Accuracy and resolution in 2D resistivity inversion

Accuracy and Resolution in 2D Resistivity Inversion

Jeffrey Z. Snyder

B.S. Geology

Duke University, 1996

Submitted to the Department of

Earth, Atmospheric, and Planetary Sciences

in partial fulfillment of the requirements for the degree of

Master of Science in Geosystems

at the

Massachusetts Institute of Technology

© Massachusetts Institute of Technology 2001. All rights reserved.

Signature of Author... . ... . . ... --Department of Earth, Atmospheric, and Planetary Sciences 11 May 2001

Certified by ... %

...--.-.--.-Frank Dale Morgan Professor of Geophysics Thesis Supervisor

Accepted by... ... ... ... .. ... Ronald G. Prinn Chairman, Department of Earth, Atmospheric, and Planetary Sciences

Accuracy and Resolution in 2D Resistivity Inversion

by

Jeffrey Z. Snyder

Submitted to the Department of

Earth, Atmospheric, and Planetary Sciences on 11 May 2001 in partial fulfillment of the requirements for the degree of

Master of Science in Geosystems

Abstract

Two-dimensional resistivity inversion employing regularization enforces a constraint of smoothness that minimizes error and avoids unrealistically complex solutions to the inverse problem. The insensitivity of surface

array data to deeper targets typically presents an under-determined problem for solution by the inversion algorithm, and the smoothing function within the algorithm tends to "smear" tomographic imagery. Together, the physical challenges of electrical resistivity as a geophysical method and the numerical challenges inherent in solving the inverse problem introduce errors in the accuracy of 2D resistivity imagery.

It is important to know what surface array geometry will provide data that yields the best results from the inversion. Testing this inversion algorithm on data from several array geometries provides the opportunity to determine the comparative performance of each survey method.

By introducing two types of resistive anomalies and varying their location within a synthetic homogeneous

half-space, it is possible to generate synthetic data with a forward model algorithm. The data from each array geometry is then inverted in order to illustrate the accuracy and resolution response of the inversion algorithm. The inversion images are converted into binary images after defining a critical resistivity that describes the contrast between background resistivity and target resistivity. The binary images are used as interpretive tools that allow the user to overcome the "smearing" introduced by the inversion.

Because of its consistent performance from the margins to the center of an array, a left-right sweep geometry combined with a pseudosection geometry appears to be the best choice for a surface array when there is no

knowledge of the subsurface structure or resistivity distribution.

The critical resistivity and the area of the anomaly are used to describe the performance of the inversion. When taken as functions of increasing depth, the critical resistivity decreases and the area of anomaly increases, providing a respective correlation with the current density and the degree of smoothness. Initial results by forming a product of critical resistivity and area suggest that it is possible to approximate the product from the original forward model, but further testing is warranted to provide more conclusive results.

Thesis Supervisor: F. Dale Morgan Title: Professor of Geophysics

Acknowledgements

I'd like to take the opportunity to thank a few folks who helped me complete this work, and this program:

Professor Dale Morgan, who offered me the latitude to define a problem and run with it. Even to the end, his efforts to excite me about the science involved in this project, and where it was leading, helped give me the added motivation to drive forward with the project.

John Sogade, who provided me with the forward and inversion codes, taught me how to use them, provided me with a sounding board for ideas about the project, and helped me to better frame the problem as I immersed myself in it. John's guidance and editing were invaluable in helping me to provide a cohesive framework for the written report.

Dan Bums, whose positive attitude and flexibility helped me to pull this together right at the end. Dan was also instrumental in getting me to look at the fundamental science in this project, which made final revisions and alterations so much easier.

My officemates, Jiganesh Patel and Stacy Archfield, who comprised the Geosystems team this year. Successful completion of a program like this requires teamwork, and their support and assistance were fantastic.

Darrell Coles, whose MatLab prowess got me out of trouble on more than one occasion, and whose experience with Geosystems and with ERL made life easier.

I could not have been at MIT without the support of my closest friends and family, so to them I must say thanks: To my parents, who supported me throughout this entire process. To my sister, Larissa, who kept an eye out for me in the last few weeks of school and knew how to keep things under control at home in South Jersey. And to the Breault family, who offered their home in Rhode Island as a safe haven that I could visit in order to escape MIT.

Very special, heartfelt thanks must go to my fiancee', Lorna. Her immeasurable strength and support has kept me in the game since I arrived at MIT. Though I'm sure that I pushed the limits of her flexibility and understanding, she stood by to provide a pillar for me to lean on. Thanks so much for being there.

Finally, thank you to the Exxon Mobil Corporation for providing MIT with funding that ultimately paid for my fellowship and allowed me to be here in the first place.

Contents

1 Introduction

1.1 Background 6

1.2 Objectives 10

2

Formulation of the Forward and Inversion Problems

2.1 Current density as a function of depth 12

2.2 The forward model 16

2.3 Selection of grid discretization 22

2.4 Formulation of the inverse problem 22

3

Accuracy and Resolution Testing

3.1 Array geometries to be tested 29

3.2 Determining the critical resistivity 35

3.2.1 Binary plotting of inversion images 35

3.3 Critical resistivity as a function of depth 39

3.4 Calculating the integrated area of the anomaly 47

3.5 Smoothness as a function of depth 54

3.6 The product of per and the integrated area 61

4

Summary and Conclusions

4.1 Summary of Testing Results 68

4.2 Future Work 70

Contents (cont.)

Appendices

Appendix 1: Imaging of a 10,000 f-m anomaly. Central location. Al-i

Appendix 2: Imaging of a 500 1-m anomaly. Central location. A2-1

Appendix 3: Imaging of a 10,000 A-m anomaly. Left flank location. A3-1 Appendix 4: Imaging of a 500 9-m anomaly. Left flank location. A4-1

Chapter 1

Introduction

Electrical resistivity surveying techniques have emerged as useful methods for

determining earth structure. By applying a known electrical current at various point source

locations on the earth's surface, surveyors attempt to infer the subsurface resistivity distribution

based on the different voltage potentials measured at the surface. Due to the sometimes wide

range of electrical resistivity values that certain materials in the earth possess, this technique

provides the surveyor with an effective geophysical method for delineating high contrast features

including reservoirs, cavities, and structure in the subsurface.

When conducting 2D resistivity surveying, researchers use linear arrays that consist of

electrical current sources and voltage potential receivers. The application of current at the surface

interacts with materials, objects, and structure in the subsurface, each with unique resistivity, to

produce voltage potential differences along the array. The locations of the voltage potential

measurements relative to the current sources dictate what area of the subsurface may possess the

resistivity that is effecting the voltage potential measurement. Varying the distance between the

current source electrodes and the magnitude of the current can change the current density and the

depth of current penetration beneath the array geometry. This in turn can affect the ability of the

voltage potential measurements to reflect the true resistivity distribution beneath the array.

Modern interpretation techniques utilize the solution of resistivity forward and inverse

problems. In our case the forward problem is the solution to the two-dimensional differential

equations that govern the relationship between applied currents and measurable potentials at the

system of blocks, each with resistivity p, (i =1 to m). The resistivities are called the model

parameters. The sole purpose of the inverse problem is to resolve the model parameters that will

fit the measured data to a prescribed tolerance and in so doing determine the subsurface

resistivity distribution.

Because of the potential to diagnose structure and material properties in the subsurface,

resistivity inversion has been investigated for applicability and effectiveness. Mufti (1976)

conducted 2-D resistivity inversion using a finite difference model utilizing a non-uniform

discretization of the earth. Later, Pelton et al (1978) constructed a faster, more computationally

efficient inversion algorithm using ridge regression and least-squares to achieve fast convergence

on a solution model that accurately represented the subsurface resistivity distribution. Tripp et al

(1984) expanded work in two-dimensional resistivity inversion by addressing the inverse

problem with a non-linear inversion method.

Recent work has attempted to optimize the inversion of surface measurements for the 2-D

and 3-D resistivity problems by utilizing computational methods that further increase the

accuracy and efficiency of algorithms while solving the inverse problem. The transmission

network analogy first developed for use in geophysical applications by Madden (1972) and then

adapted by others (Pelton et al, 1978; Tripp et al 1984; Zhang et al 1995) as a numerical

modeling approach to the resistivity inverse problem has yielded a computationally efficient

inversion algorithm that reduces computing time and minimizes error to produce accurate results.

Unfortunately, several physical and numerical challenges to solving the inverse problem

accompany any application of the resistivity method. The decrease in current density with depth

leaves the surface data relatively insensitive to deeper targets. The equivalence problem,

solutions because several subsurface resistivity distributions may be capable of producing

equivalent voltage potential measurements at the surface. The non-unique nature of the

resistivity inverse problem is further compounded by noisy surface data. This is because a

multiplicity of models often fit the surface data to prescribed tolerances. Which of the prescribed

tolerances works best is an open question.

Oftentimes, the surface potential measurements (data) are outnumbered by the model

parameters. This leads to an underdetermined inverse problem. Though determined or

overdetermined problems (where the amount of data is greater than the amount of model

parameters) are preferred, the inversion algorithm should be robust enough to accommodate the

underdetermined problems. Constraints that enforce requirements for acceptable solutions can

also be applied in order to overcome the non-uniqueness of the possible solutions to the inverse

problem.

Sometimes data seems to be insensitive to certain parameters or combinations of

parameters, which means that these parameters are irrelevant in an inversion sense. Similarly,

certain data points seem not to be affected by any of the parameters, so these are unimportant

data points. While survey arrays must be optimized to avoid unimportant data points, and models

should be chosen to avoid irrelevant parameters, situations often arise where such data points or

parameters exist in the formulation of an inverse solution. This leads to an ill-posed inverse

problem with consequent numerical instability.

Calculation of the inverse problem can yield several non-unique solutions that minimize

error in the forward model, when only one unique resistivity distribution should exist for a given

area. Assuming that there is no a priori knowledge of the subsurface resistivity distribution,

converge on inappropriate local minimum error solutions rather than global minimum solutions.

As a result, it is possible for inversion methods to yield any number of "rough" solutions that are

far more complex than the data or the reality of the earth may demand.

One approach towards overcoming the problem of non-uniqueness has been to apply

regularization to enforce a constraint of minimum roughness between nearest neighbors for each

iterative solution. This regularization pushes the inversion algorithm towards the simplest model

demanded by the data. It is equivalent to selecting a certain class of solution with prescribed

properties from amongst a variety of non-unique solutions (Shi 1998). In electromagnetic

sounding inversion, Constable et al (1987) coined the term "Occam's inversion" because their

inversion algorithm sought the simplest possible model demanded by the data. Their justification

was that "the real profile must be at least as rich as the profile found, but never less complex in

structure." LaBrecque (1996) later adapted this concept to inverting crosshole resistivity data.

While regularization yields simple, low error solutions, the affect of smoothing tends to

"smear" the results such that sharp boundaries are avoided if at all possible. There is then a loss

of accuracy and resolution in the subsurface resistivity model that becomes evident in the

resulting imagery created by the inversion.

The spatial distortion that accompanies this technique hinders appropriate interpretation of

subsurface imagery. In order to more effectively utilize the inversion method and its output

imagery, users should possess a perspective whereby they can expect the accuracy and resolution

behavior of the method in response to data produced by different anomaly types and locations

1.2 Objectives

To date, there has been little attempt to test this inversion method and standardize an

understanding of its accuracy response to different targets and locations relative to the surface

survey. Because of the physical challenges inherent with the resistivity method and the side

effects of overcoming the numerical challenges in the inverse problem, there are two primary

sources of inaccuracy and resolution loss. First, the weakening current density as a function of

depth and the insensitivity that it invokes in surface potential measurements makes it difficult to

"illuminate" relatively deep resistive or conductive anomalies within the subsurface. Second, the

nearest-neighbor approximation and smoothing introduced by the regularization of the inverse

problem creates a smearing and homogenization of deep anomalies. Understanding the roles that

these two factors play in effecting an accurate and well-resolved response from the resistivity

inverse problem will provide a perspective with which to interpret resistivity imagery, and

perhaps provide an avenue for future optimization of the inversion method.

In order to illustrate the accuracy and resolution behavior of this resistivity inversion

technique, this investigation will provide a catalog of images produced by the inversion

algorithm in response to synthetic data from two resistive anomaly types across a variety of

locations. After defining a critical threshold resistivity value in the imagery, we will use that

threshold value to create binary images as an added interpretive tool. The threshold value will

also be used to determine the integrated area of a resistive anomaly, enabling simultaneous

comparison of the current density, the degree of smoothing, and the integrated area of the

method and the numerical challenges of the inverse problem combine to affect the overall

accuracy performance of the inversion.

Finally, the testing regimen will be conducted for several surface array geometries,

expanding upon similar work conducted by Shi and Morgan (1997), in order to exhibit how

different data collection methods might provide more complete data to make the inverse problem

better determined. Because real-world applications provide little real data or imagery with which

to verify the accuracy of the inversion, the approach used here is to create synthetic surface data

using an algorithm that incorporates known electrical currents and a prescribed resistivity

distribution to solve the forward problem and produce surface potential measurements. The

synthetic data will then be utilized in the algorithm to solve the inverse problem, and the

resulting resistivity model can be compared with the initially prescribed model. By varying the

locations and the magnitudes of the prescribed resistivity anomalies, it is possible to assess some

standard behaviors of the algorithm so that we may gain a better understanding of the limitations

Chapter 2

Formulation of the Forward and Inverse Problems

2.1 Current density as a function of depth

The ability of current to penetrate the subsurface, travel through the earth, and return to the surface to yield surface data measurements seriously affects the ability of the any inversion

method to solve for deep resistive anomalies. A consideration of the 1-D case for a point current

source on the surface of a homogeneous isotropic earth illustrates how the voltage potential at a

location within the earth decreases with respect to increasing depth. Given Ohm's Law,

V= IR (2.1)

where V is the voltage potential at a given point, I is current flowing through the point, and R is

the resistance of the path. This can be expanded to consider the resistivity of the material in the

current path and the distance between the source and the point of interest, yielding

vIp 1

V = (2.2)

2;r r

Next, consider the 2-D case where there are two current source electrodes placed on the surface.

Figure 2.1(a) illustrates an example of a pair of current source electrodes and receiver potential

electrodes. The voltage difference between any two points on the surface, as a result of the

current flowing from one electrode to the other, can be given by

AV= - (2.3)

2rct r, r2 r3 r4

Where AV is the voltage potential difference between two points, P, and P2. I is the current

Power

Figure 2.1(a). A sample current electrode pair and voltage potential pair, used to illustrate equation (2.3). Reproduced

from

Power

Figure 2.1 (b). Solving for current density at depth with a dipole source at the surface. Reproduced from (Telford 1990).

Given these relations, we can also determine the horizontal current density within a

homogeneous medium with resistivity p as

Ji = (-) (2.4)

P &

x= I ){x (x-L) (2.5)

2;r r,'r

(Refer to Figure 2.1(b) for a graphic representation of (2.5)) If we wish to find the current

density at a point P in the subsurface at a depth of z, and we assume that P lies on a line

perpendicular to the midpoint between the current sources, then (2.5) becomes

J = ( ) 3> (2.6)

2;

4.

where L represents the distance between the current sources and I represents the current. Holding

the current I constant, we can solve for J, (in amperes/m2_{) as a function of z for various values of}

L. Figure 2.1(c) illustrates how the current density varies with L and z. The magnitude of the

current density at shallow depths is highest when the current source electrodes are closest

together. However, it is possible to increase the current density at depth by increasing the

distance between the current sources. The cost of achieving greater penetration is a loss of

current density in the shallow subsurface, and in all cases the general trend is for current density

Current Density I =I

10 10 10 10', 10

log Current Density, J (Mps/m2)

Figure 2.1(c). Current density as a function of depth, with I (current) being held constant.

-0

2.2 The forward model

Expansion of Ohm's Law into the three-dimensional case yields a system of nonlinear

differential equations that describe the relationship between applied surface currents, subsurface

resistivity, and surface voltage potential measurements. The system of equations is

V V(x, y, z) = -p(x, y, z)J(x, y, z) (2.7) V -J(x, y, z) = I(x, y, z) (2.8)

where again V represents the voltage potential, p is the resistivity, J is the current density, and I

is the current.

The resistivity forward model has a transmission network analog, first developed in

electrical engineering but later adapted to geophysical applications by Madden (1972). It consists

of network nodes, boundary nodes, and impedance branches. Based on this transmission network

model, the resistivity forward problem described by Equations (2.7) and (2.8) below can be

converted to a linear set of algebraic equations using Kirchoff s current law, which is then solved

using a bi-conjugate gradient algorithm. The schematic in Figure 2.2(a) exhibits the construction

of the transmission network. Note that for the geophysical resistivity problem, current sources

can be placed at network nodes, and voltage values are placed at nodes located on the top center

of each discretized block within the transmission network (Zhang et al 1995 and Shi 1998).

These expressions can then be solved numerically for the forward model by using a

discretized 3-D model of the subsurface. The 2-D forward model algorithm used in this

investigation uses a three-dimensional discretized subsurface region that is always three elements

thick in the y-direction while the x and z ranges are specified by the user and the middle slice is

z

* - network node

o - boundary node

Figure 2.2(a). A schematic of the transmission network analog as applied to the resistivity problem. Reproduced

from

In order to perform the testing and analysis of the resistivity inversion algorithm, it is necessary to produce synthetic data from a known field of structure. In this case, the forward model utilizes a 15 x 30 grid of blocks to represent a discretized subsurface cross-section. Each block represents a spatial value of size and resistivity as prescribed by the user. For the purposes of this investigation, each grid column entry will be 1 m (meter) specifying the width of the block, and each row grid entry will be 1 m specifying the thickness of each block. However, the first and thirtieth columns will possess a width of 150 m and the second and twenty-ninth

columns will possess a 50 m width in order to maintain appropriate boundary conditions that will

improve the accuracy of the forward model results. Figure 2.2(b) illustrates this spatial grid. A

100 Q-m homogeneous body will represent the background resistivity of the field of interest. The forward model allows users to choose the number of anomalies to introduce into the

cross-section; however, this study will utilize a target block of dimensions 2 m high by 4 m wide with

varying resistivity and/or position.

The forward model algorithm requires the user to provide an input data file that

represents the survey geometry. Figure 2.2(c) shows an example of a dipole-dipole

pseudosection data set, and Figure 2.2(d) is an example of a left-right sweep data set developed

at the Earth Resources Laboratory (MIT).

The user possesses the flexibility of creating synthetic data sets that mimic selected

survey geometries and of creating data sets in response to varying subsurface resistivity

distributions. It is important to note that, in the physical sense, a variety of subsurface resistivity

distributions could be constructed to yield identical surface data sets. This problem of

equivalence will take on an even greater role of importance when we consider the inverse

150 50 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 50 150 150 50 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 50 150 150 50 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 50 150 150 50 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 50 150 150 50 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 50 150 150 50 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 50 150 150 50 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 50 150 150 50 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 50 150 150 50 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 50 150 150 50 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 50 150 150 50 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 50 150 150 50 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 50 150 150 50 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 50 150 150 50 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 50 150 150 50 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 50 150

Voltage (mV) Current (mA)

Total number of electrode source pairs Initial source pair locations

Number of receiver pairs Receiver pair locations

3 4 -4.08416 1 4 5 -1.67314 1 5 6 -0.83572 1 6 7 -0.47804 1 7 8 -0.29642 1 8 9 -0.19191 1 9 10 -0.12718 1 2 3 7 4 5 -4.08416 1 5 6 -1.67314 1 6 7 -0.83572 1 7 8 -0.47804 1 8 9 -0.29642 1 9 10 -0.19191 1 10 11 -0.12718 1

Voltage (mV)

Total number of electrode source pairs _{3 4} Initial source pair locations

Number of receiver pairs Receiver pair locations

Current (mA) 2 3 -4.08416 1 3 4 -1.67314 1 4 5 -0.83572 1 5 6 -0.47804 1 6 7 -0.29642 1 7 8 -0.19191 1 8 9 -0.12718 1 9 10 -0.10153 1 10 11 -4.08416 1 11 12 -1.67314 1 12 13 -0.83572 1 13 14 -0.47804 1 14 15 -0.29642 1 15 16 -0.19191 1 16 17 -0.12718 1

Figure 2.2(d). Example offorward model data format for dipole-dipole left-right sweep array. (Note: Potential values are not representative of actual data. Figure has been provided only to

2.3 Selection of grid discretization

In order to conduct tests on the behavior of the inversion, it was necessary to establish an appropriate grid discretization for use in the forward model. In this investigation, we have chosen the simplest approach by using equivalent grid discretizations in both the forward and inversion grids. The forward model produces data based on a 15 x 30 element grid as illustrated in Figure 2.2(b), where each grid element within the boundaries is 1 m by 1 m in size. The inversion algorithm solves for model parameters based on the same grid discretization.

2.4 Formulation of the inverse problem

For the purposes of numerical calculations, the subsurface of the earth is discretized into a grid similar to that shown in Figure 2.4(a). Each block in the grid represents a resistivity value and a spatial dimension, such that the discretized subsurface electrical resistivity represents the parameters of the model, m, that are to be inverted. The data, d, constitutes all of the measured voltage-current pairs that are taken along a survey line at the surface.

The non-linear resistivity inverse problem requires solution of an iterative linear system of equations. This system of linear equations may be ill-posed because the initial guess for a solution in the iterative inversion algorithm may be far from the truth, and some of the data and/or model parameters may be irrelevant or unimportant to the solution of the inverse problem.

Without a priori knowledge of the subsurface structure forcing the inversion to converge

on a particular solution, it would be quite possible for the inversion to yield an unrealistically

complex, or "rough" solution. Though such a solution may be a minimum error result, the

complexity or roughness may be unnatural or perhaps more complex than the original data

Surface data observations, d

I I I I I I I I I I I I I

- I -, -, -I- 1-I-u - I - ~- I-I-I-I - I

-model parameters, m representing resistive blocks within the discretized subsurface.

Figure 2.4(a). Schematic comparison between surface data measurements and model parameters. 4-- - - -- - m - -

--

-

-

-

-

---

4.

-seek a low error solution while also enforcing a constraint of smoothing on the inversion.

Assuming that simplest solutions and the structures they represent are appropriate for

geophysical interpretation of high contrast structure in the real earth, the nearest-neighbor

regularization imposes a condition of uniqueness on the solution to the inverse problem by

requiring that it be the simplest low-error solution demanded by the data (Shi 1998).

To see how this regularization can be imposed, first consider a generic linear

algebraic case:

d =Gm + e (2.9)

For the resistivity inverse problem, d represents the surface data measurements (voltage

potentials), G represents the forward model that relates current, voltage, and resistivity in space;

m represents the unknown resistivity distribution in the subsurface; and e is the misfit error.

The Tikhonov regularization requires the definition and minimization of a function that

combines data misfit and the roughness of the model:

T = (d -Gm)) T _{R -'(d - Gm))+ r(m - m}

)T L L(m - mo) (2.10)

where ![ function that must be minimized, R is the data covariance matrix, r is the regularization

parameter, and L is a linear operator (Tikhonov 1977). In (2.10), the first term represents the data

misfit produced by the model and the second term represents the "stabilizing functional" that

defines the spatial roughness of the model parameters so that (2.10) can be redefined as

T = s, + rs2 (2.11)

The inversion algorithm employs a non-linear conjugate gradient method that iteratively

attempts to find a solution model to the inverse problem while minimizing the function V. Each

of .For relatively high values of , s, dominates the behavior of the algorithm in attempting to

minimize . For low values of r, sj dominates the inversion algorithm.

If [ _{approaches a minimum value for both the first and second terms, the algorithm will} stop iterating in order to prevent it from relaxing the smoothness constraint while in pursuit of

even lower error solutions. This ensures that the algorithm converges on the smoothest, low-error

solution. Figure 2.4(b) illustrates the behavior of V, z , s1, and S2 with advancing iterations.

While the smoothing constraint yields a simple, low error solution, the effect of the

smoothing also tends to "smear" the results, such that there is a loss in accuracy and resolution

that is evident in the subsurface imagery attained by the inversion. Figure 2.4(c) is a comparison

between imagery from a forward model and the related inversion. The top image is a graphic

representation of a single resistive block anomaly that was used to create synthetic surface data

in the forward model algorithm. The bottom image is the graphic representation of the inversion

results. The inversion tends to smear the anomaly downward, and it changes the values of the

resistivity in the target. The smearing in the imagery illustrates the effect that the

nearest-neighbor smoothing constraint has when producing the simplest model. The decrease in

resistivity values can be attributed to both the nearest neighbor smoothing during the inversion

and the increasing insensitivity of the surface data to relatively deeper targets.

Because of this spatial distortion, it is important for users of this inversion technique to

understand the accuracy and resolution response of the inversion based on the current density

provided by different survey geometries and the location of anomalies within the subsurface.

The design of this inversion algorithm for 2D resistivity produces a distinctive

manifestation in the output imagery, in that anomalies in the 'inverted' images are smeared as a

1 2 3 4 5 6 7 8 9 10 11

Iteration

Forward: Resistivity (Ohm-m) 0 10000 2 8000 4 E 6 6000 4000 10 12 2000 14 0 5 10 15 20 25 Distance (m)

inversion: Resistivity (Ohm-m) 0 7000 2 6000 4 5000 4000 'M 8 ₃₀₀₀ 10 2000 12 1000 14 0 5 10 15 20 25 Distance (m)

Figure 2.4(c). A comparison between a synthetic forward model (top) and the modelproduced by inversion (bottom) of the data generated by the forward model.

in this inversion method, the inversion creates a smearing of resistivity anomalies in the imagery

that can lead to misinterpretation by the user. This smearing effect is further compounded by the

decrease in data density and sensitivity for deeper targets. Due to the physics that govern the

flow of electrical current in the solid earth, the current density from surface sources decays with

depth. This combination of regularization and data density distorts accuracy and resolution with

increasing depth.

Sasaki (1992) undertook a resolution analysis of an inversion method that dealt with

borehole and surface potential measurements. The analysis he presented focused primarily on

factors internal to the inversion algorithm, such as grid discretization and damping coefficient

(T), _{with some investigation into electrode placement within the survey array. However,}

systematic testing of the inversion in response to resistive anomalies that vary with location and

magnitude has yet to be done, and in this case may yield some consistent qualitative standards of

interpretation and performance, particularly when compared with the amount of smoothing

imposed by the inversion and the current density. Furthermore, there may also be ways to

increase the accuracy of the inversion by providing it better data in order to better determine the

inverse problem. Applying different array geometries may yield better current density or greater

sampling at depth. The remainder of this investigation will consist of a series of simple yet

elegant analytical tests that will qualify the accuracy behavior of the inversion in response to

Chapter 3

Accuracy And Resolution Testing

3.1 Array geometries to be tested

To illustrate the potential for improvement in inversion results, five surface array

geometries will be used to produce synthetic data from the forward model. Recall that our

forward model will utilize a 15 x 30 discretized grid to represent the subsurface, and the first, second, twenty-ninth, and thirtieth columns are use only to provide boundaries to the grid. Also,

referring back to the transmission network analogy reminds us that voltage values can be

assigned to the top center of each block and current values can be assigned to any network node.

For the purposes of creating synthetic data sets, current source locations and receiver locations

can be at any location between 3 and 28.

In this case, the dipole-dipole pseudosection array uses 23 current source pair locations.

Starting at the extreme left side of a field to be imaged, two adjacent electrodes (positions 3 and

4) are selected as the current source pair. Then, the receiver pairs are staggered across the

remainder of the electrodes along the survey line, with the terminus being the 2 8th position. The source pairs are then shifted one position to the right, and again the receiver pairs extend from

the source pair to the terminus. With this array geometry, a complete survey has a total of 276

potential measurements. Figure 3.1 displays a simple schematic of the survey geometry.

The left-right sweep begins with the current source pair placed at the extreme ends of the

survey line, such that one current electrode starts at the 3 position and the other begins at the 28

position. Potential measurements are then taken at all of the electrode positions between the

current source electrodes. After the measurements are taken, the left hand electrode is then

measurements are only taken between the source electrodes. This procedure continues until the

survey has utilized a total of 50 source pair locations for a total of 552 measurements. Figure 3.1(b) is a schematic of the left-right sweep.

The dipole-dipole middle sweep begins with the current source electrodes paired at the

extreme ends of the array line. The mid-point of the line is designated as an axis of symmetry.

All receiver electrodes are paired so that they are equidistant from the mid-point of the array. This procedure requires a total of 13 source pair locations for a total of 156 measurements.

Figure 3.1(c) is a schematic of the middle-sweep.

The final two arrays to be tested will be hybrids of those mentioned. One array will be a

left-right-pseudo, the other will be a left-right-middle sweep. For example, adding the data

collected from a left-right-sweep survey to a pseudosection survey gives us the left-right-pseudo

hybrid.

Figure 3.1(d) is a test matrix that describes the different parameters that will be varied in

r&"ti

,

9,

9

.

n=1

n=2

n=3

Figure 3.1(a). The dipole-dipole pseudosection array geometry. The current source dipole is located at the top left hand side of the survey line, and the receiver dipoles are paired to the

right. Reprinted from (Shi 1998).

rtv V1

1 2 3

1 2 3

1 2 3

Figure 3.1(b). The left-right sweep. The current source electrodes begin at the extreme ends of the survey geometry, and all of the receiver pairs are between the source pairs. The left hand electrode is then swept to the right towards the right electrode. The process is repeated in order

Figure 3.1(c). The middle-sweep. The current source electrodes are paired, beginning at the extreme ends of the survey geometry. Receiver pairs are centered, symmetric to the mid-point of

Array Geometry Varied Pseudosection Left-Right Sweep Middle Sweep Left-Right-Pseudo Left-Right-Middle 2D Resistivity Inversion

Anomaly Type Anomaly Size

Varied Constant 10,000 92m 4 m wide x 2 m thick 500 Qm 1 Q-m Test Matrix Anomaly Depth Varied 1 m to 8 m depth Location Varied Center of grid Left flank of grid

Background Resistivity: Constant, 100 Q-m.

Figure 3.1(d). Matrix for varied parameters in 2D resistivity inversion accuracy testing. Note that the 1 -m has only been tested for centrally located anomalies.

3.2 Determining the critical resistivity

In order to determine the size and magnitude of the resistive anomaly produced by the

inversion algorithm, it is necessary to define a threshold resistivity value for each model that the

inversion produces. This critical resistivity, per, can be used as a baseline to create binary plots and integrated area plots, and it can also be used as a proxy for the decay of resistivity values that

can be witnessed for deeper targets in the forward model. Using the definition provided by Beard

and Morgan (1991) for resistive anomalies set against relatively conductive backgrounds,

x= log10 Pmax (3.1)

x= log10 Pmin (3.2)

Ax = (xh -x,)/3 (3.3)

Pcr =1 OXh (3.4)

For conductive anomalies set against relatively resistive backgrounds, the solution for critical

resistivity becomes

Pcr = lOx+ (3.5)

The calculation of the critical resistivity value contains information about the actual resistivity

contrast in the image that can be used as a simple descriptor of the accuracy in resistivity

magnitude that the inversion produces.

3.2.1 Binary plotting of inversion images

With our definition for the critical resistivity, we conduct an element-wise comparison of

the model parameters in the inversion grid to Per. Values in the inversion grid that are greater

than Per will are assigned the value of 1 (resistive), and those values that are less than Per received

the original forward model resistivity distribution that was used to create synthetic data, and the

solution model that was found by the inversion.

All of the figures for these tests can be found in the appendices. In each appendix, there are test results for each of the five arrays (pseudosection, right sweep, middle sweep,

left-right-pseudo sweep, and left-right-middle sweep). The appendices have been ordered according

to the following test examples: Appendix 1 possesses test results for a 10,000 Q-m target that

was located near the surface, along the midpoint of the array line. Appendix 2 is for a 500 Q m

anomaly located near the surface along the midpoint. Appendices 3 and 4 are for 10,000 2-m

and 500 Q-m targets, respectively, that were located near the left margin. Appendix 5 possesses

results from the 1 9-m case for centrally located anomalies.

Based on the inversion images and the binary plots in Appendices 1 and 2, it would

appear at first glance that the pseudosection array actually produced better imagery than the

left-right sweep, particularly for the anomalies on the centerline. The left-left-right-pseudo also produced

an accurate response that was comparable to the pseudosection alone. For a shallow target, the

inversion placed the anomaly at the appropriate depth with the appropriate size in response to

both the pseudosection data (Fig. Al-1) and the left-right-pseudo data (A 1-4), but there appears

to be some loss of resolution in the left-right sweep test case (Fig. A1-2). As evidenced in the

binary plots, the left-right-middle sweep (Fig. A 1-5) was nominally less effective than the

pseudosection and left-right-pseudo sweep, particularly for the 10,000 Q m target. The middle

sweep was very inaccurate for all centerline targets (Figs. Al-3, Al-8, A2-3, A2-8), producing

heavily smeared anomalies and significantly reduced resistivity values.

If we recall from Figure 2.1(c), the current density was greatest at the surface when L, the distance between the current source electrodes, was minimized. This explains the comparatively

better performance of the pseudosection array in response to centerline shallow targets, as the

distance between current source electrodes was always 1 meter. The increased current density in

shallow areas beneath the midpoint of the array made the surface potential measurements more

sensitive to resistive targets located there.

The images in Appendices 3 and 4 represent repetitions of the same experiments except

that the anomaly was located along the left margin of the subsurface grid. The utility of the

left-right sweep, and particularly the left-left-right-pseudo sweep becomes immediately apparent by their

ability to better resolve anomalies along the flanks. The left-right-pseudo sweep also produced a

closer approximation of the magnitude of resistivity.

The pseudosection array, with its apparently triangular capture zone, completely lost

accuracy and the ability to resolve any targets near the margins. The inversion rendered tear-drop

shaped anomalies smeared down and towards the center and much lower resistivity values in

response to data from the pseudosection array. The middle-sweep appeared to be very inaccurate

when applied alone; the results it provided permitted the inversion to render anomalies on both

flanks when only one target was introduced in the forward model. It would appear that the

left-right-middle sweep was not made more effective by the added presence of the middle-sweep

data.

By these images, we can see how the different array geometries used current source placement and voltage receiver placement to determine current density and subsurface sampling.

In the case of the pseudosection geometry, it accurately (relative to the other geometries)

captured shallow resistive targets near the centerline, but the geometry limited its ability to

Appendix 5 contains images from tests of a conductive (1 t-m) target set against a relatively resistive background. The imagery tends to be much more difficult to interpret, as the

inversion tended to return a more smeared anomaly than we saw in the cases of resistive targets.

While the resistive targets tended to be underestimated by the inversion, the conductive targets

tended to be overestimated in magnitude. However, the determination of the critical resistivity

value and the creation of the binary plot appeared to be a relatively robust approach towards

interpreting the imagery for the conductive targets. The performance of the inversion appeared to be consistent with that witnessed for the resistive targets with one notable exception: though the

left-right-pseudo sweep offered the best performance, the left-right sweep outperformed the

pseudosection for the conductive target.

To summarize, the performance of the inversion response to the following surveys

geometries can be ranked as follows:

For capturing a resistive anomaly near the array midpoint 1. Left-Right-Pseudo Sweep

2. Pseudosection

3. Left-Right Sweep

4. Left-Right-Middle Sweep

5. Middle Sweep

For capturing a resistive anomaly near the array margins 1. Left-Right-Pseudo Sweep

2. Left-Right Sweep

3. Left-Right-Middle Sweep

4. Pseudosection

5. Middle Sweep

For capturing a conductive anomaly near the array midpoint 1. Left-Right-Pseudo Sweep

2. Left-Right Sweep

3. Left-Right-Middle Sweep

4. Pseudosection

5. Middle Sweep

3.3 Critical resistivity as a function of depth

It is convenient that pc, is determined by the resistivity values within the solution to the inverse problem because the critical resistivity can also be used as proxy for the behavior of the

inversion. The response of the inversion tends to yield decaying resistivity values as a function of depth; the full field plots in the appendices exhibit this behavior. However, plotting per as a function of the depth of the anomaly can yield insight into the decay in resistivity that the

inversion produces. The decay in resistivity is in response to the lack of sensitivity in the data for

deeper targets. A plot of pr for a 10,000 Q-m and 500 &-m, centrally located anomaly in

response to pseudsection array data can be found in Figure 3.3(a). A similar plot for the inversion

results borne from left-right sweep data can be found in Figure 3.3(b), and 3.3(c) refers to the

left-right-pseudo sweep. All experiments were repeated for anomalies on the flanks, and the

results are plotted in Figures 3.3(d)-(f).

(Further testing will be performed on the 10,000 Q-m and 500 Q-m targets. The middle

and left-right-middle sweeps will be omitted from further testing. The results in Section 3.2

suggested that the left-right-middle sweep was not competitive with the left-right-pseudo sweep, and the middle sweep is quite inaccurate as a stand-alone survey.)

Recall that in Figure 2.1(c), we illustrated how current density decays with increasing

depth. In response to data from each array geometry, the critical resistivity value decays with

increasing depth, though the 500 9-m cases decay more gradually than the 10,000 Q-m cases. It

appears that the critical resistivity shares a directly proportional relationship with the natural

behavior of current density in the subsurface, but the rate and magnitude of decay of the critical

In comparing the quality of the results for these three array types, it appears that the

left-right-pseudo sweep data produced consistently better results, particularly because pcr reached an absolute maximum in response to the left-right-pseudo sweep data and because per was

consistently higher as a function of depth. The significance of this higher critical resistivity value

is that it describes, essentially, a higher magnitude of contrast between the target anomaly and

the background and thus a more accurate response. Physically, this is most likely a function of

the increased current density that the larger distances between current source electrodes tend to

produce for deeper targets.

It is interesting to note that the pseudosection produced a critical resistivity value for the

500 Q-m at 1 m that was a near match for the critical resistivity value in the forward model (Figure 3.3(a), lower plate). These results agree with the concept that the shorter dipole lengths

in the pseudosection array produced higher current density at shallow depths, thus providing

more current to interact with resistive targets in the shallow subsurface.

When targets on the margins were considered, the results show that the pseudosection

data produced a rapid decay for per. The left-right sweep and the left-right-pseudo sweep

produced more gradual decay with increasing depth, consistent with the results from Section 3.3

that indicated better capture in the margins by the left-right sweep and its hybrid, the

10,000 ohm-m Anomaly 500 1000 1500 2000 Resistivity (Ohm-m) 500 ohm-m Anomaly 120 140 160 180 200 220 Resistivity (Ohm-m) 240 260 280

Figure 3.3(a). Critical resistivity as ajfunction of depth. The inversion algorithm was run using synthetic dipole-dipole pseudosection data generated in response to the two centrally located

anomaly types. Notice the behavior of the critical resistivity with respect to depth, and the similarity with current density as a function of depth (Figure 2.1(c)).

S-2 E 0 c -3 .- 6 IL --C, 2500 -1 E 0--3 0-5 .r -4 1L CD--8' 10 F - Inversion Rho ocr __Forward Rhoc 0 300

10,000 ohm-m Anomaly 500 1000 1500 2000 Resistivity (Ohm-m) 500 ohm-m Anomaly -1 $ -2-E 0 -4 0-5 o --8 _0- -Q. -8 L 0 -1 E 0 -3 -l 4 0-5 10 180 200 220 Resistivity (Ohm-m) 240 260 280

Figure 3.3(b). Critical resistivity as afuinction of depth. The inversion algorithm was run using synthetic left-right sweep data generated in response to the two centrally located anomaly types. Notice the behavior of the critical resistivity with respect to depth, and the similarity with current

density as a

function

160 2500 0 120 - --I --_-- _{Inversion Rhocr} Forward Rhoor -r -140 300

10,000 ohm-m Anomaly 2500 500 1000 1500 2000 Resistivity (Ohm-m) 500 ohm-m Anomaly ___ _{Inversion Rhocr} -- _{Forward Rhoor} -r -160 180 200 220 Resistivity (Ohm-m) 240 260 280 300

Figure 3.3(c). Critical resistivity as a

function

similarity with current density as a function of depth (Figure 2.1(c)).

-1 -E 0 C -3 IL 5 0 c 6-o -7 -8 -0 -6 -7 -81

10

-10,000 ohm-m Anomaly 500 1000 1500 2000 2500 Resistivity (Ohm-m) 500 ohm-m Anomaly I I I F 180 200 220 Resistivity (Ohm-m) 240 260 280 300

Figure 3.3(d). Critical resistivity as afunction of depth. Dipole-dipole pseudosection data in response to margin-located anomalies.

' -2 E 0--3 o -4 I-5 5-6 . 0 --2 E 0--3 0-5 -6 8-7 -8 '0 Inversion _Rhoc Forward Rho I-I a a 120 140 160

10,000 ohm-m Anomaly 500 1000 1500 2000 Resistivity (Ohm-m) 500 ohm-m Anomaly -1 S-2 E 0-c -3 -z 4 0-5 0 -6 (. o -7 -8 -1 S-2 E 0 C -3 -5 4 0 U) -6 o -7 -8

i

Figure 3.3(e). Critical resistivity as a

function

10,000 ohm-m Anomaly 500 1000 1500 2000 Resistivity (Ohm-m) 2500 500 ohm-m Anomaly 120 140 160 180 200 220 240 260 280 Resistivity (Ohm-m) 300

Figure 3.3(f). Critical resistivity as a finction of depth. Left-right-pseudo sweep data in response to margin-located anomalies. -1 E 0 C -3 0 -5 o. 0. o -7 -8 E-2 E 0 c -3 0-5 .- 6 0. c-7 -810 100

3.4 Calculating integrated area of anomaly

Using the critical resistivity values and the binary plots created from the inversion results, it is also possible to characterize the smoothing imposed by the algorithm by integrating the area

of the anomaly and plotting it as a function of depth. Figures 3.4(a)-(c) are integrated area plots

based for anomalies produced by the inversion in response to 10,000 Q-m and 500 Q-m,

centrally located anomalies that were used to produce synthetic pseudosection array, left-right

sweep, and left-right-pseudo sweep array data, respectively. Figures 3.4(d)-(f) are integrated area

plots based on targets located near the margins.

Notice that as depth increases in each case, and as pcr decreases in Figures 3.3(a)-(f), the area of the anomaly increases due to smearing. (This inversely proportional relationship between

the critical resistivity and the integrated area will be addressed in section 3.6.)

This smearing is notably more erratic for the margin-located anomalies that were imaged

with pseudosection data, but such behavior is expected when compared with the results in

Sections 3.2 and 3.3, further reinforcing the concept that pseudosection subsurface illumination

10,000 ohm-m Anomaly 10 20 30 40 50 60 Area (m 3) 500 ohm-m Anomaly S2 -E 0-C -3 4 LL .- 6-Q. $ 78 -0 -1-0-3 -32 -C -6-a) o -7 -8 -0

Figure 3.4(a). Integrated area of anomaly. The anomaly size is calculated based on the size of the binary plot (created with the critical resistivity value Pcr ). The inversion was run in response

to synthetic dipole-dipole pseudoseclion array data generated for centrally located anomalies.

10 20 30 40 50 60

10,000 ohm-m Anomaly 10 15 20 25 30 35 40 45 50 55 Area (m 3) 500 ohm-rn Anomaly -1- S-2-E 0-C-3 0-5- 0-6- -7--8 L 5 -1 -$ 2-E 0-C -3 0-5- 0-6-o -7 -8 -0

Figure 3.4(b). Integrated area of anomaly. The anomaly size is calculated based on the size of the binary plot (created with the critical resistivity value p,). The inversion was run in response

to synthetic left-right sweep array data generated for centrally located anomalies.

10 20 30 40 50

10,000 ohm-rn Anomaly 10 20 30 40 50 Area (m 3) 500 ohm-m Anomaly -1 -N -2-E 0--3 -54 a. 0-5 U-o -7 -8 -0 -1 S2 -E 0--3 5-~ 6 a. 8 7--8 -0

Figure 3.4(c). Integrated area of anomaly. The anomaly size is calculated based on the size of the binary plot (created with the critical resistivity value Pcr). The inversion was run in response

to synthetic left-right-pseudo sweep array data generated for centrally located anomalies.

10 20 30 40 50 60 70

10,000 ohm-m Anomaly 5 10 15 20 25 30 35 Area (m 3) 500 ohm-m Anomaly 5 10 15 20 25 30 35 Area (m3 )

Figure 3.4(d). Integrated area ofanomaly. Pseudosection data. Margin-located anomalies. 1 - -2- -3- -4- -5- -6- -7--8 -0 -1 E 0 c -3 0-6-Q-7 -8 0 ' I I L~ I

10,000 ohm-m Anomaly 10 15 20 25 Area (m3) 500 ohm-m Anomaly 10 15 20 25 Area (m3)

Figure 3.4(e). Integrated area of anomaly. Left-right sweep data. Margin-located anomalies.

S-2 E 0--3 o-5 -6 o -7 -8 -1 -E 0--3 -4 r 6-C. o -7 -8 -5

10,000 ohm-m Anomaly -1 -@ -2 E 0--3 -4 0-5 U-0 -6 8 7 -8 10 15 20 25 Area (m3) 500 ohm-m Anomaly -1 E 05 -LL 5 10 15 20 25 Area (m3₎

Figure 3.4(). Integrated area of anomaly. Left-right-pseudo sweep data. Margin-located anomalies.

3.5 Smoothness as a function of depth

Recall from equation (2.10), the functional f to be minimized. The second term of the

equation represents the roughness of the model and is given by

s2 = (m -m O)T LTL(m - mo) (3.6)

If we take the inverse of the roughness, then we can arrive at the smoothness of the model at the final iteration such that

Smoothness= (3.7)

S2

Plotting final iteration smoothness as a function of depth reveals how the smoothness

increases as the inversion algorithm attempts to impose minimal structure while solving an

increasingly underdetermined problem. Figure 3.5(a) is an example of this type of graph for

10,000 92-m and 500 -m anomalies that were used to produce forward model data analogous to a dipole-dipole pseudosection array geometry. Both the smoothness graph and the integrated area

graph tend to exhibit proportional behavior, such that they both appear to increase with depth.

This illustrates evidence of the smearing that the inversion algorithm imposes on anomalies.

Figure 3.5(b) and 3.5(c) are plots of inversion behavior in response to right sweep and

left-right-pseudo sweep data. Figures 3.5(d)-(f) repeat the tests in response to the geometries, but for

10,000 Ohm-m anomaly - -4 Smoothness 500 Ohm-m anomaly II I || 0 20 40 60 80 100 Smoothness 120 140 160 180

Figure 3.5(a). Final iteration smoothness as a function of depth. Pseudosection data. Centrally located anomalies. Notice that the trend is for the smoothness to increase with depth.

-6 -8 -10 -12 0 200

-10,000 Ohm-m anomaly -12 0 0 1 2 3 4 5 6 7 8 Smoothness 500 Ohm-m anomaly -2 -6 .1 - -10--12' 0 5 10 15 20 25 Smoothness

Figure 3.5(b). Final iteration smoothness as a

function

10,000 Ohm-m anomaly 0.5 1 1.5 2 2.5 Smoothness 500 Ohm-m anomaly 0.1 0.2 0.3 0.4 0.5 0.6 Smoothness

Figure 3.5(c). Final iteration smoothness as a function of depth. Left-right-pseudo sweep data. Centrally located anomalies. Notice that the trend is for the smoothness to increase with depth.

-2 E E -4 - -6--8 -0 -1 - -2--3 -4--5

-10,000 Ohm-rn anomaly I II I I--N 0 0.5 1 1.5 2 2.5 3 3.5 4 4. Smoothness 500 Ohm-m anomaly 5 10 15 20 Smoothness 25 30 35

Figure 3.5(d). Final iteration smoothness as a function of depth. Pseudosection data. Margin-located anomalies. 0 -2 -~-4 0 -2 E E -4 -6 I I I I I I I I I I