Chapter 7: Forward Modelling the Gradient Tensor Response of ...

Chapter 7: Multipoles 154

Chapter 7: Forward Modelling the Gradient Tensor

Response of Multipole Sources

7.1 Introduction

This chapter presents the mathematical relationships involving the magnetic field and the

gradient tensor components for two types of magnetic source – the quadrupole and the

octupole. They have smaller field strengths than the dipole, but this is one of the benefits of

measuring the gradient tensor of the field: weaker sources become detectable (Schmidt and

Clark, 2000). Also, the distinctive shapes of the gradient tensor components can help dictate

the type of source that may be producing a signal.

To the best of my knowledge the analytical formulae for the static magnetic quadrupole and

octupole have not been previously given in the literature. I start by deriving the formulae

required to model these fields, and express the gradient tensor components in terms of spatial

derivatives of the corresponding dipole field. I dealt with the dipole source in Chapter 2. I

present examples showing the anomaly patterns for both the magnetic field and the gradient

tensor components arising with each type of source. I am not proposing that actual geological

structures can be adequately modelled by such sources. Rather they constitute basic building

blocks which provide a useful means of understanding the complexity and possible usefulness

of the magnetic gradient tensor in exploration, especially for subtle sources. As one of the

purposes of gradiometry is to be able to discern these subtle sources, this chapter contains the

mathematics required to model these multipoles. It also contains some further examples of

eigenvalues and eigenvectors patterns of the gradient tensor around these sources. An

obvious next step is to test inversion and processing algorithms using these idealised sources.

This will be performed in the next chapter.

7.2 The Static Magnetic Quadrupole

If two magnetic dipoles of equal moment intensity are placed antiparallel to each other, they

effectively produce a magnetic quadrupole. The analytical formulae for the various responses

of a quadrupole can be determined from the dipole formulae. It is given by (Cowan, 1968):


( )dipolequad fieldfield BdB ∇•−= (7-1)

Here, d is the vector separating the centres of the two dipoles, creating a quadrupole. Figure

7.1 is a schematic diagram of the arrangement of vectors.

Figure 7.1. The magnetic field response at a point of a magnetic quadrupole can be determined from three vectors: m, d and r.

The scalar magnetic potential at a large distance from an arbitrary distribution of

magnetisation can be expanded into a series, of which the first three terms are the dipole, the

quadrupole and the octupole (Cowan, 1968). While the magnetic “strength” of a dipole is the

3-component vector m (dipole moment, units Am2), the magnetic “strength” of a quadrupole

is a 3 × 3 matrix or second rank tensor qij. This quadrupole moment tensor is equal to the

matrix product of m and d (Cowan, 1968). Each component of this tensor therefore has the

units Am3 (dipole moment (Am2) multiplied by distance (m)).

q mdx x x y x z

ij y x y y y z

z x z y z z

m d m d m dm d m d m dm d m d m d

⎡ ⎤⎢ ⎥= = ⎢ ⎥⎢ ⎥⎣ ⎦

(7-2)

In similar fashion (see later) the octupole term in the expansion has a “strength” which can be

written in terms of a third rank moment tensor. Appendix 1 analyses some mathematical

properties of the quadrupole moment tensor.


Since B is a vector field, I can split it up into its 3 Cartesian components, so equation (7-1)

can now be written:

( ) ( )( ), , , ,dquad quad quad dipole dipole dipolex y z x y zB B B B B B= − •∇ (7-3)

or in more compact component form as :

( )dipolequad ii BB ∇•−= d (7-4)

Let i be equivalent to x in the above equation. Expanding gives:

quad dipolex x y z xB d d d Bx y z

⎛ ⎞∂ ∂ ∂= − + +⎜ ⎟∂ ∂ ∂⎝ ⎠

(7-5)

So:

( )quad dipole dipole dipolex x xx y xy z xzB d B d B d B= − + + (7-6)

Therefore the three components of the total magnetic field of a quadrupole follow directly

from the gradient tensor equations for the dipole. In similar fashion, the following are

obtained:

( )quad dipole dipole dipoley x xy y yy z yzB d B d B d B= − + + (7-7)

and:

( )quad dipole dipole dipolez x xz y yz z zzB d B d B d B= − + + (7-8)

From equations (7-6) to (7-8), it is possible to calculate the TMI for a quadrupole. To

calculate the gradient tensor components, equations (7-6) to (7-8) must be differentiated with

respect to x, y and z. The following formulae are obtained:


( )dipoledipoledipolequad xxzzxxyyxxxxxx BdBdBdB ++−= (7-9)

( )dipoledipoledipolequad yyzzyyyyxyyxyy BdBdBdB ++−= (7-10)

( )dipoledipoledipolequad zzzzyzzyxzzxzz BdBdBdB ++−= (7-11)

( )dipoledipoledipolequad xyzzxyyyxxyxxy BdBdBdB ++−= (7-12)

( )dipoledipoledipolequad yzzzyyzyxyzxyz BdBdBdB ++−= (7-13)

( )dipoledipoledipolequad xzzzxyzyxxzxxz BdBdBdB ++−= (7-14)

The following relations also hold:

dipole dipole dipoleiji iij jiiB B B= = (7-15)

dipole dipole dipole dipole dipole dipoleijk jik ikj jki kij kjiB B B B B B= = = = = (7-16)

To calculate the gradient tensor response of a quadrupole, the derivatives of the equations

governing the magnetic response of a dipole are needed (equations (2-113) and (2-114)). The

following field quantities can now be presented:

( ) ( ) 320

5 7 7 9

45 1059 454

m r m rdipole

x xxxx

x xm m xBr r r r

μπ

⎛ ⎞• •= − − +⎜ ⎟⎜ ⎟

⎝ ⎠ (7-17)

( ) ( )2 30

5 7 7 9

9 45 45 1054

m r m rdipole

y yyyy

m m y y yB

r r r rμπ

⎛ ⎞• •= − − +⎜ ⎟⎜ ⎟

⎝ ⎠ (7-18)

( ) ( ) 320

5 7 7 9

45 1059 454

m r m rdipole

z zzzz

z zm m zBr r r r

μπ

⎛ ⎞• •= − − +⎜ ⎟⎜ ⎟

⎝ ⎠ (7-19)


( ) ( )⎟⎟⎠

⎞⎜⎜⎝

⎛ •+

•−−−= 9

2

77

2

750 1051515303

4 ryx

ry

rxm

rxym

rm

B yxyxxydipole

rmrmπ

μ (7-20)

( ) ( )⎟⎟⎠

⎞⎜⎜⎝

⎛ •+

•−−−= 9

2

77

2

750 1051515303

4 rzx

rz

rxm

rxzm

rm

B zxzxxzdipole

rmrmπ

μ (7-21)

( ) ( )⎟⎟⎠

⎞⎜⎜⎝

⎛ •+

•−−−= 9

2

77

2

750 1051515303

4 rxy

rx

rym

rxym

rm

B xyxyyxdipole

rmrmπ

μ (7-22)

( ) ( )⎟⎟⎠

⎞⎜⎜⎝

⎛ •+

•−−−= 9

2

77

2

750 1051515303

4 rzy

rz

rym

ryzm

rm

B zyzyyzdipole

rmrmπ

μ (7-23)

( ) ( )⎟⎟⎠

⎞⎜⎜⎝

⎛ •+

•−−−= 9

2

77

2

750 1051515303

4 rxz

rx

rzm

rxzm

rm

B xzxzzxdipole

rmrmπ

μ (7-24)

( ) ( )⎟⎟⎠

⎞⎜⎜⎝

⎛ •+

•−−−= 9

2

77

2

750 1051515303

4 ryz

ry

rzm

ryzm

rm

B yzyzzydipole

rmrmπ

μ (7-25)

( )⎟⎟⎠

⎞⎜⎜⎝

⎛ •+−−−= 9777

0 1051515154 r

xyzr

xymr

xzmr

yzmB zyx

xyzdipole

rmπ

μ (7-26)

As previously, B represents the magnetic field. The first two subscripts dictate from which

gradient tensor component the quantity has been derived and the third subscript notates the

derivative taken. An illustrative example of the magnetic field (and associated quantities)

around a quadrupole source is given below. Setting the vector m as 1Am2 in the x-direction

and d as 1m in the y-direction (i.e., a square quadrupole), placing the quadrupole at a depth of

10m, and taking measurements over an one-hundred metre squared grid, the following figures

were obtained (the physical parameters are simply chosen to best visualise the shape of the

field responses). Figure 7.2 shows the three vector components of the field, Figure 7.3 shows

the TMI of the field, and Figure 7.4 shows the gradient tensor components.


Figure 7.2. The three Cartesian components of a magnetic field around a static magnetic quadrupole. The source is in the centre of each plot at a depth of 10m. Units are in Teslas.

Figure 7.3. The total magnetic intensity of a magnetic field around a static magnetic quadrupole. The source is in the centre of the plot at a depth of 10m. Units are in Teslas.


Figure 7.4. Six of the gradient tensor components of a magnetic field around a static magnetic quadrupole. The source is in the centre of each plot at a depth of 10m. Units are in Teslas/metre.

7.3 The Static Magnetic Octupole

Just as a quadrupole can be created from two antiparallel dipoles, an octupole can be created

from two antiparallel quadrupoles. Mathematically, this is a simple extension from before. I

let s be the vector which joins the two quadrupoles, as illustrated in Figure 7.5. The

components of the octupole third rank moment tensor are therefore expressed as midjsk, where

i, j and k are any one of x, y and/or z. The components of this tensor have the units

Am4 (dipole moment (Am2) multiplied by two distance terms (m2)).

( )B s Boct quadfield field= − •∇ (7-27)


Figure 7.5. Four vectors are needed to describe the static magnetic octupole.

The three vector components become:

( )oct quad quad quadx x xx y xy z xzB s B s B s B= − + + (7-28)

( )oct quad quad quady x xy y yy z yzB s B s B s B= − + + (7-29)

( )oct quad quad quadz x xz y yz z zzB s B s B s B= − + + (7-30)

Therefore, the three vector components of an octupole (and hence the TMI) can be computed

directly from the gradient tensor components of the field due to a quadrupole. To determine

the gradient tensor components of an octupole, the derivatives of the gradient tensor

components of the quadrupole must be calculated. The following relations hold:

dipoledipoledipoledipoledipole

dipoledipoledipoledipoleoct

xxzzzzxxyzyzxxxzxzxxyzzyxxyyyy

xxxyxyxxxzzxxxxyyxxxxxxxxx

BdsBdsBdsBdsBds

BdsBdsBdsBdsB

++++

++++= (7-31)



yyzzzzyyyzyzxyyzxzyyyzzyyyyyyy

xyyyxyxyyzzxxyyyyxxxyyxxyy

BdsBdsBdsBdsBds

BdsBdsBdsBdsB

++++

++++= (7-32)




zzzzzzyzzzyzxzzzxzyzzzzyyyzzyy

xyzzxyxzzzzxxyzzyxxxzzxxzz

BdsBdsBdsBdsBds

BdsBdsBdsBdsB

++++

++++= (7-33)



xyzzzzxyyzyzxxyzxzxyyzzyxyyyyy

xxyyxyxxyzzxxxyyyxxxxyxxxy

BdsBdsBdsBdsBds

BdsBdsBdsBdsB

++++

++++= (7-34)



yzzzzzyyzzyzxyzzxzyyzzzyyyyzyy

xyyzxyxyzzzxxyyzyxxxyzxxyz

BdsBdsBdsBdsBds

BdsBdsBdsBdsB

++++

++++= (7-35)



xzzzzzxyzzyzxxzzxzxyzzzyxyyzyy

xxyzxyxxzzzxxxyzyxxxxzxxxz

BdsBdsBdsBdsBds

BdsBdsBdsBdsB

++++

++++= (7-36)

This means that I must differentiate equations (7-17) to (7-26) with respect to x, y and z. The

following formulae are therefore needed to calculate the responses of a magnetic octupole:

( )

( ) ( ) ⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

•−

•

++•

−−=

11

4

9

2

9

3

770

945630

42045180

4r

xr

xr

xmrr

xm

B

xx

xxxxdipole rmrm

rm

πμ

(7-37)

( )

( ) ( ) ⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

•−

•

++•

−−=

11

4

9

2

9

3

770

945630

42045180

4r

yr

yr

ymrr

ym

B

yy

yyyydipole rmrm

rm

πμ

(7-38)

( )

( ) ( )⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

•−

•

++•

−−=

11

4

9

2

9

3

770

945630

42045180

4r

zr

zr

zmrr

zm

B

zz

zzzzdipole rmrm

rm

πμ

(7-39)

( ) ( )⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

•−+

•

++−−=

11

3

9

3

9

9

2

770

945105315

3154545

4r

yxr

xmr

xyr

yxmr

xmr

ym

By

xyx

xxxydipole rmrmπμ

(7-40)


( ) ( ) ⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

•−+

•

++−−=

11

3

9

3

9

9

2

770

945105315

3154545

4r

zxr

xmr

xzr

zxmr

xmr

zm

Bz

xzx

xxxzdipole rmrmπμ

(7-41)

( ) ( )⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

•−+

•

++−−=

11

3

9

3

9

9

2

770

945105315

3154545

4r

xyr

ymr

xyr

xymr

xmr

ym

Bx

yyx

yyyxdipole rmrmπμ

(7-42)

( ) ( )⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

•−+

•

++−−=

11

3

9

3

9

9

2

770

945105315

3154545

4r

zyr

ymr

yzr

zymr

zmr

ym

Bz

yyz

yyyzdipole rmrmπμ

(7-43)

( ) ( ) ⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

•−+

•

++−−=

11

3

9

3

9

9

2

770

945105315

3154545

4r

xzr

zmr

xzr

xzmr

xmr

zm

Bx

zzx

zzzxdipole rmrmπμ

(7-44)

( ) ( )⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

•−+

•

++−−=

11

3

9

3

9

9

2

770

945105315

3154545

4r

yzr

zmr

yzr

yzmr

zmr

ym

By

zyz

zzzydipole rmrmπμ

(7-45)

( )

( ) ( ) ( ) ⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

•−

•+

•+

++•

−−−=

11

22

9

2

9

2

9

2

9

2

7770

945105105

210210153030

4r

yxr

yr

xr

yxmr

xymrr

ymr

xm

B

yxyx

xxyydipole rmrmrm

rm

πμ

(7-46)

( )

( ) ( ) ( )

2 2

7 7 7 9 90

2 2 2 2

9 9 11

30 2101530 210

4 105 105 945

m r

m r m r m rdipole

y yz z

yyzz

m y m yzm z m y zr r r r rB

y z y zr r r

μπ

⎛ ⎞•− − − + +⎜ ⎟

⎜ ⎟=⎜ ⎟• • •

+ + −⎜ ⎟⎝ ⎠

(7-47)


( )

( ) ( ) ( ) ⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

•−

•+

•+

++•

−−−=

11

22

9

2

9

2

9

2

9

2

7770

945105105

210210153030

4r

zxr

zr

xr

zxmr

xzmrr

zmr

xm

B

zxzx

xxzzdipole rmrmrm

rm

πμ

(7-48)

( )

( )⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

•−++

+•

+−−=

11

2

99

2

9

2

9770

945210105

1051051515

4r

yzxr

xyzmr

zxmr

yxmr

yzr

ymr

zm

Bxy

zzy

xxyzdipole rm

rm

πμ

(7-49)

( )

( ) ⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

•−++

+•

+−−=

11

2

99

2

9

2

9770

945210105

1051051515

4r

zxyr

xyzmr

zymr

xymr

xzr

xmr

zm

Byx

zzx

xyyzdipole rm

rm

πμ

(7-50)

( )

( )⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

•−++

+•

+−−=

11

2

99

2

9

2

9770

945210105

1051051515

4r

xyzr

xyzmr

yzmr

xzmr

xyr

xmr

ym

Bzx

yyx

xyzzdipole rm

rm

πμ

(7-51)

Analysing in detail the relations (7-37) to (7-51), the following generalisations are made:

( )

( ) ( ) ⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

•−

•

++•

−−=

11

4

9

2

9

3

770

945630

42045180

4r

ir

ir

imrr

im

B

ii

iiiidipole rmrm

rm

πμ

(7-52)

( ) ( )⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

•−+

•

++−−=

11

3

9

3

9

9

2

770

945105315

3154545

4r

jir

imr

ijr

jimr

imr

jm

Bj

iji

iiijdipole rmrmπμ

(7-53)

( )

( ) ( ) ( ) ⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

•−

•+

•+

++•

−−−=

11

22

9

2

9

2

9

2

9

2

7770

945105105

210210153030

4r

jir

jr

ir

jimr

ijmrr

jmr

im

B

jiji

iijjdipole rmrmrm

rm

πμ

(7-54)


( )

( )⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

•−++

+•

+−−=

11

2

99

2

9

2

9770

945210105

1051051515

4r

jkir

ijkmr

kimr

jimr

jkr

jmr

km

Bij

kkj

iijkdipole rm

rm

πμ

(7-55)

The following inter-relationships also exist:

dipole dipole dipole dipoleiiij iiji ijii jiiiB B B B= = = (7-56)

dipole dipole dipole dipole dipole dipoleiijj ijij jiij jiji jjii ijjiB B B B B B= = = = = (7-57)

dipole dipole dipole dipole dipole dipole

dipole dipole dipole dipole dipole dipole

ijki ikji jiki jkii kiji kjii

ijik iijk jiik ikij iikj kiij

B B B B B B

B B B B B B

= = = = =

= = = = = = (7-58)

Finally, recalling equation (5-1) from Chapter 5 which states that the components of the

gradient tensor satisfy Laplace’s equation, the following can also be assumed:

0dipole dipole dipolexxxx xxyy xxzzB B B+ + = (7-59)

0dipole dipole dipolexxyy yyyy yyzzB B B+ + = (7-60)

0dipole dipole dipolexxzz yyzz zzzzB B B+ + = (7-61)

0dipole dipole dipolexxxy xyyy xyzzB B B+ + = (7-62)

0dipole dipole dipolexxyz yyyz yzzzB B B+ + = (7-63)

0dipole dipole dipolexxxz xyyz xzzzB B B+ + = (7-64)

By setting the vector s to be a unit vector in the vertical direction, a cubic octupole is created.

Plots showing the individual magnetic field components, the total magnetic field and gradient


tensor components around a magnetic octupole are shown in Figures 7.6 to 7.8. The search

area and associated parameters from Figures 7.2 to 7.4 are re-used here.

Figure 7.6. The three vector components of a magnetic field around a static magnetic octupole. Units are in Teslas.

Figure 7.7. The total magnetic intensity of a magnetic field around a static magnetic octupole. Units are in Teslas.


Figure 7.8. Six of the gradient tensor components of a magnetic field around a static magnetic octupole. Units are in Teslas/metre.

7.4 Eigenanalysis

Previously, I have calculated the eigenvalues and eigenvectors around various sources and

found some interesting characteristic patterns emerge. In Figures 7.9 and 7.10 I have plotted

the eigenvalues around the quadrupole and octupole sources. The two sets of graphs

produced consist of distinct, similar shapes, and as expected the magnitude of the response

has decreased roughly an order of magnitude for the magnetic octupole compared to the

quadrupole source.

Figure 7.9. The three eigenvalues around a quadrupole source have distinct shapes. Bquaeva1 is the 1st eigenvalue corresponding to a magnetic quadrupole source, and Bquaeva2 and Bquaeva3 are the 2nd and 3rd respectively.


Figure 7.10. The three eigenvalues around an octupole source have a similar appearance to those for the quadrupole, but the responses are roughly an order of magnitude smaller in size. Bocteva1 is the 1st eigenvalue corresponding to a magnetic octupole source, and Bocteva2 and Bocteva3 are the 2nd and 3rd respectively.

As mentioned in Chapter 6, there will always be some ambiguity with the eigenvectors: the

negative of an eigenvector still satisfies the condition for being an eigenvector itself. This is

one of the primary reasons why solutions from different algorithms that calculate the

eigenvectors of a matrix often differ. For this reason, the following plot shows all six

eigenvectors for the quadrupole case (three positive and three negative) projected onto the x-y

plane.

Figure 7.11. All six eigenvectors (three positive and three negative) create a distinctive pattern around a quadrupole source. The quadrupole source is at a depth of 10m at the centre of the grid.


Note that although there is a distinct pattern of vectors for this particular source (and enough

information to decide where the source probably is), if only a few measurements were made,

the position of the source would be harder to locate. Figure 7.12 shows the eigenvectors

around an octupole source.

Figure 7.12. All six eigenvectors (three positive and three negative) create a distinctive pattern around an octupole source, similar to a quadrupole. The source is at a depth of 10m at the centre of the grid.

7.5 Discussion

Dipole, quadrupole and octupole expressions are found as the first three terms of a multipole

expansion (Cowan, 1968). Such an expansion normally represents the scalar potential (rather

than the actual field) for such magnetic sources. Further manipulation (taking the gradient)

must be performed to yield the vector field quantities. Mathematically, the formulae derived

as part of a multipole expansion are not easily modelled, but are useful in determining how

the fields behave.

For example, the terms of the multipole expansion determine the order of the potentials. A

dipole (2-pole) is the first term, followed by the quadrupole (4-pole), then the octupole (8-

pole), followed by a 16-pole, 32-pole, 64-pole, etc. From the examples in this paper, the


signal due to each source diminishes as the pole distribution becomes more complex.

Whatever the TMI of the dipole, the quadrupole is approximately two orders of magnitude

smaller, and the octupole 10 times smaller again. The octupole is essentially 4 alternating bar

magnets. Placing them close together has the effect of nullifying the overall, far-field

response. The individual Cartesian component and gradient tensor responses also decrease in

intensity as the pole order gets higher. The shapes of the responses are markedly different,

being simplest for the dipole.

The field signature is probably the most direct way of determining the source type. However,

care must be taken with this, as many of the quadrupole maps look similar to the octupole

maps, albeit at a much reduced amplitude. To best determine what is causing the field

response an inversion must be run on the data. The next chapter will introduce a method for

determining a possible source type from only few measurements of the field.

The multipole responses calculated and presented in this chapter are noise-free. In reality

there will always be some noise present, representing measurement errors and other

uncertainties. The responses given here are very small (as low as 10-16 Teslas/metre in the

octupole scenario) as the source only had the strength of a small bar magnet, and was placed

10 metres below the surface. Actual geophysical values would be scaled up significantly in

accordance with the dimensions and magnetisation of the geological structures producing the

anomaly. Currently, the most accurate magnetic gradiometers hope to achieve resolutions of

around 0.01nT/m (Schmidt et al., 2004).

While many of the forward modelling formulae presented in this chapter are quite lengthy,

many of them can be shortened. As an example, if an octupole has only dy and sz components

(m being in any orientation and all other components equal to zero), then equation (7-31) is

reduced substantially to the following:

dipoleoct xxyzyzxx BdsB = (7-65)

Such simplifications should accelerate forward modelling significantly, and the next chapter

will show how this could be applied to an inversion routine.


7.6 Conclusions

The formulae describing the magnetic response of a static magnetic quadrupole and octupole

have been derived and illustrated. This includes the total field, the individual Cartesian

components, and all components of the gradient tensor. It is shown that derivatives of the

gradient components for a magnetic dipole are required in order to calculate the components

of the quadrupole and octupole sources. The shapes of the anomaly curves as well as the

magnitudes of the various responses are shown to be distinctive for the various sources. The

next step is to develop inversion algorithms for extracting information on the location, depth,

and strength of the various sources from analysis of the various components.

Chapter 8: Automated Inversion 172

Chapter 8: Automated Inversion for Multipole Sources

8.1 Introduction

Inversion theory often involves formulating the problem to be solved as a matrix equation

(Menke, 1984; Scales and Tenorio, 2001). Representing the collected field data as a vector d,

and the model data to be determined as a vector s, the following equation holds:

d As= (8-1)

The matrix A is referred to as the design matrix to the problem, and its size is determined by

the number of field data collected and the number of inversion model parameters. In reality,

we want to determine s from d and A:

1s A d−= (8-2)

Note that if and only if there are the same number of field measurements as there are model

parameters, will A be a square matrix. As only the inverse of a square matrix can be

calculated, this must be the case for this style of inversion to work. If A is not a square

matrix, a generalised matrix is formed by pre-multiplying both sides of equation (8-1) by the

transpose, AT, and then inverting the generalised matrix. The relationship becomes (Scales

and Tenorio, 2001):

( ) 1s A A A dT T−

= (8-3)

In all cases, the design matrix must be constructed prior to the inversion. It is rare that the

design matrix is easily determined, as (certainly for gravity and magnetics) there is no direct

linear relation between the variables (say x, y, z, mx, my, mz) and the design matrix. Many

problems can be linearised through creating an integral using the Green’s function, but this

process is intensive (Glenn et al., 1973) and I would like to introduce a simpler scheme.

Two vectors are required to describe the magnetic field around a magnetic dipole: the dipole

moment and a displacement vector (representing the distance and direction from the dipole to


the field point). From measurements of the magnetic field at the surface of the Earth, it is

possible to determine the possible positions of magnetic dipoles in the subsurface. The

process of determining these positions is often referred to as dipole tracking. Generally, this

is an over-determined problem, as there are 6 parameters to solve (assuming a single dipole)

and many measurements.

There exist several dipole-tracking routines in the literature (Schmidt et al., 2004), but no

multipole-tracking routines (to my knowledge). Dipole-tracking routines generally involve an

optimisation technique, searching the solution space either locally or globally. The inversion

process is generally based on the Total Magnetic Intensity (TMI) of the field (Blakely, 1996),

but in recent years the gradient tensor components are starting to be used (Schmidt et al.,

2004). With computers getting more powerful, it is possible to implement a search routine

that performs many calculations. The multipole-tracking algorithm I have constructed utilises

the gradient tensor components of the field (the TMI and three components of the total field

vector could also be used in conjunction) and is a combined global/local inversion procedure.

The technique has been designed to accommodate the fact that the gradient tensor

components can be measured independently, and used as part of an automated inversion.

Optimisation techniques that invert to a dipole are built from various assumptions. The

technique of a matrix inversion (equations (8-1) to (8-3)) requires a complete data set, and the

dipole must lie in the search region. Given the statistical nature of the technique presented

here, areas with a multipole present should be easily located.

A description of the inversion process follows.

8.2 Qualitative Method

I begin by defining a 3-D search volume. Before any measurements of the gradient tensor are

made, a simulated subsurface region representing the search volume (or area) is formed. Each

grid point in that volume represents a possible position in which a dipole could exist. It is

possible to calculate a dipole moment for each point in the subsurface using the set of linear

equations given in section 8.3. When a single measurement of the magnetic gradient tensor is

made, the dipole moment is calculated at each point in the region, such that it satisfies that

single measured magnetic gradient field point. The dipole moment components are calculated


from a set of linear equations, the equations being for the magnetic response of a magnetic

dipole.

Having calculated these “possible” dipole moments, I then take a second measurement of the

magnetic field at a neighbouring point along the survey line and the process is repeated (the

search volume must remain the same). Now, it is not physically possible to have two dipoles

with different dipole moments occupying the same position in space. So if the dipole moment

calculated at position x in the subsurface from one observation point is not equal to the dipole

moment calculated at position x from another observation point, then a magnetic dipole does

not exist there. If the dipole moment calculated at a given subsurface point remains constant

for all the gradient tensor measurements made of the field along the entire survey line, this is

taken as the solution.

Instead of discarding solutions that do not remain constant, all the solutions are kept stored in

the computers memory, and a simple statistic (variance) is formed to indicate the “constancy”

of the dipole moments. This takes into account the fact that a dipole may not lie on an exact

search position in the selected search region, and the fact that noise will be present in the data

(Scales and Snieder, 1998). The data are presented as a subsurface section where each point

represents the variation in dipole moment values. Areas with small variation correspond to

areas likely to contain a dipole.

As mentioned earlier, it is sometimes desirable to consider possible magnetic quadrupoles in a

field area, and the dipole technique just described can be extended to the quadrupole case as

well. The major difference is with the number of linear equations used. While the magnetic

dipole moment vector has three unknowns, the quadrupole moment tensor has nine

unknowns. With only five linear equations (the independent components of the gradient

tensor), the dipole scenario is over-determined and the quadrupole scenario is under-

determined. However, for very simple quadrupole orientations, five equations are enough.

Even for more complicated orientations, it is still possible to use only the five equations.

In order to determine the components of the dipole moment vector or quadrupole moment

tensor, the linear equations are taken and converted to reduced row echelon form (Anton and

Rorres, 1994). This can be done automatically in many data processing packages. For the

magnetic dipole this will calculate the three components of the dipole moment, and for the


magnetic quadrupole this will create a 7 × 5 matrix of which some simplification will be

required before calculating the quadrupole moment.

8.3 Quantitative Method

The basic method is to fully explore all grid points (possible source positions) in the

subsurface volume (global inversion method) and to calculate the dipole moment from given

field measurements at each recording position (local inversion method). Values will only be

the same for each recording position if the source actually occupies that subsurface point.

Start with a single measurement of the magnetic gradient tensor at a particular point along the

ground surface. Assuming that this measurement arises from a single magnetic dipole located

at position r(x,y,z) in the subsurface, there are five independent components (see Chapter 2 for

derivation). These five equations can be presented as a series of linear equations. As only

three equations are needed to define the three components (mx, my and mz) of the dipole, here

I have selected Bxx, Bxy and Bxz. The equations are:

3 2 2

5 7 5 7 5 70

9 15 3 15 3 15 4dipolex y z xx

x x y x y z x zm m m Br r r r r r

πμ

⎛ ⎞ ⎛ ⎞ ⎛ ⎞− + − + − =⎜ ⎟ ⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠ ⎝ ⎠ (8-4)

2 2

5 7 5 7 70

3 15 3 15 15 4dipolex y z xy

y x y x xy xyzm m m Br r r r r

πμ

⎛ ⎞ ⎛ ⎞ ⎛ ⎞− + − + − =⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠⎝ ⎠ ⎝ ⎠

(8-5)

2 2

5 7 7 5 70

3 15 15 3 15 4dipolex y z xz

z x z xyz x xzm m m Br r r r r

πμ

⎛ ⎞ ⎛ ⎞⎛ ⎞− + − + − =⎜ ⎟ ⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠ ⎝ ⎠

(8-6)

By making the following nine substitutions, the above equations can be simplified.

3

5 7

9 15x xar r

= − 2

5 7

3 15y x ybr r

= − 2

5 7

3 15z x zcr r

= − (8-7)

0

4dipolexxd Bπ

μ=

2

5 7

3 15x xyer r

= − 7

15xyzfr

= − (8-8)


0

4dipolexyg Bπ

μ=

2

5 7

3 15x xzhr r

= − 0

4dipolexzi Bπ

μ= (8-9)

Equations (8-4) to (8-6) can therefore be generalised into the form:

x y zm a m b m c d+ + = (8-10)

x y zm b m e m f g+ + = (8-11)

x y zm c m f m h i+ + = (8-12)

This is effectively a set of linear equations which can be row-reduced (via some automated

process) so that:

1 0 00 1 00 0 1

x

y

z

a b c d mb e f g mc f h i m

⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥⇒⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦

(8-13)

From the single measurement, a surrounding window is created in which we will determine

the dummy solutions for each candidate source position. Effectively, values for x, y, z and Bij

are substituted into equations (8-4) to (8-6). This is done for every possible source position in

the search region.

Therefore there are magnetic dipole moments assigned at each point within the subsurface

that satisfy the measured field point. By taking further measurements at different field points

(positions at or near the Earth’s surface) and repeating the process, multiple solutions are

obtained, but only the magnetic dipole moments that remain constant for all field points will

be the “true” solutions.

The process is similar for the magnetic quadrupole. The equations are lengthier, and the

derivatives of the formulae governing the magnetic response of a magnetic dipole are

required. The five equations governing the magnetic response of a magnetic quadrupole can

be simplified using substitutions similar to the dipole case above. In order to express the


quadrupole equations in terms of the quadrupole moment tensor, the following 19 identities

(denoted A to S) are obtained:

2 4

5 7 9

9 90 105x xAr r r

= − + 3

7 9

45 105xy x yBr r

= − + (8-14)

3

7 9

45 105xz x zCr r

= − + 2 2 2 2

5 7 7 9

3 15 15 105x y x yDr r r r

= − − + (8-15)

2

7 9

15 105yz x yzEr r

= − + 2 2 2 2

5 7 7 9

3 15 15 105x z x zFr r r r

= − − + (8-16)

3

7 9

45 105xy xyGr r

= − + 2

7 9

15 105xz xy zHr r

= − + (8-17)

2 4

5 7 9

9 90 105y yIr r r

= − + 3

7 9

45 105yz y zJr r

= − + (8-18)

2 2 2 2

5 7 7 9

3 15 15 105y z y zKr r r r

= − − + 2

7 9

15 105xy xyzLr r

= − + (8-19)

3

7 9

45 105yz yzMr r

= − + 3

7 9

45 105xz xzNr r

= − + (8-20)

0

4quadxxO Bπ

μ−

= 0

4quadyyP Bπ

μ−

= 0

4quadxyQ Bπ

μ−

= (8-21)

0

4quadyzR Bπ

μ−

= 0

4quadxzS Bπ

μ−

= (8-22)

The following equations therefore represent the magnetic response of a magnetic quadrupole,

where the midj terms represent the components of the quadrupole moment tensor:

x x x y x z y x y y y z z x z y z zAm d Bm d Cm d Bm d Dm d Em d Cm d Em d Fm d O+ + + + + + + + = (8-23)


x x x y x z y x y y y z z x z y z zDm d Gm d Hm d Gm d Im d Jm d Hm d Jm d Km d P+ + + + + + + + = (8-24)

x x x y x z y x y y y z z x z y z zBm d Dm d Em d Dm d Gm d Hm d Em d Hm d Lm d Q+ + + + + + + + = (8-25)

x x x y x z y x y y y z z x z y z zEm d Hm d Lm d Hm d Jm d Km d Lm d Km d Mm d R+ + + + + + + + = (8-26)

x x x y x z y x y y y z z x z y z zCm d Em d Fm d Em d Hm d Lm d Fm d Lm d Nm d S+ + + + + + + + = (8-27)

Again, this can be converted to reduced row echelon form automatically and solutions found.

As there are more variables than solutions, it is expected that the quadrupole moment tensor

cannot be fully resolved for all complex quadrupole orientations. However, the above system

of equations can be simplified by taking into account the fact that the quadrupole moment

tensor is equal to its transpose (see Appendix 1 for discussion and proof). The above

equations can be therefore be written:

x x x y x z y y y z z zAm d Bm d Cm d Dm d Em d Fm d O+ + + + + = (8-28)

x x x y x z y y y z z zDm d Gm d Hm d Im d Jm d Km d P+ + + + + = (8-29)

x x x y x z y y y z z zBm d Dm d Em d Gm d Hm d Lm d Q+ + + + + = (8-30)

x x x y x z y y y z z zEm d Hm d Lm d Jm d Km d Mm d R+ + + + + = (8-31)

x x x y x z y y y z z zCm d Em d Fm d Hm d Lm d Nm d S+ + + + + = (8-32)

There are now five equations in six unknowns. A way to remove the extra unknown would be

to simplify the above system by removing a column. If I assume that (say) the quadrupole is

lying flat in the x-y plane, then it would be possible to remove all the components involving

mz or dz. This will give an over-determined problem and allow the system to be solved.


8.4 Method and Results

In this section I will illustrate the inversion procedure on three different examples. For all the

examples, a measurement of the field is made at the first point on the x-axis (x equal to 1), and

the dipole and quadrupole moment tensors are calculated for all the subsurface points (for x

equal to 1 to 50, and z equal to -1 to -40). The multipole moment for each point is stored in

the computer’s memory. The second measurement is made at the next point (x equal to 2).

The dipole and quadrupole moments are then calculated for each subsurface point. Again, the

values are stored in computer memory. As this process continues, a larger and larger number

of multipole moments are associated with each point in the region. Therefore, a plot of the

standard deviation at each point in the region should reveal the multipole as an area with

small variation. By plotting the inverse of the standard deviation, it is possible to associate

highs with possible sources. Once x reaches the final point (x equal to 50) the process is

terminated.

For the first example, the dipole source is placed at a depth of 20 metres (z equal to –20), at

20 metres along the x-axis. The dipole is given a moment of 1Am2 parallel to the x-axis. The

inversion routine designed to locate a single dipole is run on this data set.

For the second example, two antiparallel dipoles are placed near each other (at x coordinate

25 metres and z coordinate 20 metres), effectively creating a square quadrupole with equal

strength in the x and y direction (mxdy = 10Am3). The inversion routine designed to locate a

single dipole source is also run on this data set.

The above data set is also used to test the quadrupole inversion technique. For the quadrupole

inversion process undertaken here, the system of linear equations is simplified by removing

the mzdz term, creating five equations in five unknowns.

The three examples using the inversion routine are shown in Figures 8.1 to 8.3. Figure 8.1 is

a plot of the inverse of the standard deviation of calculations of mz as a function of x and z.

Note that there is a peak at the (20,20) position, suggesting a dipole source at this locale

(showing the inversion works for this scenario). The mean dipole moment here is mx = 1, as

required.


Figure 8.2 is a plot of the inverse of the standard deviation of calculations of mz for two

dipoles, separated by a small distance. The dipole inversion routine does not determine the

position of these sources, instead exhibiting a region of high values around another area

(30,25) extending in both directions.

Figure 8.3 shows the same example, but this time allowing the inversion routine to locate a

single quadrupole and not a single dipole. The peak at the position (20,20) suggests that a

quadrupole is present here. The average value of mxdy at this point is 10Am3, as required by

the forward model.

Figure 8.1. The inverse of the standard deviation of calculated dipole moment values shows a high in an area that a magnetic dipole is present.


Figure 8.2. The inverse of the standard deviation of calculated dipole moment values shows a region of high values in an area where two magnetic dipoles are present, but does not succeed in locating the position of the quadrupole source.

Figure 8.3. The inverse of the standard deviation of calculated quadrupole moment values shows a high (the red peak) in the area in which a magnetic quadrupole is present.


8.5 Discussion

The inversion technique described in this chapter is a combination of local and global search

techniques. It is a local search technique in the sense that I choose the window size and use a

simple linear inversion to compute the dipole moment from the gradient tensor values. It is a

global search technique in the sense that every single point in the solution space (subsurface

volume) is “searched” as a possible source location.

If further components of the field were measured (say Bx, By and Bz) in addition to the 5

gradient tensor components, then there would be 8 linear equations. Ideally 6 measurements

for the quadrupole case are needed (see Appendix 1), so 8 would certainly reduce some of the

uncertainty.

One obvious extension of this work is to look at an octupole source. While the dipole

moment is a three component vector, and the quadrupole moment is a nine component tensor

(with 6 components being independent, see Appendix 1), the octupole moment will have 27

components (10 components of which are independent, see Appendix 1). An octupole

consists of two quadrupoles, where one of the quadrupoles is reversed. The application of an

inversion routine to locate the magnetic octupole would be largely under-determined, with 10

unknowns and 5 equations. Even if we take the additional components Bx, By and Bz, this is

only 8 components of the 10 needed, and we would have to make some assumptions about the

nature of the octupole (e.g., define the shape of the octupole) before inverting to it. It would

not have to be a particularly simple octupole, as even a cubic octupole only has the one

coefficient (mxdysz), but we are restricted to these simple cases rather than a general case.

Figure 8.4 shows an octupole with mxdysx=1Am4 and mxdysy=2Am4. All other components of

the moment are equal to zero. This octupole could easily be located via an inversion routine.


Figure 8.4. Combinations of the components of m, d and s describe the octupole moment. In the illustration shown here, there are two components of the octupole rank-3 tensor moment, mxdysx=1Am4 and mxdysy=2Am4.

For the data sets used, the anomaly was always placed on a search position. In reality it

would be rare for such a scenario to occur, so the routine has been tested on several

simulations whereby the source is placed in between search points. Figure 8.5 shows the

gradient tensor responses of a magnetic quadrupole, and Figure 8.6 shows the corresponding

inversion results (inverse of the standard deviation of the mxdy term). The quadrupole (mxdy =

1Am3) is placed at x position 25.5 metres, at a depth of 5.5 metres, and the search area is 50

metres along the x-axis, and 20 metres deep. The source is easily detected.

Figure 8.5. Gradient tensor response of magnetic quadrupole at horizontal position 25.5m and depth 5.5m.


Figure 8.6. Inversion results corresponding to Figure 8.4. The source is easily located as the red peak.

If the search routine misses the source entirely (e.g., the source is very small and below

detectable threshold, or the window does not cover the solution point) the inversion will not

locate the source. If the source is only just detectable, the inversion routine may still locate

something. Magnetic gradiometers have a measurement resolution of 0.01nT/m, so Figure

8.7 shows a scenario where each field measurement is rounded off to the nearest 0.01nT/m,

with the gradient tensor responses around the quadrupole source being visibly affected (i.e.,

they are not smooth curves). Figures 8.8 and 8.9 show the inversion results for two of the

quadrupole moment components. The anomaly is slightly offset in the mxdy term, but is still

detectable in the mydz term.

Figure 8.7. Gradient tensor response of magnetic quadrupole at horizontal position 25m and depth 5m with all measurements rounded off to the nearest 0.01nT/m.


Figure 8.8. Inversion results (mxdy) corresponding to Figure 8.6. The source is slightly offset, but still detectable.

Figure 8.9. Inversion results (mydz) corresponding to Figure 8.6. The source is easily located as the red peak.

In all the previous examples, no noise was present. As noise is always present in geophysical

data, it must be taken into account. However, before simply adding noise, it is important to

notice that there are two ways that noise can be included in this data. The first is to add a

different amount of noise to each gradient tensor component, as if they were being measured

separately (independently) in the field. The second is to add noise to one component, and

assume that the other components were calculated from this component. As both techniques

are used in data collection, it is necessary to analyse both cases.

To illustrate the effect of noise on the inversion process, consider a quadrupole placed at

position (25,-5) with moment mxdy = 10Am3 (the same search area as before is utilised), and a

single random value of Gaussian noise added to each component. Taking the average

amplitude of the maximum field signal I have selected a value of 1% random noise to add to


the field signals of all the gradient tensor components (approximately 0.006nT/m in this case).

I then repeat the same inversion, but this time adding separately calculated 1% random noise

to each component (approximately 0.0012nT/m for Bzz, 0.0003nT/m for Bxy and Byz, the other

components show only minor signal in this dimension). Figure 8.10 shows the inversion

results of the first scenario and Figure 8.11 shows the inversion results for the second

scenario. A source is detected in both cases, although the solution corresponding to the

second scenario is more accurate. In fact, the position of the quadrupole in Figure 8.10 is

nearly 10 metres from the actual position. This demonstrates that it is more accurate to

measure the components of the gradient tensor separately when using this inversion routine,

than it is to measure a single component to calculate the entire gradient tensor and run the

inversion.

Figure 8.10. Inversion results (mxdy) corresponding to gradient tensor data with 1% repeated noise. The source is slightly offset, but still detectable.

Figure 8.11. Inversion results (mxdy) corresponding to gradient tensor data with 1% independent noise. The source is easily detectable.


Figures 8.12 and 8.13 show corresponding inversion results when the data noise level is much

higher at 10%. Again, it is possible to delineate the source in both cases, but the situation in

which separate noise is added to each gradient tensor component yields more accurate results.

Figure 8.12. Inversion results (mxdy) corresponding to gradient tensor data with 10% repeated noise. The source is offset approximately 7 metres to the right.

Figure 8.13. Inversion results (mxdy) corresponding to gradient tensor data with 10% independent noise. The source position appears slightly closer to the surface than previously shown.

Finally, I have presented all these results (Figures 8.1 to 8.13) by showing images of the

inverse of the standard deviation for a single component of the quadrupole moment only. The

other components exhibit similar behaviour, and occasionally variance of zero at all points in

the search region. I have not found a simple way of combining the components together to

produce a single data set for interpretation. Multiplication of the data sets results in a plot

exhibiting zero at each data point, and addition of the data sets is mathematically difficult, as

the range of values for the “possibility of quadrupole” is unpredictable. For this reason, I

recommend analysing each component separately.


8.6 Conclusions

An inversion routine has been constructed and tested for some multipole sources. The

generalised octupole source contains too many variables for this style of combined global and

linearised inversion routine to be applied, unless a specific source orientation is known which

can be expressed in five components of the third rank moment tensor or less.

The inversion routine successfully locates a dipole and quadrupole source, including scenarios

whereby noise was added to the data, and where there was only small amplitude in the field

measurements. The inversion routine appears to work more accurately for gradient tensor

components collected independently of each other, rather than from a data set with repeated

noise.

I conclude that the inversion technique given in this chapter could be used in real time with

data collection. It shows promise, and with further development, may be shaped into a more

complete routine. Alternatively, again with further development, it could be used as a filter

for analysing collected data in areas of complex magnetisation.

Sources rarely occur in isolation, as we have been considering here. Any inversion routine

should be capable of dealing with multiple (interfering) sources. In some sense, the multipole

sources are composite and interfering, but they occur at the (almost) same spatial location.

The next obvious step would be to consider multiple macro-sources throughout the model

volume and to devise gradient tensor inversion methodology to recover the source positions,

shapes and magnetisation. Such research would have to form the subject of a separate PhD or

postdoctoral project.

Chapter 9: Discussion and Conclusions 189

Chapter 9: Summary and Conclusions

The objectives of this research were as follows:

• Provide a systematic working mathematical notation for potential field theory and

gradiometry, and determine the gradient field response of simple objects for forward

modelling and inversion,

• Determine what improvements gravity and magnetic gradient tensor measurements may

have for near-surface exploration,

• Investigate the application of standard filters to multi-component gradient data and to

develop new filtering techniques,

• Devise an appropriate inversion routine and determine some guidelines as to how a multi-

component inversion should be efficiently carried out, and

• Develop mathematical relationships for magnetic multipoles, and to construct an automated

inversion technique for such sources.

A mathematical treatment of potential field theory was given in Chapter 2. This provided a

basis of mathematical notation for the thesis, as well as illustrating various relationships

between gravity and magnetic field variables. The chapter also outlined the gravity and

magnetic responses of the basic “building blocks” of materials, i.e., point, rectangular prism

and magnetic dipole sources. I have demonstrated (using a generalised form of Poisson’s

relationship with the introduction of three weighted variables (α, β and γ)) that a magnetic

point source is effectively equivalent to a magnetic dipole source, allowing simplification of

complex dipolar structures such as the prismatic dipole. This chapter also outlined the

relationships between gravity (or magnetic) field components utilising Fourier transforms.

Additionally, I have introduced a new notation (the asterisk notation) that allows simplified

mathematical description of the gravity and magnetic fields around prismatic sources.

Chapter 3 illustrated that ground-based magnetic gradient tensor surveys should indeed be

successful in near-surface exploration, with the signal diminishing as the data collection

height is increased. The range of values obtained from regolith forward modelling have a

similar range of values to that which can be sensed with current acquisition systems. Ground

based gravity gradiometer surveys only have the sensitivity to detect relatively large contrasts


in density, and so would be unsuited for very detailed regolith landform mapping. Forward

modelling of a 3-D regolith situation revealed however that the very near-surface features

(within a few metres) dominated the field signal.

Chapter 4 examined the magnetic field signal further, taking into account “real” geological

variation in the physical properties of near-surface materials. It was shown that the magnetic

susceptibility of surface materials does not always correlate to definable regolith units or

landforms. Further forward modelling of the components of the magnetic gradient tensor

yielded very noisy data, suggesting that magnetic gradiometry can detect realistic regolith

features, but these features are likely to have erratic field responses. The use of magnetic

dipoles to represent ferromagnetic materials in the regolith model allowed a deeper

mineralisation feature to be detectable from surface measurements, without being covered by

the noise of the surrounding geology.

The use of standard filters on gradient tensor data in Chapter 5 illustrates how the filters can

be used to enhance anomalies within a data set. The application of Reduction to the Pole to

each of the gradient tensor components corresponding to the magnetic field around a dipole

illustrates that a “peak” will not always reveal the position of the source, and the components

should be used in conjunction to best delineate the source position. There are numerous ways

of combining the components of the gradient tensor, perhaps the most successful being the

“Tensor Analytic Signal,” a mathematical construct condensing all components of the

gradient tensor into a single data set. A basic smoothing convolution filter appears to be best

applied after the Tensor Analytic Signal is calculated.

Multiplicative combinations of the gradient tensor generally don’t yield further information

from the data sets presented here, only the combinations of the diagonal components of the

gradient tensor appear to strengthen any anomalies. Also useful is the determinant of the

gradient tensor. This signal can be modified to produce larger amplitudes by swapping

positive and negative signs in the governing equation. The inverse matrix of the gradient

tensor generally shows very little diagnostic information, although gaps (or spikes) can appear

in the data where the determinant is equal to zero (meaning no inverse exists). This can be

related to a dipole source, but is not definite proof of such a case.

Chapter 6 examined some implications of multi-component inversion for the gradient tensor,

and showed that there is no benefit from inverting to all five independent components of the


gradient tensor simultaneously. The inversion routine selected was approximately three times

quicker when inverting to a single component, and the output geological model (when

forward modelled to produce the other components of the gradient tensor) produced gradient

tensor data that matched forward modelling of the input geological model. A Genetic

Algorithm was used for this process. The geological model produced as a result of the

inversion does not always match the required model, due to the large size of the solution

space. I found that one model in ten would match the starting model. All the models fit the

field data.

I also presented a quasi-linear dipole-tracking eigenanalysis routine, and illustrated that it

does not produce a single solution for source direction and dipole moment (rather it produces

four instead). However, the calculation of eigenvalues and eigenvectors proved useful in

searching the forward models of Chapter 3, and further depth information was obtained

(palaeochannel information). This information was previously unseen in any of the forward

models and through the application of filters.

The formulae describing the magnetic response of a static magnetic quadrupole and octupole

were derived and illustrated in Chapter 7, including discussion on how to compute magnetic

fields around multipoles of higher order. The derivatives of the gradient components for a

magnetic dipole are required in order to calculate the components of the quadrupole and

octupole sources. The shapes of the anomaly curves as well as the magnitudes of the various

responses are distinctive for the various sources, with the field anomaly diminishing for

increased model complexity. Appendix 1 contains some further discussion on the nature of

multipole moments, and outlines a proof as to which components of a multipole moment are

independent.

Finally, in Chapter 8, an automated inversion routine was developed that benefits from

independent measurements of the components of the gradient tensor (c.f., the Genetic

Algorithm of Chapter 6). The inversion routine successfully locates a dipole and quadrupole

source (and simple octupoles), including cases whereby random noise was added to the data,

and where there was only a small anomaly in the field measurements. The routine takes a

single measurement of the gradient tensor and calculates all the possible components of the

multipole moment through a search area or volume. Further measurements of the gradient

tensor repeat the process and any region where the multipole moment components remain

constant is taken as a possible solution position. This inversion routine could therefore be


used automatically in real time with data collection, or as a filter for analysing collected data

to detect possible dipole or quadrupole sources.

I have therefore addressed all the objectives outlined in Chapter 1 of this thesis. Potential

field gradient tensor data have been examined through the application of forward modelling,

filters and numerous inversion techniques, and I have shown that magnetic gradiometry is

especially suited to near-surface regolith exploration.

Chapter 7: Forward Modelling the Gradient Tensor Response of ...

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