chapter 6: inversion of potential field tensor data

Chapter 6: Inversion 123

Chapter 6: Inversion of Potential Field Tensor Data

6.1 Introduction

Interpreting geophysical data to recover the subsurface structure and material properties is

rarely easy. Generally, there are many unknowns and only a small number of clues to

determine the possible solutions. The ill-posed nature of these problems is that data are

insufficient and inconsistent. The mathematical space containing all possible solutions (right

and wrong) to a problem creates what is called the “model space” of a problem. The subset of

this space containing the correct answers is called the “solution space,” and takes up a

minimal portion of the model space.

In geophysical exploration, data measured at, above, or below the ground are obtained as part

of some field survey, and extraction of the physical properties of the Earth from this data is a

mathematical process that is vital for interpretation. As there is rarely any direct solution,

geophysicists often resort to optimization (or inversion) techniques, and this is a major area of

active research (Al-Chalabi, 1971; Li and Oldenburg, 1996; Li and Oldenburg, 1998;

Oldenburg et al., 1997; Parker, 1994; Scales and Tenorio, 2001; Vozoff and Jupp, 1975; Xia

and Sprowl, 1992; Zhdanov, 2002).

Inversion techniques generally involve some sort of iterative process carried out through

many forward model runs to better match the observations. A forward model is the

“theoretical” or “computed” geophysical response (gravity or magnetics signature) of a

simulated Earth model. The majority of images shown in Chapter 3 are examples of forward

models. An inversion technique (generally) will take a forward model response, and compare

it to what is measured or observed above the ground surface in an actual geophysical survey.

The model parameters are then adjusted in a systematic way, which alters the calculated or

theoretical response. The new response is then compared to the field data. The process

continues until the difference between the field data and the forward modelled data is

minimised, subject to certain constraints, regularisation and degree of model complexity.

Mathematically, the “Earth property” variables that need to be determined can be represented

as m1, m2, …, mn, and the forward model f is a function of these variables. The forward model

can be written in functional form as:


( ) ( )kn MMMmmmf ,...,,,...,, 2121 = (6-1)

where M1 to Mk are the output values (e.g., gravity gradient tensor responses) of the forward

model. There does not have to be the same number of variables (n) as there are field points

(k). If k is less than n, the problem is under-determined, and if k is greater than n, the problem

is over-determined.

The data measured in the field need to be compared to the computed values M, and the

difference between the two minimised. If I denote the field data by D1 to Dk, and some

difference function between M and D by ψ, I have:

( ),M Dmisfit ψ= (6-2)

So the aim of the inversion is to minimise ψ. Common forms of ψ include the chi-squared

misfit function (χ2), and the Root Mean Square (RMS) misfit. The chi-squared function is

defined as:

2

2

1

ki i

i M

D Mχσ=

⎛ ⎞−= ⎜ ⎟

⎝ ⎠∑ (6-3)

where σM is the standard deviation of the forward modelled data ( 2Mσ is the variance), i.e.,

each data point is weighted by the confidence we have in it. Note that if the variance of the

forward modelled data is high, the misfit will be low, regardless of how well the data fits.

The RMS error is related to the Chi-squared misfit as follows:

kRMS

2χ= (6-4)

As this formula is basically the chi-squared formula divided by the number of data points (k),

a “good” RMS is equal to 1.

This chapter presents some experiments that test some fundamental questions relating to the

inversion of gradient tensor data. I will then present the theory of some inversion routines


before outlying the theory of Genetic Algorithms (the selected inversion routine for the

experiments in section 6.2). Some important results regarding the output of inversion routines

(namely, due to the large mathematical size of the solution space, there are significant

differences in the output models when more than one inversion is undertaken) are discussed.

I then present an analysis of the use of eigenvalues and eigenvectors for interpretation of

potential field tensor data (eigenanalysis). Finally, I will outline a dipole inversion routine,

and discuss some of its limitations. This acts as primary motivation for a new inversion

routine that will be presented in Chapter 8.

6.2 Gradient Tensor Inversion

The majority of potential field inversion routines assume that only one component of the field

(e.g., TMI) is being used. Gradient tensor data, however, have five components that can be

used for inversion. This immediately raises several questions, such as:

• Which component of the gradient tensor should be used for inversion?

• Should all the components be used simultaneously?

• Are there any benefits to running a multi-component inversion?

• Will inverting to more than one component reveal more information about the geology?

• Will the model produced as the result of a single-component inversion, when forward

modelled to calculate the remaining components of the gradient tensor, match these other

components?

I have already shown that, mathematically, the gradient tensor components are related through

the use of Fourier transforms (see equations (2-129) to (2-133)). The components can

however be measured separately; they will contain their own noise. If the components of the

gradient tensor are calculated from a single component, noise present in that single

component will propagate into all other components. Meanwhile, a single component

inversion has benefits when a combined (multiplicative or additive) inversion may be

disadvantageous because of strong noise in one component swamping the final result. The

cleanest and strongest individual component can be selected for analysis. A combined

analysis of all components offers, in principle, the greatest resolution and fidelity; it is

preferred when the data are not captured by noise. This suggests that both single- and multi-

component inversions may have their own place, depending on circumstances.


In order to test if any further information is obtained from a multi-component inversion, the

following tests have been run. A simple simulated geological scenario is created and the

gradient tensor response calculated. I then repeatedly run an inversion on a) a single

component of the gradient tensor, and b) the five components of the gradient tensor. For the

multi-component inversion the RMS which is used is the average of the five individual RMS

error values.

I used a Genetic Algorithm (the theory of which is to be discussed later in this chapter) and

ran the experiment 20 times (10 times for the single-component data, and 10 times for the

multi-component data). Each inversion was successful (i.e., a RMS of less than 1 was

obtained in each case), and Figure 6.1 shows a graph outlining how long each inversion took.

Generally, it takes approximately three times as long to complete the multi-component

inversion. However, due to the statistical nature of the inversion routine chosen, the longest

run time in this set of experiments was for one of the single-component inversions.

Figure 6.1. This graph shows that it takes approximately three times as long to run a multi-component inversion than a single-component inversion for the Genetic Algorithm used.


Since a single-component inversion only uses information from that component, it is

necessary to determine if the forward model of the geological model produced from the

inversion actually fits with the other components of the gradient tensor.

In the experiments above, a final geological model was produced by the single-component

inversion routine. This model was then used to calculate the theoretical responses for the

other gradient tensor components. In all cases they matched the “observed” data to a high

degree. This suggests that a single-component inversion is quicker and may be comparable to

a multi-component inversion.

This raises the question: which component should be used for the single component

inversion? My preference is to use the component showing the most pronounced anomaly, as

this is likely to contain the most information about the subsurface. However, if only one

component has been measured, the above experiments suggest that an inversion using the one

component will not “exclude” information from the other components.

6.3 Local and Global Inversion Techniques

Inversion routines can be subdivided into two broad groups: linear and non-linear (Parker,

1994; Scales and Tenorio, 2001). Linear inversions are generally much easier to solve; there

is effectively a linear relationship between the data and the model. However, geophysical

inversions can rarely be solved through a linear relationship and most are non-linear in nature.

Non-linear inversion techniques fall into two groups: local and global. The names relate to

the area of the solution space that is being examined to determine the solution. Local

techniques focus on a small portion of the solution space around an initial guess and develop

the solutions there in order to “move” throughout the solution space and converge on the local

minima. Global techniques search as much of the solution space as possible to find the global

minimum. The following sections describe specific inversion types falling into these

categories, and the inversion of simulated potential field data. Some mathematical

manipulation of the potential field gradient tensor is also undertaken to develop a near-linear

relationship between the gradient tensor at a point due to a dipole source.


6.3.1 Local Inversion Methods

Local inversion techniques require a starting model. The misfit of this model is determined

and then with the help of a sensitivity function (i.e., the Jacobian matrix of partial derivatives

of the theoretical data with respect to the model parameters) the model parameters are

adjusted so that a new model is produced. If the new model has a smaller misfit than the

previous model, it is kept and further adjustments are carried out on this model. If the misfit

is greater, operations revert to the earlier model.

A simple example of a local technique in two dimensions is the Descent method (Heath et al.,

2003; Zhdanov, 2002), which is a basis for many gradient-type inversion methods. Figure 6.2

illustrates this process. A model is calculated within the known constraints and the misfit of

that model is calculated. Four more models and their misfits are then calculated, each model

corresponding to adjacent points of the original model in the solution space. One of these

new four values may have a smaller misfit than the original, and is therefore selected as the

starting position for the next iteration. The process is repeated until a minimum is reached.

The variables corresponding to the smallest misfit are taken as solutions to the problem.

Figure 6.2. A Descent method in two dimensions. a) A misfit value is calculated within the constraints of the required model. b) The misfit values are calculated at each of the four surrounding points. c) The point with the smallest misfit value is selected from the four, and the process is repeated. d) The process continues until a minimum is reached, and the parameters chosen are those that fit the model best.

Note that if there are three variables to be optimised, six adjacent points in the solution space

must be determined. The logical extension is that if there are four variables, eight adjacent

points are needed, and so on.

This process gives rise to the problem of local minima in inversion routines. That is, the

topography of the solution space contains numerous valleys or local minima. A simple (one-

dimensional) solution space is shown is Figure 6.3. The variable (dipole moment magnitude)


is represented on the horizontal axis, and misfit on the vertical axis. The graph shows

examples of local minima, global minimum, local maxima, and global maximum.

It is common for local inversion routines to converge in local minima and hence select this as

the solution to the problem. Two inversion routines, the “Steepest descent” method and

“Occam’s” inversion technique attempt to overcome these difficulties by some mathematical

sophistry.

Figure 6.3. The difference in a solution space between a local minimum, global minimum, local maximum and global maximum.

The steepest descent technique is similar to the simple descent method described above, with

two major changes: The path chosen to travel through the solution space is determined by the

steepest gradient rather than a preset selection of left, right, up or down, and the step size of

each iteration can vary depending on the misfit. Generally, the larger the misfit, the larger the

“jump” that the iteration is allowed to make. While this technique is still susceptible to local

minima, the large “jumps” can often overpass these to find the global minimum. Despite

these advantages, many users of the technique find that it is time consuming; the routine must

be run several times before a global minimum is found (Heath et al., 2003; Press et al., 2002).

Occam’s Razor, or the Principle of Parsimony, is a widely accepted scientific approach

(Constable et al., 1987). The often-quoted saying is that the simplest (or smoothest) solution


is probably the correct one. Occam’s inversion is a commonly used geophysical inversion

technique that assumes that the underlying geology contains only smooth changes; i.e., no

sharp boundaries. The forward models of the underlying geology must adhere to strict rules

regarding how the physical properties (of the geology) change in each direction. I have not

chosen to utilise Occam’s inversion to invert for the near-surface magnetic structure, as it is

obvious from Chapter 4, the regolith does contain a lot of variability and therefore is not

smooth. For more information on Occam’s inversion, see Constable et al., 1987.

6.3.2 Global Inversion Methods

Methods that involve a much larger search through the solution space are called global

inversion techniques. Generally, they take much longer to compute, but are less susceptible

to local minima. Two techniques are presented here: Monte Carlo and Genetic Algorithms

(GAs). Monte Carlo is a relatively simple technique and can be classed as a subset of Gas

(Boschetti et al., 1997; Gallagher et al., 1991). Genetic Algorithms have been developed as

part of my research as a technique to invert gradient tensor data and the results will be seen in

a later section.

Monte Carlo Techniques

“Monte Carlo” is a name applied to numerous mathematical routines where randomness is

involved (Anton and Rorres, 1994). In geophysical inversion, a Monte Carlo technique

simply involves calculating many possible solutions (combinations of model parameters) and

their theoretical responses to a problem and selecting the model with the smallest misfit as the

solution. This can involve a random selection of parameters or a thorough search of the

solution space.

As this involves calculating a great number of solutions, the technique is generally not

suitable for large-scale inversions. However, for inversions involving few variables (say 2 to

4) the process is effective because not only is the solution found, but information is obtained

about the solution space. Figure 6.4 shows the solution space for a hypothetical problem

involving two variables (magnetic strength and depth) and the confidence level given to the

various models.


Figure 6.4. Monte Carlo inversion involves (a) calculating many solutions to a problem, and (b) selecting the solution with the best fit.

Genetic Algorithms

Genetic algorithms are a global search technique; hence they provide a search of a large

solution space, but only for a modest number of parameters. The process is based on the

evolutionary theory, where DNA (Deoxyribonucleic acid) strands contain information that

gets passed on to future generations. Mathematically, a population of solutions is created, the

parameters of each model acting as its “DNA.” Each model has its own misfit. Pairs of

models in the population of solutions are selected to “breed” (a process commonly referred to

as “crossover”), whereby data is exchanged between the two models, producing an offspring

population. The process is repeated for a large number of generations until a model is found

with a small enough misfit to consider the problem solved.

The application of GA’s as a non-linear inversion technique has been described by many

scientists (Bäck, 1996; Gallagher et al., 1991; Goldberg, 1989; Sen and Stoffa, 1992; Smith et

al., 1992; Stoffa and Sen, 1991). The technique can be applied to potential field data

(Boschetti et al., 1997) to determine the appropriate physical properties (e.g. density,

magnetic susceptibility, dipole moment) of the subsurface. A flow chart is given in Figure

6.5, which shows the steps used in the GA method. The process is based on (Gallagher et al.,

1991), and is described below.


Figure 6.5. This flow chart shows the steps taken in a GA, based on (Gallagher et al., 1991).

The first step is simply to import the field data into the computer. The computer program then

calculates the RMS error of this data. A matrix array is then created (the size of which

depends on the model size), such that each matrix represents a depth slice, and the value at

each point (element) in the matrix is equal to the unknown physical property that one wishes

to determine by the inversion algorithm. In order to constrain values for the inversion, pre-set

values are simply typed in, and fixed throughout the process; all other spaces are allocated a

random number. The array is repeatedly stored into the computer’s memory, creating new

random numbers each time. For each model, the chosen gradient tensor response is

calculated, and the associated RMS misfit errors are calculated at each step.

Generally, if the RMS for any of the models is less than 1%, then the problem is considered

solved. However, if not, it must still have a suitable value to be considered for crossover. If

the model is still not good enough, it is discarded and a new model takes its place. This

continues until an interim population has been created that is to be used for crossover. The

models are randomly paired off and converted to strings. The strings represent DNA strands

that are the fundamental elements of genetic inversions. Figure 6.6 shows the process of

converting a block of data (e.g. a section of subsurface) to a string of data.


Figure 6.6. Simply taking each row of data and pasting it on the end of the previous row creates a string of data from a three-dimensional block of data. The result of this is a string of data that can be used to simulate a DNA strand.

Once all the models are paired off, a random number is generated for each pair. If the number

is above a chosen limit, then the pair is allowed to crossover. If not, the pair passes onto the

next generation unaffected. Crossover is illustrated in Figure 6.7. It simply involves taking

the two strings and swapping over a portion of their data. The point at which crossover

occurs is randomly chosen, and can occur at any point along the string.

Figure 6.7. A random point is selected along the strings, and the remaining string segments are swapped.


There is a small probability that a mutation can occur, i.e. a point of data gets replaced by a

defined opposite (e.g. 9 gets replaced by 1 (10 - 9 = 1), and 4 gets replaced by 6 (10 – 4 = 6)).

The probability of mutation occurring is usually very small. However, if the chance is high

enough, all the points get mutated, and the inversion becomes a Monte Carlo style inversion,

where an exhaustive search of the solution space is performed (Press et al., 1992).

Finally, Genetic Algorithms are entirely dependent on the information contained within one

generation (i.e., one population of solutions (Gallagher and Sambridge, 1994)). Therefore if

an inversion has stopped due to a preset “iteration limit,” the information forming the final

population can be continued as the starting model for a new inversion.

6.4 Testing and Optimising the Genetic Algorithm

Algorithms have been constructed in Matlab in order to invert potential field tensor data via a

GA approach. Since the inversion creates a population of RMS errors at each iteration, it is

useful to visualise how these converge during the inversion routine. Figure 6.8 illustrates how

this works. For the example shown in the figure there is a population of four solutions, where

each “solution” corresponds to a particular specification of model parameters, with each

solution having an RMS error represented by a small circle. At the first iteration, the four

solutions can be seen to have large RMS errors, but these decrease as the inversion routine

continues.

Figure 6.8. Each circle represents the error of a potential solution at a particular iteration. The RMS errors should decrease with time.


After some initial experimentation, it became apparent that a Genetic Algorithm used to invert

to a model having as many parameters as that given in Chapter 3, would take much too long

(say many weeks). A simple, randomly created model was therefore used. The geology was

simply a rectangular prism of dimensions 3 × 3 × 2 metres, located with its top surface at a

depth of 2 metres. The prism was divided into 18 cells as shown. Measurements were taken

over the ground surface (z = 0) on a 10 × 10 metre grid. The values for magnetic

susceptibility were generated randomly (within a realistic range) and were changed for each

of the 20 inversion experiments. Figures 6.9 and 6.10 shows the simulated geology and the

gradient tensor forward model response respectively for one of the inversions. This simplistic

model was chosen as a basis for answering the questions posed at the start of the chapter.

Figure 6.9. A randomly generated geological model used as an ideal model for the inversion process. The model is three by three by two metres in volume.


Figure 6.10. Gradient tensor components of the simple geological scenario for inversion testing.

With this simple model, the inversion should run fast with less computer memory than needed

for a large-scale inversion, and therefore a large population of solutions could be used. Using

100 solutions, and allowing the routine to run for 500 iterations, the models converged to a

solution. Figure 6.11 shows the model RMS vs. iteration number as one of the inversion

routines progresses. Note how the circles (each representing the RMS of a particular model)

tend to converge on smaller RMS values as the inversion proceeds. Figure 6.12 shows the

gradient tensor responses of the successful model. This figure is similar to Figure 6.10,

although it is not exactly the same.


Figure 6.11. The inversion showed a decreasing trend in the population of RMS errors over time.

Figure 6.12. Forward modelling the successful inverse model yield results similar to the original data, shown in Figure 6.10.


The convergence of the data in Figure 6.11 suggests that the Genetic Algorithm is

successfully inverting, and comparison of the field responses (Figures 6.10 and 6.12) also

suggests that the inversion was successful. However, when comparing the original geological

model (Figure 6.9) with the geological model produced as part of the inversion (Figure 6.13),

it is immediately apparent that the models do not match. This is due to the fact that the

solution space to this problem is large, and the solution found as part of the inversion does not

necessarily need to match the original forward model to be classed as a solution. This is the

non-uniqueness problem of potential field interpretation: more than one geological model can

fit the observations (Skeels, 1947).

Figure 6.13. When compared to the original model (Figure 6.9), this geological model is quite different, although it produces very similar potential field responses (Figure 6.12).

As mentioned earlier, the inversion was run 20 times. Of these, 10 were single component

inversions, and the remaining 10 were multi-component inversions. Therefore, 20 three-

dimensional geological models were produced as a result of these experiments, with 20

corresponding (randomly generated) three-dimensional original models. A search was carried

out to determine if any of the geological models (produced from the inversion routine)

matched their original model. In all, only two of these experiments yielded results where the

inverted geological model corresponded to the actual geological model. This suggests a one

in ten chance of locating the best model through such an inversion routine.


Genetic Algorithms can therefore be used for inversion of potential field gradient tensor data

for regolith geophysical exploration. However, interpreting the resulting geological model

must be taken with care, and the results presented here suggest that GA style inversion

routines should be undertaken more than once. Due to the random nature of the routine, it is

not possible to determine how many times an inversion should be run for a particular

scenario. Proper constraining of geological information (e.g., physical properties from

surface material and drill-hole data) should help with making the inversion results more

accurate.

A common procedure in undertaking global style inversions is to run the global inversion

first, and then in the final stages “home-in” to a solution using a local inversion routine (e.g.,

the steepest descent method or Occam’s inversion). The results of the experiments above are

an example whereby a local routine was not needed, as a solution was found. In reality, the

geological models would be much larger and need to incorporate much more information. In

these cases, a localised routine may be necessary to complete the inversion. To incorporate a

local inversion routine at the end of a GA, the GA need not be run to completion, rather the

best model taken at some pre-determined point (say a selected RMS value), and the local

inversion routine undertaken on that data set. In this case all the information from the other

models as produced by the GA would be discarded, and instead of having a range of possible

solutions at the end of the routine, only one would be determined.

It is also possible to form a hybrid inversion scheme by alternating between global and local

methods. That is, taking the best model from the GA to run a local inversion routine, and

taking the best model from the local inversion to compete with further randomly generated

models as part of a GA. Such a routine would be very computationally intensive and beyond

the scope of this work.

Chapter 8 will introduce a new inversion routine designed to specifically utilise the

components of the magnetic gradient tensor.

6.5 Linear (and Near-Linear) Methods

The following section describes the development of eigenvalues and eigenvectors (often

referred to as eigenanalysis) to interpret potential field tensor data, and a technique presented


by (Schmidt et al., 2004) that describes an (almost) linear relationship between the potential

field gradient tensor and a single dipole source. I will also show how eigenvectors can be

used for interpretation of potential field gradient tensor maps.

6.5.1 Eigenanalysis of the Gradient Tensor

For any square symmetric matrix B, such as the gradient tensor matrix, there exists a set of

vectors b, called the eigenvectors of B, such that:

Bb bλ= (6-5)

where λ is referred to as the eigenvalue of the eigenvector b. For a 3 by 3 matrix there will be

three eigenvectors with three associated eigenvalues (Anton and Rorres, 1994). Note that as

the negative of an eigenvector will also satisfy equation (6-5), there is fundamental

uncertainty as to which way the eigenvectors point.

Each set of three eigenvectors sit at right angles to each other and are said to be “orthogonal.”

Generally the eigenvectors are normalised, and are therefore referred to as “orthonormal.”

The length (or magnitude) of each eigenvector is given by its corresponding eigenvalue.

The eigenvalues and eigenvectors can be determined automatically via some high-level

programming packages, or manually using known formulae (Greenhalgh et al., 2005). It is

important to note that the following method will only work for a matrix where the diagonal

components sum to zero (e.g., the gravitational or magnetic field satisfies Laplace’s equation

and so the condition holds). I first re-arrange the elements of the gradient tensor into the

intermediate quantities:

22222xzyzxyyyxxyyxx BBBBBBBp −−−−−−= (6-6)

xzyzxyzzyyxxxzyyxyzzyzxx BBBBBBBBBBBBq 2222 −−++= (6-7)


⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

⎟⎠⎞

⎜⎝⎛−

−= −

3

1

32

cosp

qα (6-8)

The three eigenvalues are then given by:

⎟⎠⎞

⎜⎝⎛−=

3cos

321

αλ p (6-9)

⎟⎠⎞

⎜⎝⎛ +−−=

33cos

322

απλ p (6-10)

⎟⎠⎞

⎜⎝⎛ −−−=

33cos

323

απλ p (6-11)

The eigenvectors can then be determined as follows (the notation ki are simply a set of

constants required to compute the final set of eigenvectors):

1 2 3 1k k k= = = (6-12)

1

74 λ−

−−=

yy

yzxy

BkBB

k 2

85 λ−

−−=

yy

yzxy

BkBB

k 3

96 λ−

−−=

yy

yzxy

BkBB

k (6-13)

( )( )( ) yzxyyyxz

xxyyxy

BBBBBBB

k−−

−−−=

1

112

7 λλλ

(6-14)

( )( )( ) yzxyyyxz

xxyyxy

BBBBBBB

k−−

−−−=

2

222

8 λλλ

(6-15)

( )( )( ) yzxyyyxz

xxyyxy

BBBBBBB

k−−

−−−=

3

332

9 λλλ

(6-16)


In order to normalise the eigenvectors, the following are needed:

27

24

2110 kkkk ++= (6-17)

28

25

2211 kkkk ++= (6-18)

29

26

2312 kkkk ++= (6-19)

The three normalised eigenvectors are then given by:

kjib10

7

10

4

10

11

ˆkk

kk

kk

++= (6-20)

kjib11

8

11

5

11

22

ˆkk

kk

kk

++= (6-21)

kjib12

9

12

6

12

33

ˆkk

kk

kk

++= (6-22)

If, however, the values of Bxy and Bxz are equal to zero, the above equations for k1 to k9 do not

hold. The revised equations for this situation are given in equations (6-23) and (6-24).

Equations (6-17) to (6-22) still hold.

1 2 3 0k k k= = = 7 8 9 1k k k= = = (6-23)

14 λ−

−=

yy

yz

BB

k 2

5 λ−

−=

yy

yz

BB

k 3

6 λ−

−=

yy

yz

BB

k (6-24)

If the values of Bxy, Bxz and Byy are equal to zero, the above equations for k1 to k9 do not hold.

The revised equations for this situation are given in equations (6-25) to (6-26). Again,

equations (6-17) to (6-22) hold true.

1 2 3 0k k k= = = 7 8 9 1k k k= = = (6-25)


yz

zz

BBk −

= 14

λ yz

zz

BBk −

= 25

λ yz

zz

BB

k−

= 36

λ (6-26)

In order to visualise the eigenvalues and eigenvectors around a magnetic source, I have

conducted the following simulation. The measurement plane is the x-y plane at a height of 10

metres from the source. The plane is of dimensions 100 by 100 metres with measurements

taken every 10 metres in each direction. The dipole is given a dipole moment equal to 5Am2

in the x direction (the x axis is vertical in the figures). The three eigenvalues are shown in

Figure 6.14. Note that eigenvalues 1 and 2 are identical apart from a sign change.

Figure 6.14. Three eigenvalues around a dipole source.

In examining the eigenvectors, it is important to view both the original and the negative of

each eigenvector, as these also satisfy the conditions for being an eigenvector. For this

reason, Figures 6.15 and 6.16 (left diagram) show the three eigenvector plots separately, and

Figure 6.16 (right diagram) shows the eigenvectors together in a single plot (all six).


Figure 6.15. The first two eigenvectors around a dipole source. Note the similarities between the plots, especially how the first plot appears to be symmetric around the line equal to 0m on the South-North axis.

Figure 6.16. The third eigenvector around a dipole source (left) and all three eigenvectors superimposed (right). The third eigenvector plot consists entirely of vectors pointing either north or south, and the superimposed plot shows a distinct 6-sided pattern around the anomaly.

There is a distinct pattern of eigenvectors around the dipole source shown in Figure 6.16. The

pattern shows areas where the eigenvectors lie exactly parallel (and therefore perpendicular)

to the x- and y-axes, and areas where they do not. The effect is of two joined crosses and this

may prove useful for interpreting gradient tensor data. It is possible to import a gradient

tensor data set and compute the eigenvectors at each point on the grid, since a 3 by 3 matrix of

measurements exist at each point. The next few figures demonstrate the application of such a


procedure to the synthetic data set of Chapter 3. The regolith model in question contained

several soil types, ordnance, mineralisation and a cadaver. The measurements were computed

for the ground level and the deduced first and second eigenvector plots are given in Figure

6.17.

6.17. The first eigenvector (left) shows various regolith features, mainly as disturbances around the boundaries of units, but there is much uncertainty in the map, and interpretation is problematic. The second eigenvector plot (right) appears to show additional features hitherto unseen in any gradient tensor maps (note the vaguely vertical features running down the right hand side of the plots).

The first eigenvector diagram reveals some interesting features. Not only do areas

corresponding to similar regolith type seem to yield a constant eigenvector, but also there is

disturbance around areas such as the landmines, and where the contact between regolith units

is dipping. The second eigenvector plot shows significant irregularities around the areas

where there are dipping contacts, and less disturbance around the channel area in the north of

the map. The plot also shows what appears to be a feature running down the east side of the

area, and another similar feature running from just right of the centre to the bottom. This may

relate to the palaeochannel feature introduced in earlier chapters, as the boundaries of the

palaeochannel (when projected vertically to the surface) roughly match these features. This

suggests that eigenvector plots can be used directly to define geological structure, although

much care must be taken. The third and combined eigenvector plots are shown in Figure

6.18.


Figure 6.18. The third eigenvector of the regolith simulation (left) reveals little correlation to regolith structure. The superimposed image (right) illustrates “distortion” around the surface regolith boundaries as well as lineations that may represent boundaries from greater depth (see also Figure 6.19).

As with the third eigenvector around the dipole, this third eigenvector shows little “structure”,

although some subtle features are present. The creek and landmines are visible, but

boundaries between regolith units are still hard to pick. Previously, I superimposed the

eigenvector data to see if any patterns were present (Figure 6.16, right), and this is repeated

here. The superimposed data allows us to see the boundaries of the regolith units and other

various features, including information from greater depth (e.g., the palaeochannel and

mineralisation). Inspection of the right hand side image of Figure 6.18 reveals the base of the

palaeochannel (see Figures 3.9 and 3.10). Figure 6.19 shows the base of the palaeochannel

superimposed onto the image in Figure 6.18. This suggests that determining the eigenvectors

from a gradient tensor survey can highlight new features previously “hidden” even after the

use of filters.

This technique of plotting the eigenvectors of the gradient tensor is effectively a quasi-

inversion scheme. It has similar characteristics to a filter (i.e., it can be applied to a data set to

enhance information) and an inversion routine.

In Chapter 7, I will examine the eigenvector fields around some different magnetic source

types.


Figure 6.19. This is the same as the previous figure, but the outline of the palaeochannel has been included. The outline corresponds to features in Figure 6.17. 6.5.2 Relationship Between Eigenvectors and a Dipole Source

Recall that there are two vectors required to determine the magnetic field response (at a point)

of a magnetic dipole: the dipole moment (m) and the displacement (distance and direction) to

the field point (r). Mathematically speaking, these two vectors define a plane, which I will

now refer to as the m-r plane. Therefore the cross product of m and r will yield a vector

normal to the m-r plane, which I shall denote by v.

v m r= × (6-27)

It can be shown that the direction of vector v is equal to the direction of the eigenvector b3

(the eigenvector corresponding to the eigenvalue with the smallest absolute value) of the

magnetic gradient tensor. The derivation will not be given here (as it is quite involved), but

can be found utilising equations (6-5) and (6-27), i.e., showing that equation (6-27) satisfies

the conditions for being an eigenvector.


Having determined that one of the eigenvectors (b3) is equal to the cross product of m and r,

and cognisant of the fact that the three eigenvectors of the gradient tensor are orthogonal, it

follows that the other two eigenvectors (b1 and b2) must lie in the m-r plane. The vectors m

and r can then be defined as linear combinations of the two remaining eigenvectors, i.e., m

and r can be determined from a measurement of the gradient tensor at a point.

Unfortunately, there is generally more than one solution to the problem (often four vectors for

m and four associated vectors for r), meaning that this technique is not strictly linear.

Schmidt et al. (2004) describes the complete process. It can be summarised as follows:

Denoting λ3 as the eigenvalue of the gradient tensor matrix at a given point having the

smallest absolute value, then if λ3=λ2=λ1/2, there are two possible solutions that are given by:

11ˆˆ br = ( ) 131

ˆsgnˆ bm λ= (6-28)

and

12ˆˆ br −= ( ) 132

ˆsgnˆ bm λ−= (6-29)

where the hat on the vector indicates that the vector is a unit vector, or normalised. The

function “sgn” is the signum function (this assigns a sign to a number, depending on the

number itself (Schmidt et al., 2004)). If however λ3=λ1=λ2/2, then the two solutions are

different, and are given by:

21ˆˆ br = ( ) 231

ˆsgnˆ bm λ= (6-30)

and

22ˆˆ br −= ( ) 232

ˆsgnˆ bm λ−= (6-31)

However, the general case is that |λ3| < |λ1| and |λ3| < |λ2|, and the polar angles φ and θ must

be introduced to solve the problem.


⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎟⎟⎠

⎞⎜⎜⎝

⎛ −+=

−

−21

23

22

21

31

23

cosλλλ

λφ (6-32)

( )

( ) ( ) ⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

+⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛+

= −

φφλλ

φθ2

2

3

1

1

sincos2

sincos (6-33)

Use of these formulae leads to the following four solutions for normalised m and r:

( ) ( ) 211ˆsinˆcosˆ bbr θθ += (6-34)

( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( )( ) 211ˆcossinsincosˆsinsincoscosˆ bbm θφθφθφθφ ++−= (6-35)

12 ˆˆ rr −= 12 ˆˆ mm −= (6-36)

( ) ( ) 213ˆsinˆcosˆ bbr θθ −= (6-37)

( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( )( ) 213ˆcossinsincosˆsinsincoscosˆ bbm θφθφθφθφ −++= (6-38)

34 ˆˆ rr −= 34 ˆˆ mm −= (6-39)

These eight equations give normalised (unit) vectors, one of which will always point towards

the source (r) and one that will represent the orientation of the dipole (m). Therefore it is not

possible to determine the dipole moment strength from this routine, only the direction of the

moment.

This theory is now tested and demonstrated on a simulated data set. A dipole is placed in the

centre of a 50 by 50 metre grid, and measurements taken around it. The dipole is at a depth of

5 metres, and has a dipole moment of 5Am2 in the North direction (i.e., a bar magnet oriented

in North-South direction). The maps for r1 and m1 are shown in Figure 6.20.


Figure 6.20. Not all the vectors determined as r1 (left) point directly to or from the source in the centre of the grid, and not all the vectors determined as m1 (right) represent the orientation of the dipole.

These two plots represent only one of the solutions for r and m determined by the algorithm.

Note that while sometimes the vector r points to the source (in the centre of the grid), and

while sometimes the vector m is vertical (the direction of the dipole moment), they do not

always point in the correct direction. Furthermore, if we did not know the position or

orientation of the dipole, we wouldn’t be able to determine them from these Figures. The

remainder of the solutions are shown in Figures 6.21 to 6.23.





While it is not possible to easily determine which of the solutions is correct, a distinct pattern

emerges by superimposing the images. Figures 6.24 (left) shows the four solutions for r, and

here it is obvious where the source is located. Note that the arrows point away from the

source, as the vector represents the direction to the source from the field point. Similarly, the

four solutions for m have been superimposed (Figure 6.24, right). At each field point, there is


a vector facing directly north or south, as this is the orientation of the dipole, although there

are still many vectors pointing in erroneous directions. A simple re-calculation of this data set

(removing any vectors that do not remain constant) should leave only vectors pointing in the

correct direction.

Figure 6.24. The superimposed images showing the position of the source from each field point.

6.6 Discussion and Conclusions

In this chapter I have introduced inversion theory and focussed on three methods for

interpreting potential field gradient tensor data. Using a simple genetic algorithm inversion

routine, I have conducted experiments on multi-component data sets and shown that

(generally) a single-component inversion produces results that are comparable to multi-

component inversion, but with computational economies. While this may suggest that

measurements of all the gradient tensor components do not necessarily yield additional

geological information than a single component, Chapter 8 will outline a new inversion

routine that benefits from independent measurement of the gradient tensor components.

Furthermore, in order to run a single component inversion, a particular single component

must first be selected. I have suggested that the gradient tensor component with the largest

amplitude (or highest signal to noise ratio) contains the most geological information, and

therefore should be used in preference to the other components for a single component

inversion. However, selecting the gradient tensor response with the largest amplitude is not


possible unless all of the gradient tensor components have been measured or computed. As

outlined in section 2.4, it is possible to take a map of a single gradient tensor component and

from it calculate the remaining gradient tensor components. As noise can be repeated

throughout the remaining components in this process, I would advocate measuring all the

components in the first place.

Genetic Algorithms have been tested and optimised such that they can invert to simple

geological situations. However, these inversions produce geological models that generally do

not match the original geological model, even though their gradient tensor responses match

the simulated field measurements. This suggests that inversion results must be interpreted

with care, and constraint of geological information is vital.

The use of eigenvalues and eigenvectors has been developed as a tool for interpreting gradient

tensor data, specifically determining source direction and delineating geologic boundaries for

the 3-D simulated regolith model given in Chapter 3. The use of the superposition of

eigenvectors yields plots whereby geological information from depth is possible. I have been

able to detect the palaeochannel introduced in Chapter 3 through the use of eigenvectors.

This palaeochannel has remained hidden through the forward modelling of Chapters 3 and 4,

and the entire filter processes of Chapter 5.

A second eigenvector technique was applied and I have examined the dipole inversion theory

developed by (Schmidt et al., 2004), and shown that while the proper dipole and the dipole

orientation is amongst the solutions, it is not possible to determine directly which of the four

solutions is the correct one. I have shown that superposition of the solutions may help with

this problem, and for the case where the vector corresponding to the orientation of the dipole,

rejection of vectors that do not remain constant should yield the correction orientation. In

Chapter 8 I will develop a routine that will find the position and dipole moment of a buried

dipole, extend this to the case where more than one dipole is present and extend this to some

more complex multipole sources, to be introduced in the next chapter.

chapter 6: inversion of potential field tensor data

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