chapter 7: forward modelling the gradient tensor response of
TRANSCRIPT
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):
Chapter 7: Multipoles 155
( )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.
Chapter 7: Multipoles 156
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:
Chapter 7: Multipoles 157
( )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)
Chapter 7: Multipoles 158
( ) ( )⎟⎟⎠
⎞⎜⎜⎝
⎛ •+
•−−−= 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.
Chapter 7: Multipoles 159
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.
Chapter 7: Multipoles 160
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)
Chapter 7: Multipoles 161
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)
dipoledipoledipoledipoledipole
dipoledipoledipoledipoleoct
yyzzzzyyyzyzxyyzxzyyyzzyyyyyyy
xyyyxyxyyzzxxyyyyxxxyyxxyy
BdsBdsBdsBdsBds
BdsBdsBdsBdsB
++++
++++= (7-32)
Chapter 7: Multipoles 162
dipoledipoledipoledipoledipole
dipoledipoledipoledipoleoct
zzzzzzyzzzyzxzzzxzyzzzzyyyzzyy
xyzzxyxzzzzxxyzzyxxxzzxxzz
BdsBdsBdsBdsBds
BdsBdsBdsBdsB
++++
++++= (7-33)
dipoledipoledipoledipoledipole
dipoledipoledipoledipoleoct
xyzzzzxyyzyzxxyzxzxyyzzyxyyyyy
xxyyxyxxyzzxxxyyyxxxxyxxxy
BdsBdsBdsBdsBds
BdsBdsBdsBdsB
++++
++++= (7-34)
dipoledipoledipoledipoledipole
dipoledipoledipoledipoleoct
yzzzzzyyzzyzxyzzxzyyzzzyyyyzyy
xyyzxyxyzzzxxyyzyxxxyzxxyz
BdsBdsBdsBdsBds
BdsBdsBdsBdsB
++++
++++= (7-35)
dipoledipoledipoledipoledipole
dipoledipoledipoledipoleoct
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)
Chapter 7: Multipoles 163
( ) ( ) ⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜
⎝
⎛
•−+
•
++−−=
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)
Chapter 7: Multipoles 164
( )
( ) ( ) ( ) ⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜
⎝
⎛
•−
•+
•+
++•
−−−=
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)
Chapter 7: Multipoles 165
( )
( )⎟⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜⎜
⎝
⎛
•−++
+•
+−−=
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
Chapter 7: Multipoles 166
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.
Chapter 7: Multipoles 167
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.
Chapter 7: Multipoles 168
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.
Chapter 7: Multipoles 169
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
Chapter 7: Multipoles 170
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.
Chapter 7: Multipoles 171
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
Chapter 8: Automated Inversion 173
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
Chapter 8: Automated Inversion 174
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
Chapter 8: Automated Inversion 175
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)
Chapter 8: Automated Inversion 176
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
Chapter 8: Automated Inversion 177
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)
Chapter 8: Automated Inversion 178
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.
Chapter 8: Automated Inversion 179
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.
Chapter 8: Automated Inversion 180
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.
Chapter 8: Automated Inversion 181
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.
Chapter 8: Automated Inversion 182
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.
Chapter 8: Automated Inversion 183
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.
Chapter 8: Automated Inversion 184
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.
Chapter 8: Automated Inversion 185
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
Chapter 8: Automated Inversion 186
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.
Chapter 8: Automated Inversion 187
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.
Chapter 8: Automated Inversion 188
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
Chapter 9: Discussion and Conclusions 190
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
Chapter 9: Discussion and Conclusions 191
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
Chapter 9: Discussion and Conclusions 192
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.