (a) n=6, (b) n=12 and (c) n=18. Note that as the ‘n’ factor increases, the

zone of high positive sensitive values becomes increasingly concentrated in a shallower

zone below the P1-P2 dipole. ............................................................................................ 92

Figure 69. Example of apparent resistivity pseudosection with pole-dipole array with large

‘n’ values. Note that the anomaly due to a small near-surface high resistivity block

becomes greater as the ‘n’ factor increases. This means that the sensitivity of the array to

the near-surface region between the P1-P2 potential dipole becomes greater as the ‘n’

factor increases. ................................................................................................................. 93

Figure 70. Diagrammatic illustration of differences in objective function shapes for the pole-

pole array and dipole-dipole array data sets leading to different models obtained from

optimization routine. ......................................................................................................... 95

Figure 71. The I.P. values for some rocks and minerals. ......................................................... 97

Figure 72. The Cole-Cole model. (a) Simplified electrical analogue circuit model (after

Pelton et al. 1978). = resistivity, m = chargeability, = time constant, c = relaxation

constant. (b) Amplitude and phase response to sine wave excitation (frequency domain).

(c) Transient response to square wave current pulse (time domain). Most I.P. receivers

measure the integral of the decay voltage signal over a fixed interval, mt, as a measure of

the I.P. effect. .................................................................................................................... 97

Figure 73. Magusi River massive sulphide ore body inversion models. (a) Apparent resistivity

pseudosection. Resistivity inversion models obtained using the (b) perturbation and (c)

complex resistivity methods. (d) Apparent I.P. (metal factor) pseudosection. I.P. models

obtained using the (e) perturbation and (f) complex resistivity methods. ....................... 101

Figure 74. Sketch of separated cable spreads setup used (after Dahlin and Loke, 2015). ..... 102

Figure 75. Resistivity and chargeability pseudosection from field demo at 3rd IP workshop at

Ile d’Oleron (after Dahlin and Loke, 2015). .................................................................. 102

Figure 76. The possible arrangements of the electrodes for the pole-pole array in the cross-

borehole survey and the 2-D sensitivity sections. The locations of the two boreholes are

shown by the vertical black lines. ................................................................................... 105

Figure 77. A schematic diagram of two electrodes below the surface. The potential measured

at P can be considered as the sum of the contribution from the current source C and its

image C’ above the ground surface. ................................................................................ 106

Figure 78. The 2-D sensitivity patterns for various arrangements with the pole-bipole array.

The arrangement with (a) C1 and P1 in first borehole and P2 in second borehole, (b) C1

in the first borehole and both P1 and P2 in the second borehole, (c) all three electrodes in

the first borehole and (d) the current electrode on the ground surface. .......................... 107

Figure 79. The 2-D sensitivity patterns for various arrangements of the bipole-bipole array.

(a) C1 and P1 are in the first borehole, and C2 and P2 are in second borehole. (b) C1 and

C2 are in the first borehole, and P1 and P2 are in second borehole. In both cases, the

distance between the electrodes in the same borehole is equal to the separation between

the boreholes. The arrangements in (c) and (d) are similar to (a) and (b) except that the

distance between the electrodes in the same borehole is half the spacing between the

boreholes. ........................................................................................................................ 108

Figure 80. Possible measurement sequences using the bipole-bipole array. Other possible

measurements sequences are described in the paper by Zhou and Greenhalgh (1997). . 110

Figure 81. Several possible bipole-bipole configurations with a single borehole. (a) The C1

and C2 electrodes at depths of 3 and 2 meters respectively below the 0 meter mark. (b)

The C1 and P1 electrodes at depths of 3 and 2 meters respectively below the 0 meter

mark. (c). The C1 electrode is at a depth of 3 meters below the 0 meter mark while the

C2 electrode is on the surface. (d). The C1 electrode is at a depth of 3 meters below the 0

meter mark while the P1 electrode is on the surface. ...................................................... 112

Figure 82. A pole-bipole survey with a single borehole. The C1 electrode is at a depth of 3

meters below the 0 meter mark. ...................................................................................... 112

Figure 83. Test of optimized cross-borehole arrays with a synthetic model. (a) Two-layer test

model with conductive and resistive anomalies. Inversion models for (b) optimized data

set with all arrays, (c) 'standard' data set and (d) the reduced optimized data set that

excludes arrays with both current (or potential) electrodes in the same borehole. All the

data sets have 1875 data points. The outlines of the rectangular blocks showing their true

positions are also shown. ................................................................................................. 114

Figure 84. Schematic diagram of the MERIT method with the electrodes are planted along

the surface and directly below using the direct push technology (after Harro and Kruse,

2013)................................................................................................................................ 115

Figure 85. Inversion models for the different data sets for the data collected with electrodes at

surface and 7.62 m depth with 4 m horizontal spacing. Models for the (a) standard arrays

(405 data points), optimized arrays with (b) 403 and (c) 514 data points....................... 116

Figure 86. Landslide field example, Malaysia. (a) The apparent resistivity pseudosection for a

survey across a landslide in Cangkat Jering and (b) the interpretation model for the

subsurface. ....................................................................................................................... 117

Figure 87. Industrial pollution example, U.K. (a) The apparent resistivity pseudosection from

a survey over a derelict industrial site, and the (b) computer model for the subsurface. 118

Figure 88. Mapping of holes in a clay layer, U.S.A. (a) Apparent resistivity pseudosection for

the survey to map holes in the lower clay layer. (b) Inversion model and (c) sensitivity

values of the model cells used by the inversion program. .............................................. 119

Figure 89. Water infiltration mapping, U.K. (a) The apparent resistivity and (b) inversion

model sections from the survey conducted at the beginning of the Birmingham

infiltration study. This shows the results from the initial data set that forms the base

model in the joint inversion with the later time data sets. As a comparison, the model

obtained from the inversion of the data set collected after 10 hours of irrigation is shown

in (c). ............................................................................................................................... 120

Figure 90. Time-lapse sections from the infiltration study. The sections show the change in

the subsurface resistivity values with time obtained from the inversion of the data sets

collected during the irrigation and recovery phases of the study. ................................... 121

Figure 91. Hoveringham pumping test, U.K. (a) The apparent resistivity pseudosection at the

beginning of the test. The inversion model sections at the (b) beginning and (v) after 220

minutes of pumping. ........................................................................................................ 122

Figure 92. Percentage relative change in the subsurface resistivity values for the

Hoveringham pumping test. To highlight the changes in the subsurface resistivity, the

changes in the model resistivity are shown. Note the increase in the model resistivity

below the borehole with time. ......................................................................................... 122

Figure 93. Use of Archie’s Law for the Hoveringham pumping test. Sections showing the

relative desaturation values obtained from the inversion models of the data sets collected

during the different stages of the Hoveringham pumping test. Archie’s Law probably

gives a lower limit for the actual change in the aquifer saturation. ................................ 123

Figure 94. Groundwater survey, Nigeria. (a). Apparent resistivity pseudosection. (b) The

inversion model with topography. Note the location of the borehole at the 175 meters

mark. ................................................................................................................................ 124

Figure 95. The inversion model after 4 iterations from an underwater riverbed survey by

Sage Engineering, Belgium. ............................................................................................ 125

Figure 96. Thames River (CT, USA) survey with floating electrodes. (a) The measured

apparent resistivity pseudosection. Inversion models obtained (b) without constraints on

the water layer, and (c) with a fixed water layer. ............................................................ 126

Figure 97. Survey to map oil sands, Alberta, Canada. (a) Location of major tar sands deposits

in Alberta, Canada. (b) Example of resistivity log and geologic column of Athabasca oil

sands. (c) 2-D resistivity model from imaging survey (Kellett and Bauman, 1999). ..... 127

Figure 98. A simple arrangement of the electrodes for a 3-D survey. ................................... 129

Figure 99. Two possible measurement sequences for a 3-D survey. The location of potential

electrodes corresponding to a single current electrode in the arrangement used by (a) a

survey to measure the complete data set and (b) a cross-diagonal survey. ..................... 129

Figure 100. 3-D sensitivity plots for the pole-pole array. The plots are in the form of

horizontal slices through the earth at different depths. ................................................... 130

Figure 101. 3-D sensitivity plots for the pole-dipole array with n=1 in the form of horizontal

slices through the earth at different depths. The C1 electrode is the leftmost white cross.

......................................................................................................................................... 132

Figure 102. 3-D sensitivity plots for the pole-dipole array with n=4 in the form of horizontal

slices through the earth at different depths...................................................................... 132

Figure 103. 3-D sensitivity plots for the dipole-dipole array with n=1 in the form of

horizontal slices through the earth at different depths. The C2 electrode is the leftmost

white cross. ...................................................................................................................... 134

Figure 104. 3-D sensitivity plots for the dipole-dipole array with n=4 in the form of

horizontal slices through the earth at different depths. The C2 electrode is the leftmost

white cross. ...................................................................................................................... 134

Figure 105. The 3-D sensitivity plots for the Wenner alpha array at different depths. ......... 135

Figure 106. The 3-D sensitivity plots for the Wenner-Schlumberger array with the n=4 at

different depths. ............................................................................................................... 135

Figure 107. The 3-D sensitivity plots for the Wenner gamma array at different depths. ...... 136

Figure 108. Using the roll-along method to survey a 10 by 10 grid with a multi-electrode

system with 50 nodes. (a) Surveys using a 10 by 5 grid with the lines orientated in the x-

direction. (b) Surveys with the lines orientated in the y-direction. ................................. 137

Figure 109. A 3-D model with 4 rectangular prisms in a 15 by 15 survey grid. (a) The finite-

difference grid. (b) Horizontal apparent resistivity pseudosections for the pole-pole array

with the electrodes aligned in the x-direction.................................................................. 139

Figure 110. The model used in 3-D inversion. ...................................................................... 141

Figure 111. Inversion models for the Vetlanda landfill survey data set. (a) Using standard

inversion settings. (b) With a higher damping factor for the topmost layer. (c) Using

diagonal roughness filter in the horizontal (x-y) directions. (d) Using diagonal roughness

filters in the vertical (x-z and y-z) directions as well. (e) Using the roughness filter in all

directions. ........................................................................................................................ 143

Figure 112. Types of 3-D roughness filters. (a) With components in the x- and y- directions

only for the horizontal filter. (b) With components in the diagonal directions in the x-y

plane for the horizontal filter. (c) Applying the roughness filter with the corner model

cells as well. Only two (out of eight) corner cells are shown. ........................................ 144

Figure 113. Inversion models for the Vetlanda landfill survey data set using model cells of

equal lengths in the x- and y- directions. (a) Using standard inversion settings. (b) With a

higher damping factor for the topmost layer. (c) Using diagonal roughness filter in the

horizontal (x-y) directions. (d) Using diagonal roughness filters in the vertical (x-z and y-

z) directions as well. ........................................................................................................ 145

Figure 114. Synthetic model for long electrodes survey. 3-D model using cells of low

resistivity (0.01 .m) that are marked in red to simulate cased wells. ........................... 146

Figure 115. Comparison of inversion models using point electrodes with and without long

electrodes. (a) Inversion model for pole-pole data set using only surface point electrodes.

(b) Inversion model for pole-pole data set using 121 point and 3 long electrodes. ........ 147

Figure 116. 3-D data grid formats.......................................................................................... 148

Figure 117. Methods to model the effect of an electrode using the finite-difference and finite-

element methods. ............................................................................................................. 149

Figure 118. The use of an appropriate mesh spacing to obtain sufficient accuracy for

electrodes in the same array that close together. ............................................................. 149

Figure 119. Map with survey lines and infrastructure at the Hanford site. ........................... 150

Figure 120. Types of model grids for the Hanford survey data set. (a) Using a 5 meters

spacing model for the entire area. (b) Using a 4 meters spacing model for the entire area.

(c) Using a mixture of 4 and 5 meters spacing model grid. ............................................ 151

Figure 121. Inversion model for Hanford survey site. ........................................................... 152

Figure 122. Arrangement of survey lines using a 3 cable system with the Abem SAS

instrument. ....................................................................................................................... 153

Figure 123. Inversions model for (a) combined Wenner-Schlumberger and dipole-dipole data

set, (b) optimized data set. The actual positions of the blocks are marked by black

rectangles. ........................................................................................................................ 153

Figure 124. Horizontal sections showing the model resolution for the (a) comprehensive data

set, (b) standard arrays and (c) optimized arrays. The electrode positions are marked by

small green crosses in the top layer in (a). ...................................................................... 154

Figure 125. Vertical cross-sections showing the model resolution for the comprehensive,

‘standard’, small and large optimized data sets............................................................... 155

Figure 126. The synthetic test model with two 1000 ohm.m rectangular blocks (marked by

black rectangles) embedded in a 100 ohm.m background medium. ............................... 155

Figure 127. The inversion models for the synthetic model with two 1000 ohm.m rectangular

blocks (marked by black rectangles) embedded in a 100 ohm.m background medium. 157

Figure 128. Comprehensive data set point-spread-function plots on a vertical x-z plane

located at y=4.5 m for a model cell at different depths with centre at x=4.5m and y=4.5m.

......................................................................................................................................... 157

Figure 129. Model resolution sections with circular perimeter for (a) comprehensive data set

with 180300 arrays, optimized data sets with (b) 946 and (c) 2000 arrays. .................... 158

Figure 130. Inversion results for survey with a circular perimeter using optimized arrays with

(a) 946 and (b) 2000 data points...................................................................................... 158

Figure 131. The (a) initial and (b) perturbed synthetic test models with apparent resistivity

pseudosections and inversion models assuming a constant electrode spacing and flat

surface. (c) The inversion model obtained with the algorithm that allows the electrodes

to shift. ............................................................................................................................. 163

Figure 132. (a) 3-D synthetic test model with a rectangular survey grid. In the perturbed

model, the resistivities of the smaller blocks were changed from 400 and 20 ohm.m to

350 and 25 ohm.m. (b) The survey grid for the perturbed model. The four electrodes

shifted are marked by red circles. Electrode 1 was shifted 0.3 m in the x-direction,

electrode 2 moved 0.3 m in the y-direction, electrode 3 moved -0.2 m in both x and y-

directions while electrode 4 was shifted vertically upwards by 0.4 m. ........................... 164

Figure 133. Inversion models for the (a) initial and (b) perturbed data sets with fixed

electrodes in a rectangular grid. The true positions of the bands and prisms are marked by

black lines. The positions of the shifted electrodes are marked by small crosses in the top

layer in (b). ...................................................................................................................... 164

Figure 134. Inversion models for the perturbed data set using different relative damping

factors for the electrodes positions vector with a homogenous half-space starting model.

......................................................................................................................................... 165

Figure 135. Surface x-z profiles along the line y=10 m for inversions using a (a) homogenous

half-space and (b) initial data set starting models. .......................................................... 165

Figure 136. Inversion models for the perturbed data set using different relative damping

factors for the electrodes positions vector with the inversion model from the initial data

set as the starting model. ................................................................................................. 165

Figure 137. (a) Arrangement of electrodes in the Birmingham 3-D field survey. (b)

Horizontal and (c) vertical cross-sections of the model obtained from the inversion of the

Birmingham field survey data set. The locations of observed tree roots on the ground

surface are also shown..................................................................................................... 166

Figure 138. Example L-curve plots. (a) A plot of the model roughness versus the data misfit

for the Birmingham survey data set for a number of damping factor values. (b) A plot of

the curvature of the curve in (a) for the different damping factor values. The ‘optimum’

damping factor is at the maximum curvature point......................................................... 167

Figure 139. The 3-D model obtained from the inversion of the Lernacken Sludge deposit

survey data set displayed as horizontal slices through the earth. .................................... 168

Figure 140. A 3-D view of the model obtained from the inversion of the Lernacken Sludge

deposit survey data set displayed with the Slicer/Dicer program. A vertical exaggeration

factor of 2 is used in the display to highlight the sludge ponds. Note that the color contour

intervals are arranged in a logarithmic manner with respect to the resistivity. ............... 168

Figure 141. Map of the Panama Canal region with the survey area marked (Noonan and

Rucker, 2011). ................................................................................................................. 169

Figure 142. The model grid used for the Panama Canal floating electrodes survey with 20 by

20 meters cells. The location of the electrodes along the surveys lines are shown as

colored points. ................................................................................................................. 170

Figure 143. The inversion model for the Panama Canal floating electrodes survey. The first 4

layers correspond to the water column. ........................................................................... 170

Figure 144. Geological map of the Copper Hill area (White et al. 2001). ............................. 171

Figure 145. Electrodes layout used for the 3-D survey of the Copper Hill area. ................... 171

Figure 146. The I.P. model obtained from the inversion of the Copper Hill survey data set.

Yellow areas have chargeability values of greater than 35 mV/V, while red areas have

chargeability values of greater than 45 mV/V (White et al., 2001). ............................... 172

Figure 147. Location of uranium mines in the Athabasca basin, Saskatchewan (Bingham et

al., 2006). ........................................................................................................................ 173

Figure 148. Geological model of uranium deposit (Bingham et al., 2006). .......................... 173

Figure 149. Example of inversion model from the Midwest deposit area showing the

resistivity at a depth of about 200 meters (Bingham et al., 2006). ................................. 174

Figure 150. A complex time-lapse field survey example. (a) Map of Cripple Creek survey

site. (b) Overhead view of the inversion model grid with electrodes layout. (c) Iso-surface

contours for the -4% resistivity change at different times after the injection of the sodium

cyanide solution (that started at 2.8 hours from the first data set in snapshots used). t1=

1.1 hours, t2= 2.4 hours, t3= 3.7 hours, t4= 4.9 hours. (d) Overhead view of iso-surfaces.

......................................................................................................................................... 175

Figure 151. Geological map of (a) south-west South Australia, (b) the Burra area, and (c) a

plot of survey electrodes and model cells layout. ........................................................... 176

Figure 152. Burra survey (a) resistivity and (b) I.P. inversion model layers. ........................ 177

Figure 153. The model (a) resistivity and (b) I.P. resolution index, and (c) VOI values. The

red arrows at the left side of (c) shows the position of the vertical slice shown in Figure

154. .................................................................................................................................. 178

Figure 154. Vertical cross-sections of the (a) resistivity model resolution index, (b) I.P

resolution index and (c) VOI in the X-Z plane 0.8 km north of the origin. .................... 179

Figure 155. 2-D test model with conductive overburden. ..................................................... 192

Figure 156. Models obtained with different relative damping weights αs. ............................ 193

Figure 157. Models for the Burra data set with a relative damping weight of 0.5 for αs. ..... 194

Figure 158. I.P. vertical sections along the y-direction (at x=1650 to 1700 m) Burra data set

(a) without (αs=0.0) and (b) with (αs=0.5) a reference model constraint. ....................... 194

Figure 159. A long electrode in a (a) homogeneous medium, (b) two-layer medium with a

low resistivity lower layer and (c) partly in air. .............................................................. 196

Figure 160. Example of non-conventional electrodes arrangements. (a) Offset pole-dipole

arrangement, (b) distributed pole-dipole arrangement and (c) measured resistance

components for a vector array potential triplet. .............................................................. 197

Figure 161. Layout of electrodes in field survey with the inverse model grid. ..................... 198

Figure 162. Resistivity and I.P inverse models for vector array field data set. ..................... 199

Figure 163. Synthetic model with a low resistivity structure. Apparent resistivity

pseudosections for (a) Wenner alpha, (b) Wenner beta, (c) dipole-dipole, (d) Wenner-

Schlumberger arrays (e) forward pole-dipole, (f) reverse pole-dipole, (g) forward pole-

dipole (with n=1), (h) reverse pole-dipole (with n=1) arrays. (i) The synthetic model with

a near-surface low resistivity structure in the middle. .................................................... 200

Figure 164. Arrangement of the electrodes for (a) Wenner alpha, (b) Wenner beta, (c) dipole-

dipole, (d) Wenner-Schlumberger, (e) forward pole-dipole, (f) reverse pole-dipole, (g)

forward pole-dipole (with n=1), (h) reverse pole-dipole (with n=1) arrays. ................... 201

Figure 165. Inverse models for the apparent resistivity data for the (a) Wenner alpha, (b)

Wenner beta, (c) dipole-dipole, (d) Wenner-Schlumberger, (e) forward pole-dipole, (f)

combined forward and reverse pole-dipole, (g) forward pole-dipole (with n=1), (h)

combined forward and reverse pole-dipole (with n=1) arrays. ....................................... 202

Figure 166. Lambayanna survey (a) map and (b) inverse model grid (segments shown by

coloured grid lines).......................................................................................................... 203

Figure 167. Lambayanna survey inverse models using (a) standard monolithic inversion with

a single mesh (b) 2x4 segmented mesh, and (c) electrodes binned to every 0.5 m. ....... 206

Figure 168. 3-D plot of Lambayanna survey inverse model. ................................................ 207

Figure 169. (a) Plot of the transformed I.P. compared to the original I.P. values. (b)

Difference between the transformed and original I.P values. ......................................... 209

Figure 170. Measured apparent resistivity and I.P. pseudosections for the Magusi River

survey, with inverse models using the different I.P. model constraints. ......................... 210

Figure 171. (a) L-curve and (b) curvature curve for the Filborna data set. (c) L-curve and (d)

curvature curve for the Lambayanna data set.................................................................. 215

Figure 172. Inverse models for Lambayanna field data set for iterations 3 to 8 using a slow

cooling sequence of damping factors. ............................................................................. 216

List of Tables

Table 1. 1-D inversion examples using the RES1D.EXE program. ........................................ 11

Table 2. The median depth of investigation (ze) for the different arrays (Edwards, 1977). L is

the total length of the array. Note identical values of ze/a for the Wenner-Schlumberger

and pole-dipole arrays. Please refer to Figure 4 for the arrangement of the electrodes for

the different arrays. The geometric factor is for an "a" value of 1.0 meter. For the pole-

dipole array, the array length ‘L’ only takes into account the active electrodes C1, P1 and

P2 (i.e. it does not take into account the remote C2 electrode). ........................................ 29

Table 3. Forward modeling examples. ..................................................................................... 50

Table 4. Methods to remove bad data points ........................................................................... 54

Table 5. Tests with different inversion options ....................................................................... 63

Table 6. Tests with different topographic modeling options ................................................... 68

Table 7. Tests with option to fix the model resistivity ............................................................ 70

Table 8. Inversion times for the long Redas survey data set using different settings. ............. 79

Table 9. Inversion times for different line lengths. .................................................................. 79

Table 10. Tests with 2-D I.P. inversion ................................................................................. 103

Table 11. A few borehole inversion tests............................................................................... 110

Table 12. 3-D forward modeling examples ........................................................................... 138

Table 13. 3-D inversion examples ......................................................................................... 142

Table 14. Comparison of different methods for the inversion of the Lambayanna data set. . 205

Table 15. Different types of mass storage drives (2020). ...................................................... 206

Copyright (1996-2020) M.H.Loke

1

1 Introduction to resistivity surveys

1.1 Introduction and basic resistivity theory

The resistivity method is one of the oldest geophysical survey techniques (Loke, 2011).

The purpose of electrical surveys is to determine the subsurface resistivity distribution by

making measurements on the ground surface. From these measurements, the true resistivity of

the subsurface can be estimated. The ground resistivity is related to various geological

parameters such as the mineral and fluid content, porosity and degree of water saturation in the

rock. Electrical resistivity surveys have been used for many decades in hydrogeological,

mining, geotechnical, environmental and even hydrocarbon exploration (Loke et al., 2013a).

The fundamental physical law used in resistivity surveys is Ohm’s Law that governs

the flow of current in the ground. The equation for Ohm’s Law in vector form for current flow

in a continuous medium is given by

J = E (1.1)

where is the conductivity of the medium, J is the current density and E is the electric field

intensity. In practice, what is measured is the electric field potential. We note that in

geophysical surveys the medium resistivity , which is equals to the reciprocal of the

conductivity (=1/), is more commonly used. The relationship between the electric potential

and the field intensity is given by

E= −

(1.2)

Combining equations (1.1) and (1.2), we get

J= −

(1.3)

In almost all surveys, the current sources are in the form of point sources. In this case, over an

elemental volume

V surrounding the a current source I, located at

( )

sss zyx ,,

the relationship

between the current density and the current (Dey and Morrison, 1979a) is given by

)()()(. sss zzyyxx

V

I−−−

= J

(1.4)

where

is the Dirac delta function. Equation (3) can then be rewritten as

( ) ( )

)()()(,,,, sss zzyyxx

V

I

zyxzyx −−−

=•−

(1.5)

This is the basic equation that gives the potential distribution in the ground due to a

point current source. A large number of techniques have been developed to solve this equation.

This is the “forward” modeling problem, i.e. to determine the potential that would be observed

over a given subsurface structure. Fully analytical methods have been used for simple cases,

such as a sphere in a homogenous medium or a vertical fault between two areas each with a

constant resistivity. For an arbitrary resistivity distribution, numerical techniques are more

commonly used. For the 1-D case, where the subsurface is restricted to a number of horizontal

layers, the linear filter method is commonly used (Koefoed, 1979). For 2-D and 3-D cases, the

finite-difference and finite-element methods are the most versatile. In Chapter 2, we will look

at the use of a forward modeling computer program for 2-D structures.

The more complicated cases will be examined in the later sections. First, we start with

the simplest case with a homogeneous subsurface and a single point current source on the

ground surface (Figure 1). In this case, the current flows radially away from the source, and the

potential varies inversely with distance from the current source. The equipotential surfaces

have a hemisphere shape, and the current flow is perpendicular to the equipotential surface.

The potential in this case is given by

r

I

2

=

(1.6)

Copyright (1996-2020) M.H.Loke

2

where r is the distance of a point in the medium (including the ground surface) from the

electrode. In practice, all resistivity surveys use at least two current electrodes, a positive

current and a negative current source. Figure 2 show the potential distribution caused by a pair

of electrodes. The potential values have a symmetrical pattern about the vertical place at the

mid-point between the two electrodes. The potential value in the medium from such a pair is

given by

−=

21

11

2CC rr

I

(1.7)

where rC1 and rC2 are distances of the point from the first and second current electrodes.

In practically all surveys, the potential difference between two points (normally on the

ground surface) is measured. A typical arrangement with 4 electrodes is shown in Figure 3.

The potential difference is then given by

+−−

=

22211211

1111

2PCPCPCPC rrrr

I

(1.8)

The above equation gives the potential that would be measured over a homogenous half space

with a 4 electrodes array.

Actual field surveys are conducted over an inhomogenous medium where the

subsurface resistivity has a 3-D distribution. The resistivity measurements are still made by

injecting current into the ground through the two current electrodes (C1 and C2 in Figure 3),

and measuring the resulting voltage difference at two potential electrodes (P1 and P2). From

the current (I) and potential (

) values, an apparent resistivity (

a) value is calculated.

I

k

a

=

(1.9)

where

+−−

=

22211211

1111 2

PCPCPCPC rrrr

k

k is a geometric factor that depends on the arrangement of the four electrodes. Resistivity

measuring instruments normally give a resistance value, R =

/I, so in practice the apparent

resistivity value is calculated by

a = k R (1.10)

The calculated resistivity value is not the true resistivity of the subsurface, but an

“apparent” value that is the resistivity of a homogeneous ground that will give the same

resistance value for the same electrode arrangement. The relationship between the “apparent”

resistivity and the “true” resistivity is a complex relationship. To determine the true subsurface

resistivity from the apparent resistivity values is the “inversion” problem. Methods to carry out

such an inversion will be discussed in more detail at the end of this chapter.

Figure 4 shows the common arrays used in resistivity surveys together with their

geometric factors. In a later section, we will examine the advantages and disadvantages of some

of these arrays.

There are two more electrical based methods that are closely related to the resistivity

method. They are the Induced Polarization (IP) method, and the Spectral Induced Polarization

(SIP) (also known as Complex Resistivity (CR)) method. Both methods require measuring

instruments that are more sensitive than the normal resistivity method, and with significantly

higher currents. IP surveys are comparatively more common, particularly in mineral

exploration surveys. It is able to detect conductive minerals of very low concentrations that

Copyright (1996-2020) M.H.Loke

3

might otherwise be missed by resistivity or EM surveys. Commercial SIP surveys are

comparatively rare, although it is a popular research subject. Both IP and SIP surveys use

alternating currents (in the frequency domain) of much higher frequencies than standard

resistivity surveys. Electromagnetic coupling is a serious problem in both methods. To

minimize the electromagnetic coupling, the dipole-dipole (or pole-dipole) array is commonly

used.

Figure 1. The flow of current from a point current source and the resulting potential

distribution.

Figure 2. The potential distribution caused by a pair of current electrodes. The electrodes are

1 m apart with a current of 1 ampere and a homogeneous half-space with resistivity of 1 m.

Copyright (1996-2020) M.H.Loke

4

Figure 3. A conventional array with four electrodes to measure the subsurface resistivity.

Figure 4. Common arrays used in resistivity surveys and their geometric factors. Note that the

dipole-dipole, pole-dipole and Wenner-Schlumberger arrays have two parameters, the dipole

length “a” and the dipole separation factor “n”. While the “n” factor is commonly an integer

value, non-integer values can also be used.

Copyright (1996-2020) M.H.Loke

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1.2 Electrical properties of earth materials

Electric current flows in earth materials at shallow depths through two main methods.

They are electronic conduction and electrolytic conduction. In electronic conduction, the

current flow is via free electrons, such as in metals. In electrolytic conduction, the current flow

is via the movement of ions in groundwater. In environmental and engineering surveys,

electrolytic conduction is probably the more common mechanism. Electronic conduction is

important when conductive minerals are present, such metal sulfides and graphite in mineral

surveys.

The resistivity of common rocks, soil materials and chemicals (Keller and Frischknecht,

1966; Daniels and Alberty, 1966; Telford et al., 1990) is shown in Figure 5. Igneous and

metamorphic rocks typically have high resistivity values. The resistivity of these rocks is

greatly dependent on the degree of fracturing, and the percentage of the fractures filled with

ground water. Thus a given rock type can have a large range of resistivity, from about 1000 to

10 million m, depending on whether it is wet or dry. This characteristic is useful in the

detection of fracture zones and other weathering features, such as in engineering and

groundwater surveys.

Sedimentary rocks, which are usually more porous and have higher water content,

normally have lower resistivity values compared to igneous and metamorphic rocks. The

resistivity values range from 10 to about 10000 m, with most values below 1000 m. The

resistivity values are largely dependent on the porosity of the rocks, and the salinity of the

contained water.

Unconsolidated sediments generally have even lower resistivity values than

sedimentary rocks, with values ranging from about 10 to less than 1000 m. The resistivity

value is dependent on the porosity (assuming all the pores are saturated) as well as the clay

content. Clayey soil normally has a lower resistivity value than sandy soil. However, note the

overlap in the resistivity values of the different classes of rocks and soils. This is because the

resistivity of a particular rock or soil sample depends on a number of factors such as the

porosity, the degree of water saturation and the concentration of dissolved salts.

The resistivity of groundwater varies from 10 to 100 m. depending on the

concentration of dissolved salts. Note the low resistivity (about 0.2 m) of seawater due to

the relatively high salt content. This makes the resistivity method an ideal technique for

mapping the saline and fresh water interface in coastal areas. One simple equation that gives

the relationship between the resistivity of a porous rock and the fluid saturation factor is

Archie’s Law. It is only applicable for certain types of rocks and sediments, particularly those

that have a low clay content. The electrical conduction is assumed to be through the fluids

filling the pores of the rock. Archie's Law is given by

m

w

a−

=

(1.11)

where

is the rock resistivity,

w is fluid resistivity,

is the fraction of the rock filled with the

fluid, while a and m are two empirical parameters (Keller and Frischknecht, 1966). For most

rocks, a is about 1 while m is about 2. For sediments with a significant clay content, other more

complex equations have been proposed (Olivar et al., 1990).

The resistivities of several types of ores are also shown. Metallic sulfides (such as

pyrrhotite, galena and pyrite) have typically low resistivity values of less than 1 m. Note that

the resistivity value of a particular ore body can differ greatly from the resistivity of the

individual crystals. Other factors, such as the nature of the ore body (massive or disseminated)

have a significant effect. Note that graphitic slate has a low resistivity value, similar to the

metallic sulfides, which can give rise to problems in mineral surveys. Most oxides, such as

hematite, do not have a significantly low resistivity value. One exception is magnetite.

The resistivity values of several industrial contaminants are also given in Figure 5.

Copyright (1996-2020) M.H.Loke

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Metals, such as iron, have extremely low resistivity values. Chemicals that are strong

electrolytes, such as potassium chloride and sodium chloride, can greatly reduce the resistivity

of ground water to less than 1 m even at fairly low concentrations. The effect of weak

electrolytes, such as acetic acid, is comparatively smaller. Hydrocarbons, such as xylene

(6.998x1016 m), typically have very high resistivity values. However, in practice the

percentage of hydrocarbons in a rock or soil is usually quite small, and might not have a

significant effect on the bulk resistivity. However when the concentration of the hydrocarbon

is high, such as the commercial oil sands deposits in Canada, the resistivity method has proved

to be a useful exploration method for such deposits (see section 7.10 for an example).

Figure 5. The resistivity of rocks, soils and minerals.

1.3 1-D resistivity surveys and inversions – applications, limitations and pitfalls

The resistivity method has its origin in the 1920’s due to the work of the Schlumberger

brothers. For approximately the next 60 years, for quantitative interpretation, conventional

sounding surveys (Koefoed, 1979) were normally used. In this method, the center point of the

electrode array remains fixed, but the spacing between the electrodes is increased to obtain

more information about the deeper sections of the subsurface.

The measured apparent resistivity values are normally plotted on a log-log graph paper.

To interpret the data from such a survey, it is normally assumed that the subsurface consists of

horizontal layers. In this case, the subsurface resistivity changes only with depth, but does not

change in the horizontal direction. A one-dimensional model of the subsurface is used to

interpret the measurements (Figure 6a). Figure 7 shows an example of the data from a sounding

survey and a possible interpretation model. This method has given useful results for geological

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situations (such the water-table) where the one-dimensional model is approximately true.

The software provided, RES1D.EXE, is a simple inversion and forward modeling

program for 1-D models that consists of horizontal layers. In the software package, several files

with extensions of DAT are example data files with resistivity sounding data. Files with the

MOD extension are model files that can be used to generate synthetic data for the inversion

part of the program. As a first try, read in the file WENNER3.DAT that contains the Wenner

array sounding data for a simple 3-layer model.

Figure 6. The three different models used in the interpretation of resistivity measurements.

Figure 7. A typical 1-D model used in the interpretation of resistivity sounding data for the

Wenner array.

The greatest limitation of the resistivity sounding method is that it does not take into

account lateral changes in the layer resistivity. Such changes are probably the rule rather than

the exception. The failure to include the effect of such lateral changes can results in errors in

the interpreted layer resistivity and/or thickness. As an example, Figure 8 shows a 2-D model

where the main structure is a two-layer model with a resistivity of 10 m and a thickness of 5

meters for the upper layer, while the lower layer has a resistivity of 100 m. To the left of the

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center point of the survey line, a low resistivity prism of 1 m is added in the upper layer to

simulate a lateral inhomogeneity. The 2-D model has 144 electrodes that are 1 meter apart. The

apparent resistivity pseudosections for the Wenner and Schlumberger array are also shown. For

the Schlumberger array, the spacing between the potential electrodes is fixed at 1.0 meter for

the apparent resistivity values shown in the pseudosection. The sounding curves that are

obtained with conventional Wenner and Schlumberger array sounding surveys with the mid-

point at the center of the line are shown in Figure 9. In the 2-D model, the low resistivity

rectangular prism extends from 5.5 to 18.5 meters to the left of the sounding mid-point. The

ideal sounding curves for both arrays for a two-layer model (i.e. without the low resistivity

prism) are also shown for comparison. For the Wenner array, the low resistivity prism causes

the apparent resistivity values in the sounding curve (Figure 9a) to be too low for spacing

values of 2 to 9 meters and for spacings larger than 15 meters. At spacings of between 9 to 15

meters, the second potential electrode P2 crosses over the low resistivity prism. This causes the

apparent resistivity values to approach the two-layer model sounding curve. If the apparent

resistivity values from this model are interpreted using a conventional 1-D model, the resulting

model could be misleading. In this case, the sounding data will most likely to be interpreted as

a three-layer model.

The effect of the low resistivity prism on the Schlumberger array sounding curve is

slightly different. The apparent resistivity values measured with a spacing of 1 meter between

the central potential electrodes are shown by black crosses in Figure 9b. For electrode spacings

(which is defined as half the total length of the array for the Schlumberger array) of less than

15 meters, the apparent resistivity values are less than that of the two-layer sounding curve. For

spacings greater than 17 meters, the apparent resistivity values tend to be too high. This is

probably because the low resistivity prism lies to the right of the C2 electrode (i.e. outside the

array) for spacings of less than 15 meters. For spacings of greater than 17 meters, it lies between

the P2 and C2 electrodes. Again, if the data is interpreted using a 1-D model, the results could

be misleading. One method that has been frequently recommended to “remove” the effect of

lateral variations with the Schlumberger array is by shifting curve segments measured with

different spacings between the central potential electrodes. The apparent resistivity values

measured with a spacing of 3 meters between the potential electrodes are also shown in Figure

9b. The difference in the sounding curves with the spacings of 1 meter and 3 meters between

the potential electrodes is small, particularly for large electrode spacings. Thus any shifting in

the curve segments would not remove the distortion in the sounding curve due to the low

resistivity prism. The method of shifting the curve segments is probably more applicable if the

inhomogeneity lies between the central potential electrodes, and probably ineffective if the

inhomogeneity is beyond the largest potential electrodes spacing used (which is the case in

Figure 8). However, note that the effect of the prism on the Schlumberger array sounding curve

is smaller at the larger electrode spacings compared with the Wenner array (Figure 9). The

main reason is probably the larger distance between the P2 and C2 electrodes in the

Schlumberger array.

A more reliable method to reduce the effect of lateral variations on the sounding data

is the offset Wenner method (Barker, 1978). It makes use of the property that the effect of an

inhomogeneity on the apparent resistivity value is of opposite sign if it lies between the two

potential electrodes or if it is between a potential and a current electrode. For the example

shown in Figure 8, if the low resistivity body lies in between a current and potential electrode

(the P2 and C2 electrodes in this case), the measured apparent resistivity value would be lower.

If the low resistivity body lies in between the P1 and P2 electrodes, it will cause the apparent

resistivity value to be higher. The reason for this phenomenon can be found in the sensitivity

pattern for the Wenner array (see Figure 23a). By taking measurements with different positions

for the mid-point of the array, the effect of the low resistivity body can be reduced.

Copyright (1996-2020) M.H.Loke

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Figure 8. A 2-D two-layer model with a low resistivity prism in the upper layer. The calculated

apparent resistivity pseudosections for the (a) Wenner and (b) Schlumberger arrays. (c) The

2D model. The mid-point for a conventional sounding survey is also shown.

Another classical survey technique is the profiling method. In this case, the spacing

between the electrodes remains fixed, but the entire array is moved along a straight line. This

gives some information about lateral changes in the subsurface resistivity, but it cannot detect

vertical changes in the resistivity. Interpretation of data from profiling surveys is mainly

qualitative.

The most severe limitation of the resistivity sounding method is that horizontal (or

lateral) changes in the subsurface resistivity are commonly found. The ideal situation shown in

Figure 6a is rarely found in practice. As shown by the examples in Figure 8 and Figure 9,

lateral changes in the subsurface resistivity will cause changes in the apparent resistivity values

that might be, and frequently are, misinterpreted as changes with depth in the subsurface

resistivity. In many engineering and environmental studies, the subsurface geology is very

complex where the resistivity can change rapidly over short distances. The 1-D resistivity

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sounding method would not be sufficiently accurate for such situations.

Figure 9. Apparent resistivity sounding curves for a 2-D model with a lateral inhomogeneity.

(a) The apparent resistivity curve extracted from the 2D pseudosection for the Wenner array.

The sounding curve for a two-layer model without the low resistivity prism is also shown by

the black line curve. (b) The apparent resistivity curves extracted from the 2-D pseudosection

for the Schlumberger array with a spacing of 1.0 meter (black crosses) and 3.0 meters (red

crosses) between the potential electrodes. The sounding curve for a two-layer model without

the low resistivity prism is also shown.

To use the RES1D.EXE program for the exercises in the table below, as well as the

other programs that we shall use in the later sections, follows the usual sequence used by

Windows XP/Vista/7/8/10. Click the ‘Start’ button, followed by ‘Programs’ and the look for

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the RES1D folder in the list of installed programs. Alternatively, you can create a shortcut icon

on the Windows Desktop.

Table 1. 1-D inversion examples using the RES1D.EXE program.

Data set and purpose

Things to try

WENNER3.DAT – A simple

synthetic data file for a 3 layer

model.

(1). Read in the file, and then run the “Carry out inversion”

step.

WENN_LATERAL.DAT and

SCHL_LATER.DAT –

Wenner and Schlumberger

array sounding data shown in

Figure 9 that are extracted

from the 2-D pseudosections.

(1). Read in the files, and then invert the data sets.

(2). Compare the results with the true two-layer model

(that has resistivities of 10 m and 100 m for the first

and second layers, and thickness of 5 meters for the first

layer).

WENOFFSET.DAT – A field

data set collected using the

offset Wenner method.

(1). Read in the files, and then invert the data set.

IPTESTM.DAT – A 1-D

sounding data file with IP

measurements as well to round

things up.

(1). Read in the files, and then invert the data set.

To obtain a more accurate subsurface model than is possible with a simple 1-D model,

a more complex model must be used. In a 2-D model (Figure 6b), the resistivity values are

allowed to vary in one horizontal direction (usually referred to as the x direction) but assumed

to be constant in the other horizontal (the y) direction. This approximation is reasonable for

survey lines that are perpendicular to the strike of an elongated structure. The most realistic

model would be a fully 3-D model (Figure 6c) where the resistivity values are allowed to

change in all 3 directions. The use of 2-D and 3-D surveys and interpretation techniques will

be examined in detail in the following chapters.

1.4 Basic Inverse Theory

In geophysical inversion, we seek to find a model that gives a response that is similar

to the actual measured values. The model is an idealized mathematical representation of a

section of the earth. The model has a set of model parameters that are the physical quantities

we want to estimate from the observed data. The model response is the synthetic data that can

be calculated from the mathematical relationships defining the model for a given set of model

parameters. All inversion methods essentially try to determine a model for the subsurface

whose response agrees with the measured data subject to certain restrictions and within

acceptable limits. In the cell-based method used by the RES2DINV and RES3DINV programs,

the model parameters are the resistivity values of the model cells, while the data is the measured

apparent resistivity values. The mathematical link between the model parameters and the model

response for the 2-D and 3-D resistivity models is provided by the finite-difference (Dey and

Morrison, 1979a, 1979b) or finite-element methods (Silvester and Ferrari 1990).

In all optimization methods, an initial model is modified in an iterative manner so that

the difference between the model response and the observed data values is reduced. The set of

observed data can be written as a column vector y given by

),.....,,(col 21 m

yyy=y

(1.12)

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where m is the number of measurements. The model response f can be written in a similar form.

),.....,,(col 21 m

fff=f

(1.13)

For resistivity problems, it is a common practice to use the logarithm of the apparent resistivity

values for the observed data and model response, and the logarithm of the model values as the

model parameters. The model parameters can be represented by the following vector

),.....,,(col 21 n

qqq=q

(1.14)

where n is the number of model parameters. The difference between the observed data and the

model response is given by the discrepancy vector g that is defined by

g = y - f (1.15)

In the least-squares optimization method, the initial model is modified such that the

sum of squares error E of the difference between the model response and the observed data

values is minimized.

=

== n

ii

Tgg 1

2

gE

(1.16)

To reduce the above error value, the following Gauss-Newton equation is used to

determine the change in the model parameters that should reduce the sum of squares error

(Lines and Treitel 1984).

gJΔqJJ TT =

i

(1.17)

where q is the model parameter change vector, and J is the Jacobian matrix (of size m by n)

of partial derivatives. The elements of the Jacobian matrix are given by

j

i

ij q

f

J

=

(1.18)

that is the change in the ith model response due to a change in the jth model parameter. After

calculating the parameter change vector, a new model is obtained by

kk1k Δqqq +=

+

(1.19)

In practice, the simple least-squares equation (1.17) is rarely used by itself in

geophysical inversion. In some situations the matrix product

JJT

might be singular, and thus

the least-squares equation does not have a solution for q. Another common problem is that

the matrix product

JJT

is nearly singular. This can occur if a poor initial model that is very

different from the optimum model is used. The parameter change vector calculated using

equation (1.17) can have components that are too large such that the new model calculated with

(1.19) might have values that are not realistic. One common method to avoid this problem is

the Marquardt-Levenberg modification (Lines and Treitel, 1984) to the Gauss-Newton

equation that is given by

( )

gJΔqIJJ T

k

T=+

(1.20)

where I is the identity matrix. The factor

is known as the Marquardt or damping factor, and

this method is also known as the ridge regression method (Inman 1975) or damped least-

squares method. The damping factor effectively constrains the range of values that the

components of parameter change vector can q take. While the Gauss-Newton method in

equation (1.17) attempts to minimize the sum of squares of the discrepancy vector only, the

Marquardt-Levenberg method also minimizes a combination of the magnitude of the

discrepancy vector and the parameter change vector. This method has been successfully used

in the inversion of resistivity sounding data where the model consists of a small number of

layers. For example, it was used in the inversion of the resistivity sounding example in Figure

7 with three layers (i.e. five model parameters). However when the number of model

parameters is large, such as in 2-D and 3-D inversion models that consist of a large number of

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small cells, the model produced by this method can have an erratic resistivity distribution with

spurious high or low resistivity zones (Constable et al., 1987). To overcome this problem, the

Gauss-Newton least-squares equation is further modified so as to minimize the spatial

variations in the model parameters (i.e. the model resistivity values change in a smooth or

gradual manner). This smoothness-constrained least-squares method (Ellis and Oldenburg

1994a, Loke 2011) has the following mathematical form.

( )

k

T

k

TqFgJΔqFJJ −=+

, (1.21)

where

z

T

zy

T

yx

T

xCCCCCCF zyx

++=

and Cx, Cy and Cz are the roughness filter matrices in the x-, y- and z-directions that couples

the model blocks in those directions (Figure 10, Figure 112).

x,

y and

z are the relative

weights given to the roughness filters in the x-, y- and z-directions. One common form of the

roughness filter matrix is the first-order difference matrix (deGroot-Hedlin and Constable

1990) that is given by

−

−

−

=

0

C

..

..

..

..

0....01100

0......0110

0......0011

(1.22)

Figure 10. Coupling between neighboring model cells through the roughness filter in a 2-D

model. (a) In the horizontal and vertical diretcions only, and (b) in diagonal diretcions as well.

Equation 1.21 also tries to minimize the square of the spatial changes, or roughness, of

the model resistivity values. It is in fact an l2 norm smoothness-constrained optimization

method. This tends to produce a model with a smooth variation of resistivity values. This

approach is acceptable if the actual subsurface resistivity varies in a smooth or gradational

manner. In some cases, the subsurface geology consists of a number of regions that are

internally almost homogeneous but with sharp boundaries between different regions. For such

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cases, the inversion formulation in (1.21) can be modified so that it minimizes the absolute

changes in the model resistivity values (Claerbout and Muir, 1973). This can sometimes give

significantly better results. Technically this is referred to as an l1 norm smoothness-constrained

optimization method, or more commonly known as the blocky inversion method. A number of

techniques can be used for such a modification. One simple method to implement an l1 norm

based optimization method using the standard least-squares formulation is the iteratively

reweighted least-squares method (Wolke and Schwetlick, 1988; Farquharson and Oldenburg,

1998). The optimization equation in (1.21) is modified to

( )

kRd

T

kRd

TqFgRJΔqFJRJ

−=+

, (1.23)

with

zm

T

zym

T

yxm

T

xR CRCCRCCRCF zyx

++=

where Rd and Rm are weighting matrices introduced so that different elements of the data misfit

and model roughness vectors are given equal weights in the inversion process.

Equation (1.23) provides a general method that can be further modified if necessary to

include known information about the subsurface geology. As an example, if it is known that

the variations in the subsurface resistivity are likely to be confined to a limited zone, the

damping factor values can modified (Ellis and Oldenburg 1994a) such that greater changes are

allowed in that zone. If the errors in the data points are known, a diagonal weighting matrix

can be used to give greater weights to data points with smaller errors.

One important modification to the least-squares optimization method is in time-lapse

inversion using the following equation (Kim et al. 2009, Loke et al. 2014a).

( )

( )

1it

T

iRiid

T

iit

T

Riid

T

irMRMαFλgRJΔrMRMαFλJRJ −

+−=++

(1.24)

M is the difference matrix applied across the time models with only the diagonal and one sub

diagonal elements having values of 1 and -1, respectively, while r is the combined resistivity

model for all the time series. is the temporal damping factor that gives the weight for

minimizing the temporal changes in the resistivity compared to the model roughness and data

misfit. Note the smoothness-constraint is not only applied in space through the F matrix, but

also across the different time models through the M matrix. This is illustrated in Figure 11.

One common variation is to apply a constraint such that the model is ‘close’ to a

reference model, qR. Equation (1.24) then then modified to the following form.

( )

( )

RkRd

T

kR

TqqIFgRJΔqFJJ −+−=+ )(

(1.25)

qR is usually a homogeneous background model. This equation imposes and additional

constraint the new model is close to the reference model with the damping factor weight α. Its

effect is similar to the Marquardt constraint in equation 1.20, and it prevents very large

deviations from the reference model.

The smoothness-constrained least-squares optimization method involves the damping

factor term

. This term balances the need to reduce the data misfit (so that the calculated

apparent resistivity values are as close as possible to the measured values) while producing a

model that is ‘reasonably’ smooth and perhaps more geologically realistic. A smaller value of

term will generally produce a model with a lower data misfit but usually at the expense of

larger variations in the model resistivity values. If the error of the measured apparent resistivity

values is known, a prudent approach might be to select a model where the data misfit is similar

to the known measurement errors. However, for most field data sets, the measurement error is

not known. There are generally two methods to automatically select the ‘optimum’ damping

factor for such cases, the GCV and L-curve methods (Farquharson and Oldenburg, 2004). For

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the inversion of a single data set, either method can be used. However, for time-lapse data sets

(Loke et al. 2014a), the L-curve method has a clear advantage due to the sparse block structure

of the Jacobian matrix (Figure 11b). The GCV method requires a matrix inversion. However,

the inverse of a sparse matrix is usually a full matrix. This makes it impractical to use for time-

lapse models with many time-series measurements such that the combined model might have

hundreds of thousands of model parameters (i.e. the number of model parameters for a single

data set multiplied by the number of time series data sets).

Figure 11. (a) Coupling between corresponding model blocks in two time-lapse models using

a cross-model time-lapse smoothness constraint. (b) Example Jacobian matrix structure for five

time series data sets and models. Each grey rectangle represents the Jacobian matrix associated

with a single set of measurements.

1.5 2-D model discretization methods

In the previous section, we have seen that the least-squares method is used to calculate

certain physical characteristics of the subsurface (the “model parameters”) from the apparent

resistivity measurements. The “model parameters” are set by the way we slice and dice the

subsurface into differen