Chapter 1: Introduction to Gravity, Magnetics and Regolith ... · PDF fileChapter 1: Introduction 1 Chapter 1: Introduction to Gravity, Magnetics and Regolith Exploration 1.1 Introduction

Chapter 1: Introduction 1

Chapter 1: Introduction to Gravity, Magnetics and

Regolith Exploration

1.1 Introduction

Many different geophysical techniques exist for imaging the subsurface in search of minerals

and petroleum. These include gravity (Griffin, 1949; LaFehr, 1980), magnetic (Gunn, 1998;

Jensen, 1965), seismic (Reynolds, 1998; Wang, 2002), electrical (Lilley, 1991; Mauriello and

Patella, 1999; Meidav, 1960), electromagnetic (Bogoslovsky et al., 1977; Constable and

Heinson, 2004; Gelius, 1996) and radar (Griffin and Pippett, 2002; Telford et al., 1996). All

rely on some contrast in the physical properties of the Earth. For example, the gravity method

depends on changes in the density of rock units. Metallic ore bodies often have a high density

contrast with the surrounding geology, and produce a positive anomaly in the gravitational

field.

These geophysical techniques are not used exclusively for mineral and petroleum exploration.

They can also be applied to environmental and civil engineering investigations. The gamma-

ray spectrometry technique is another geophysical tool for exploring the very near surface

(Minty, 1997; Minty et al., 1997; Wilford, 2002; Wilford et al., 1997). Some environmental

applications of geophysics include exploring for groundwater and locating the water table

(Meidav, 1960; Titov et al., 2005), tracing pollution plumes (Atekwana et al., 2004; Carcione

et al., 2000), locating shallow faults (Wang, 2002) and searching for UXO (Gamey et al.,

2002). Civil engineers may use geophysics to detect the depth of bedrock, locate subsurface

voids and tunnels, assess rock strength and rippability, delineate rock fracture patterns, and

determine bearing capacity for bridge and building foundations (Butler, 1984; Daniels, 1988;

Mahrer and List, 1995; Rechtien et al., 1995; Thapa et al., 1997). Geophysics can also be

used for archaeological surveys (Clark, 1986; Osten-Woldenburg et al., 2006) and for the

detection of grave sites (Bevan, 1991; Davenport et al., 1988).

In the examples above, the objective is to determine the physical properties of the subsurface

to relatively shallow depths. The zone of unconsolidated and weathered geological material is

often referred to as the regolith. In this situation, seismic and electrical methods often prove


most useful. They can yield useful information at this scale. Methods such as gravity and

magnetics are traditionally more suited to greater depths, as the instrumentation is usually part

of an airborne system that covers considerable ground per survey in reconnaissance fashion.

The instrumentation is generally not sensitive enough at airborne heights to detect near-

surface targets, which may appear as noise in the data.

However, geophysical tools that have been developed recently can measure extremely

accurately the gradients of gravity and magnetic fields (Gamey et al., 2002; Hammond and

Murphy, 2003; Lee, 2001; Schmidt et al., 2004). These tools may be accurate enough to

gather useful information in near-surface exploration. This dissertation is a study into gravity

and magnetic gradiometry for exploration within and beneath the regolith.

In this introductory chapter I first consider the problem of exploration under regolith cover

(section 1.2). I then discuss the use of gravity and magnetic techniques for geophysical

exploration (section 1.3). The chapter concludes with a statement of objectives and an outline

of the thesis.

1.2 Exploration under Cover

A layer of material (soil and weathered rock) known as the regolith covers much of

Australia’s bedrock (Doyle and Lindeman, 1985). Information about this material (and what

lies beneath it) is vital to mineral explorers (Hill, 2003), civil engineers (Butler, 1984) and

land users in general (Carlson and Zonge, 2003). The regolith is often defined as “everything

between fresh rock and fresh air” (Eggleton, 2001). It is now known that the regolith can play

host to mineral deposits as well as key to mineral indicators. The Cooperative Research

Centre for Landscape Environments and Mineral Exploration (CRC LEME) is a research

group dedicated to the understanding of regolith processes and the associated implications for

mineral exploration and environmental monitoring. Much work is done in determining the

evolution of a landscape, including the definition of and mapping of landscape units, and

chemical analyses of surface materials, especially vegetation (Hill et al., 2003; Pirson, 1946).

CRC LEME supported this geophysical research project.

Geophysics can play an important part in exploring the regolith (Gunn et al., 1997; Smyth and

Barrett, 1994). In fact, near-surface geophysics has been used extensively in the past to


determine the physical properties of near-surface materials (Papp, 2002). This has been

mostly done using so-called active methods where the “artificial” field is created by the user

and impressed into the ground. This allows some control over the depth of investigation. For

example, seismic refraction (Dentith et al., 1992; Drummond, 2002; Telford et al., 1996) is

such a technique based on a sound wave field created by explosives or weight drops. It can

be used to determine variation in depth to bedrock along a profile (assuming there is a change

in the elasticity or sound wavespeed at the boundary). Electrical resistivity is another

commonly used technique in engineering and environmental geophysics often used to detect

depth to the water table and other subsurface boundaries (Meidav, 1960; Titov et al., 2005).

The field in this case is DC electricity from a current source. Ground Penetrating Radar

(GPR) uses high frequency (50 - 500MHz) EM waves to determine the dielectric properties of

near-surface materials (Kearey and Brooks, 1996).

Unlike these active methods, which are very restricted in their aerial coverage, the gravity and

magnetic techniques offer the potential to explore the Earth on a regional scale for

reconnaissance, and pinpoint target areas for follow-up investigations. The regolith is known

to obscure the gravity and magnetic signature and so it is important to thoroughly analyse the

capability of gravity and magnetic methods given the recent advances in tensor gradiometry.

1.3 Gravity and Magnetic Techniques for Geophysical Exploration

Gravity and magnetic exploration techniques are both passive in that they exploit a naturally

existing field of the Earth. The measured quantity is the integrated effect of the subsurface.

The gravity and magnetic fields are both irrotational vector fields described by potential

theory, and are connected through Poisson’s relationship. The actual geophysical field

technique is also similar in that both entail an instrument (gravity meter or magnetometer)

being taken into the field and data sampled and recorded at spot locations along a line or

series of lines. In the following sections I discuss, in a qualitative sense, gravity and

magnetics as tools for geophysical exploration. A mathematical treatment will be given in

Chapter 2.


1.3.1 Gravity as an exploration tool

A brief history of gravity techniques

Extensive reviews relating to the history of understanding of gravity can be found in many

texts (Telford et al., 1996) so I shall not repeat a detailed account here. However, I mention

some important historical figures as the units and concepts that have been named after them

will be used in this thesis.

Galileo Galilei (1564-1642) was the first person to scientifically investigate gravity. He

dropped objects of different masses from the top of the Leaning Tower of Pisa around the

year 1589. Galileo was trying to determine a relationship between the mass of an object and

its velocity change (Halliday et al., 1997). He found that objects fall at a similar acceleration

regardless of their mass (although wind resistance can play a part, depending on how the mass

is distributed, i.e., the shape of the object). The basic unit of gravity, the Gal (equal to 1

centimetre per second squared), was named in honour or Galileo.

Sir Isaac Newton (1643-1727) developed the universal law of gravitation. This law states that

the force acting between two masses is inversely proportional to the square of the distance of

separation, and proportional to the product of the masses. The proportionality constant

became known as the Universal Gravitational Constant.

Pierre Bouguer (1698-1758) travelled to the Earth’s equator for an experiment to measure the

length of a degree of meridian. While there, he undertook experiments in determining the

horizontal gravitational attraction of a mountain. Bouguer also undertook work relating to

basic gravitational relationships, the density of the Earth, and changes in gravity due to

elevation change. Measurements of the acceleration due to gravity must go through a series

of corrections before they can be used for interpretation, as the gravity field is affected not

only by subsurface material. Tidal effects, latitude, elevation and surface topography also

contribute to the acceleration due to gravity at a point (Telford et al., 1996). The corrected

gravity reading is often referred to as the “Bouguer” anomaly, and takes into account all the

above factors, including errors in the instrument (i.e., instrument drift). It is generally

expressed in mGals (1mGal = 10-5ms-2) or gravity units (1GU = 0.1mGal). The “free-air”

gravity anomaly is another form of corrected gravity data that takes into account station

elevation but does not include the mass of material between the surface topography and the


reference datum level – the so-called Bouguer correction (Telford et al., 1996). It is generally

used in marine surveying.

Baron Roland von Eötvös (1848-1919) invented the torsion balance, a device that could

measure curvature (2nd derivative) of the gravitational field. This device was much more

accurate than previous gravity meters, and a major hydrocarbon discovery (The Nash salt

dome in Brazoria County) was attributed to this instrument. The torsion balance became

immensely popular, until 1940, when gravimeters returned as the primary tool. Gravity

gradients have the units of Eötvös, in honour of him.

In 1934, Lucien LaCoste and his thesis supervisor Arnold Romberg developed the zero-length

spring, which led to the development of the LaCoste and Romberg gravimeter in 1939 (Ander

et al., 1999). Rather than absolute gravity such as determined with a pendulum, they

measured relative gravity, or the difference compared to some reference station. The LaCoste

and Romberg gravimeter is still a standard tool for basic geophysical exploration today,

although many improvements have been made to the design (Aiken et al., 1998; Chapin,

1998; LaFehr, 1980). Other instruments like the Worden gravimeter have also been

developed and used extensively for many decades with little change in field technique or

measurement, apart from automation and digital recording (Nabighian et al., 2005). Finally,

the Scintrex CG series of gravity meters are perhaps the most popular today, due to features

such as an internal GPS, real time free-air and Bouguer corrections, online near zone terrain

correction and their accuracy (resolution of 1μGal). More information regarding Scintrex

instruments can be found on the Internet (http://www.scintrex.com/gravity.html).

Recent Developments in Gravity Recording Techniques

Perhaps the most significant breakthrough in gravity surveying occurred in 1983, when it

became possible to take gravity measurements from an airborne platform (Hammer, 1983).

While at the time this was heavily disputed (Pearson, 1984), airborne gravity has slowly

emerged as a major tool in the geophysical industry. Airborne gravity data is difficult to

collect due to a large signal-to-noise ratio and due to the non-inertial (accelerating) frame of

reference (Bell et al., 1991). It is not possible to cancel out the acceleration of the platform.

Gravity data can now be collected at a far greater rate than before, and large-scale surveys are

now commonplace (Bell et al., 1991; Biegert and Millegan, 1998). The minimum error


involved in measurements of the field in airborne gravity surveys was in the order of 2.7mGal

(Bell et al., 1991).

With airborne gravity possible, several companies started looking into measuring the gravity

gradients from an airborne platform. Gravity gradients are measured in mGals/m, or Eötvös

(1Eö = 10-4mGal/m = 10-9s-2). As the acceleration of the aircraft does not have to be taken

into account to the same degree when measuring gradients, this proved much easier than

determining the acceleration due to gravity. To date, the two most successful airborne gravity

developments are the FALCON™ project of BHP Billiton, and the BellGeospace

technologies system (Lane, 2004). Both systems use a combination of accelerometers and

GPS (Global Positioning System) to accurately determine the gravitational gradients.

At any point in a gravitational field, there exist five spatial gradients (see later). Each of these

gradients is a component of the gradient tensor of the gravitational field. There is also a

gradient tensor associated with each point in the magnetic field, and it too comprises five

independent components.

The FALCON™ System

The FALCON™ project was the world’s first-ever airborne gravity gradiometer (Dransfield

et al., 2001). First flown in 1999, the data it produced compared well to known gravity and

has since been used to fly numerous surveys around the world (Dyke et al., 2002).

FALCON™ measures two gradient components (denoted gxz and gyz) of the gravitational field

of the Earth, and uses these to compute the vertical gradient component of the field (the

mathematics for this process will be described in section 2.4). This calculated gradient is

often denoted as gzz, which is different to the gravitational field quantity gz measured by

gravimeters (the mathematical differences between these two quantities will be also be

discussed in Chapter 2).

The actual FALCON™ tool takes readings every 0.1 seconds. If the platform is moving at

200m/s (approx 700km/hr), a reading is taken roughly every 20 metres. Filters are normally

applied to suppress high spatial frequency data, such as noise. As an example, a 125 metre

wavelength filter is used (Dransfield, 1994) for a flight height of 80 metres (i.e., spikes in the


data with a wavelength of less than 125 metres are removed). Geologically, a feature with a

wavelength of less than 125 metres would be considered to be near the surface, and would

therefore not be examined for deeper exploration.

The FALCON™ system has been used to detect Kimberlite pipes (a key rock type associated

with diamonds) in South Australia in recent years. The system has also extensively explored

for diamonds in Canada (Nowak, 2002).

Bell Geospace

Bell Geospace, which developed the Air-FTG™ system was the first company to use full

tensor gradiometry from an airborne platform for geophysical exploration (Hammond and

Murphy, 2003). The system was used successfully for marine surveys over the years 1998 to

2002, before being adapted for airborne use in 2002. The system is currently being used in

Africa and North America.

The system can be used for both regional exploration, and very detailed surveying. For

regional surveys, the distance between flight lines can be of the order of 2000 metres, while

detailed surveys may have a flight line separation of 50 metres (Murphy, 2004). The

measurements from the system have (after processing) a standard deviation of noise around 5

to 6 Eötvös, a higher resolution compared to numerous other systems (Lane, 2004). The

aircraft is typically flown at a height of 80 metres, although lower flight heights are

commonly used (e.g., 60m, 15m). Data are heavily filtered (Murphy, 2004) and wavelengths

of less than 400 to 600 metres are generally removed as part of the processing.

The system has been used in a variety of applications. One example was to map a known salt

caprock feature in the USA. The resulting maps feature a characteristic rectangular prism

response that is commonly used to locate geological features. A rectangular prism can be

modelled so that it approximates the shape of a salt dome. The maps are given in Figure 1.1,

after (Murphy, 2004). Note that there are six gradient maps shown here (denoted Txx, Txy,

etc…). These correspond to the different directional gradients that correspond to a potential

field (altogether called the gradient tensor). The double subscripts relate to the direction of

the gradient. A formal definition of the gradient tensor will be given in Chapter 2.


The agreement between the vertical prism response and the observed caprock gradient tensor

response can be appreciated from Figure 1.1. Both Txx and Tyy show a peak on the anomaly,

with a trough on either side (right to left for Txx and above and below for Tyy). Txy shows a

“saddle-point” with two peaks and two troughs diagonally opposite each other. Txz and Tyz

show a peak and a trough either side of the anomaly (left to right for Txz and above and

below for Tyz). Finally, Tzz shows a peak directly above the target.

Figure 1.1. The circled areas of the gravity gradients in (a) show similar characteristics of the gravity gradients of a vertical prism in (b).

1.3.2 Magnetics as an Exploration Tool

Like the gravitational field, the magnetic field is a vector field (it has magnitude and direction

at a point) and it is a potential field (a mathematical construct presented in Chapter 2).

However, the magnetic field is essentially dipolar in nature (unlike the gravity field). That is,

the gravitational field of the Earth points to the centre (of mass) of the Earth, while the

magnetic field can be viewed as similar to a bar magnet at the Earth’s centre. Figure 1.2 is an

illustration of this. The direction of the field at a point is defined as the direction a

hypothetical magnetic north pole would take if placed at that point in the field. So the Earth’s

South Pole is actually in the direction of the North geographic pole.


Figure 1.2 (a) The gravity field lines point to the centre of mass of the Earth. (b) The magnetic field lines of the Earth are dipolar in nature.

Just as density changes give rise to anomalies of the gravitational field, magnetic

susceptibility changes produce anomalies in the magnetic field. Also, it is possible for

ferromagnetic material to produce its own permanent field (a magnetic polarisation of the

volume).

Historical Background

Again, some important historical figures are mentioned here in connection with the

development of the theory of magnetism. The various units used are named after them.

Andre-Marie Ampere, in 1820, related magnetic fields to electric currents in terms of force.

Today, electrical currents are measured in units of amperes, and the “strength” of a magnet

(the dipole moment) has the units of ampere metres squared (Am2).

In 1828, Carl Friedrich Gauss began research into magnetism and went on to invent the first

magnetometers, along with Wilhelm Weber. The American inventor Nikola Tesla also made

important contributions to the science, inventing the Tesla coil, and an alternating-current

dynamo. Depending on which system is being used, magnetic field strength either has the

units of Gauss, Weber per metre squared, or Teslas (1T = 1Wbm-2 = 104 gauss). The SI unit

for magnetic field strength is the Tesla.


James Clerk Maxwell assembled the mathematical theory of electromagnetism in 1864,

showing the inter-relations of the electric and magnetic fields. Maxwell’s four equations

remain the cornerstone of electromagnetism today, and will be used later in this thesis.

Recent Developments in Magnetic Recording Techniques

The use of magnetics for geophysical exploration is commonplace (Bellott et al., 1991; Negi

et al., 1983; Reynolds et al., 1991; Reynolds et al., 1990), and there are three main types of

magnetometers commonly used today. These are the fluxgate, proton-precession and

optically pumped magnetometers.

The fluxgate magnetometer was developed during the Second World War as a device for

detecting submarines. It contains two coils of wire and analyses the difference in electrical

current to determine the total magnetic field at a point. The fluxgate magnetometer can give

the directional components of the field (to be defined in Chapter 2), as well as the magnitude

of the total field. It can resolve features to around 1nT (Telford et al., 1996).

The proton-precession magnetometer works on the principle of nuclear magnetic resonance.

The instrument contains hydrogen nuclei (protons) that precess around the Earth’s magnetic

field. The protons have an angular velocity (precessional frequency) which is proportional to

the magnitude of the Earth’s magnetic field. The proportionality constant is known to an

accuracy of 0.001% and so measurements are quite precise. The instrument has superior

(increased) resolution compared to the fluxgate magnetometer, improving accuracy by up to

two orders of magnitude (Jensen, 1965). However, the instrument only measures the

magnitude of the total field.

Optically pumped magnetometers operate on the principle of altering energy levels of

electrons in atoms by the application of some energy source. Specific elements (cesium,

rubidium, sodium and helium in particular) have been selected that respond to the energy

levels produced by the Earth’s magnetic field. It is possible to measure total field anomalies

to an accuracy of around 0.01nT. Cesium vapour magnetometers are a common type of

optically pumped magnetometer (Hardwick, 1984).


Magnetic field data can be collected from airborne or marine platforms, or collected as part of

a ground survey. Airborne surveys have the benefit of being able to cover a large area in a

relatively short time, and can be used to cover areas which cannot be reached on ground (e.g.,

over water and in mountainous terrain). An important factor in airborne surveys is the speed

at which the instrument records. The distance the aircraft travels over the measurement

period must be small otherwise there is spatial smearing of the data.

As aircraft produce their own magnetic fields, the magnetometer is usually towed some

distance below the aircraft in a container known as a “stinger”. Alternatively, the sensors can

be placed on wing tips or the tail of the aircraft, with shielding provided by coils of electric

current, or permanent magnets appropriately placed. Data are almost always collected from

flights of parallel lines, with perpendicular lines (tie-lines) being used to help correlate the

data. Data must undergo several corrections before they are useful for interpretation. This

includes correcting for altitude differences, instrument drift and differences in line crossover

points. The Earth’s magnetic field changes with time and this must be corrected as part of

processing by means of using a fixed magnetic base station.

Ground based magnetic surveys can be similar to airborne surveys in that parallel lines of data

are often collected, with cross-lines to help tie the data together. Ground based surveys

however, take a longer time to undertake and for regional work, airborne surveys are

generally more feasible.

The development of SQUIDs (Superconducting Quantum Interference Devices) have allowed

extremely accurate measurements of the Earth’s magnetic field (Zimmerman and Campbell,

1975). SQUIDs are now being used in the development of instruments that measure the

gradient of the Earth’s magnetic field at a point, and hence the gradient tensor. Among the

organisations that are developing tools that measure the gradient of the Earth’s magnetic field

are ORNL (Oak Ridge National Laboratories) in the USA and the CSIRO (Commonwealth

Scientific and Industrial Research Organisation) in Australia.

The ORNL System

The Oak Ridge National Laboratory has undertaken much research in developing a system for

measuring the gradients of the Earth’s magnetic field. Their primary tool, the ORAGS-VG


(Oak Ridge Airborne Geophysical System – Vertical Gradient) has been used to detect

unexploded ordnance from a helicopter platform at a height of 1 to 2 metres (Doll et al.,

2006). The system comprises 8 cesium vapour magnetometers positioned to measure the

gradients of the field as differences between the meters. The data are recorded at a 1200Hz

sample rate. The system also utilises DGPS (Differential Global Positioning System),

allowing the spatial positioning of data to be accurate to within ± 50cm. The altitude of the

helicopter is measured with a laser altimeter, and the orientation of the craft measured with a

fluxgate magnetometer. The pitch, roll and yaw of the craft are measured at 2Hz by a system

of four GPS units (Doll et al., 2003; Gamey et al., 2002). The ORNL are also developing

other instruments, including the “HH” and the “Arrowhead,” both designed for total field

measurements.

Figure 1.3 shows some results from the Badlands Bombing Range (Doll et al., 2006). The

label VG-MEAS refers to the measured vertical gradient. The map shows numerous labelled

objects (unexploded ordnance (UXO)) that the system was designed to detect. The UXO are

characterised by pink peaks, the largest being present in the right hand bottom corner. Each

UXO is labelled, giving its depth (in feet) and orientation.

Figure 1.3. The ORAGS system is being used to detect unexploded ordnance. Each anomaly is labelled and is revealed as a pink peak on the map. Adapted from (Doll et al., 2006).


CSIRO’s GETMAG project

The GETMAG (Glass Earth Tensor Magnetic Airborne Gradiometer) project has been

developed at the CSIRO in Sydney over the last few years and is currently undergoing testing

for field applications. The GETMAG system measures the directional gradients of the Earth’s

magnetic field to an accuracy of 0.01nT/m.

Figure 1.4 shows data collected from the Tallawang South magnetite skarn anomaly. Each

measurement combines 128 samples taken at a sample rate of 200Hz. This shows not only

the measured components, but also the computed values of the gradients for purposes of

comparison. These computed values have come from the Total Magnetic Intensity (TMI) of

the field, which has also been measured. As can be seen on the graph, there is close

agreement between the computed (derived) and the measured data. More information can be

found in Schmidt et al. (2004).

Figure 1.4. Data collected by the CSIRO as part of the GETMAG project. Several different components of the field are shown here.


1.3.3 Advantages of Gradiometry over Total Field Techniques

There are numerous advantages to measuring the gradients of gravity or magnetic fields rather

than just the total field (Schmidt and Clark, 2006). These include:

• Increased resolution of near-surface features and decreased resolution of regional anomalies

• Tighter wavelengths around compact sources

• Easier detection and delineation of pipe-like sources

• Determination of local strike direction (not vertical TMI gradients)

• Determination of which side of a line the source lies (not vertical TMI gradients)

• Rejection of geomagnetic variations

• Less sensitive to aeroplane rotation noise

• Constraining of interpolation between flight lines

• IGRF (International Geomagnetic Reference Field) corrections not necessary

• Direct indication of the Euler structural index (when combined with measurements of the

total field)

There are also some specific benefits to measuring magnetic gradients (Schmidt and Clark,

2006):

• Magnetic tensor gradiometry provides the benefits of vector surveys without the

disadvantages of extreme sensitivity to orientation

• Gradient elements are true potential fields, with mathematical properties that allow

processing techniques common to total field methods, apply to them as well (e.g. field

continuation and reduction to the pole)

• Very rapid sampling rate of SQUID sensors allows unaliased detection of high-frequency

aircraft noise and efficient removal by filtering. Total field magnetometers have much slower

sampling

• Redundancy of tensor components gives inherent error correction and noise estimates

• Wide ranges of new types of processed data are possible

• Direct calculation of the three-dimensional Analytic Signals (Analytic Signals will be

defined in the next Chapter)

• Invariant quantities exist that have higher resolving power than the Analytic Signal

(Pedersen and Rasmussen, 1990)


• Each gradient tensor component represents a directional filter, emphasizing structures in

particular orientations

• Measurement of the gradient tensor allows calculation of parameters unaffected by aliasing

across flight lines

• Superior Euler deconvolution solutions from measured tensor elements with improved

accuracy using true measured gradients along and across lines

• Gradient tensor elements are independent of skewing caused by geomagnetic field direction

which leads to ease of interpretability

• Combination of tensor components gives information on magnetization directions

• Measurement of the full gradient tensor allows rotation of coordinate system, yielding

transformed tensor components that emphasize specified structural orientations

• Direct calculation of compact source magnetic moments is possible

• Improved resolution of pipe-like bodies

• Improved resolution of sources subparallel to flight path

• Improved delineation of North-South elongated sources at low latitudes

• Spinoff applications to downhole magnetics and remote determination of source magnetic

properties in situ.

1.4 Processing of Gradient Tensor Data

Tools for measuring the components of the gravity and magnetic gradient tensor are now

available for geophysical exploration (Doll et al., 2006; Lee, 2001; Murphy, 2004; Schmidt et

al., 2004). With the technology to measure the gradient tensor components comes the need to

understand what is being measured, and how it relates to the physical properties of the

subsurface.

There are numerous ways to analyse gravity and magnetic data. The direct determination of

subsurface properties (e.g., density, magnetic susceptibility) from the observed potential field

measurements is referred to as inversion. A lesser objective, of improving the appearance and

“interpretability” of the data, is referred to as filtering and processing. A variety of such

techniques have been around for a long time (e.g., reduction to the pole, upward continuation,

2nd vertical derivatives, see Chapter 5). These methods have not been fully adapted to date for

the analysis of gradient tensor data. An extremely useful technique of data analysis, which is


implicit in inversion, is the ability to forward model the response of a given subsurface model.

Little gradient tensor modelling so far has been reported in the literature. Generally speaking,

forward modelling and filtering are relatively rapid interpretation techniques, while inversion

is a much more involved and computationally intensive process.

Many software packages exist that allow routine scalar processing of gravity and magnetic

data. Few treat the full gradient tensor. The two main companies in Australia that produce

potential field analysis programs are Encom Pty Ltd and Intrepid Pty Ltd. Some of the

relevant Encom packages are listed here.

• Profile Analyst allows forward modelling, filtering and inversion of gravity and magnetic

field data. It can also incorporate other data, such as known geology and geochemistry.

• ModelVision Pro is widely considered to be the most advanced gravity and magnetic

processing program allowing three-dimensional forward modelling and inversion of gravity

and magnetic data. This program also allows the inclusion of known geology and

geochemistry as constraints for large-scale inversions.

• Noddy, developed by Dr Mark Jessell at Monash University, is used to simulate volumes of

structural geology, visualise them, and determine their gravity and magnetic responses.

Intrepid Geophysics offers software that performs similar tasks, with some differences in

gridding and inversion routines.

• Intrepid is used to grid unevenly spaced data (i.e., data collected as part of a field survey), to

run filters on the data, and for inversion.

• 3D GeoModeller allows detailed geological modelling to produce gravity and magnetic

responses. Geology can be modelled in either two- or three-dimensions.

• Potent is a program that allows gravity and magnetic forward modelling of simple geometric

objects, and an inversion routine for field data.

Other widely used programs include:

• Oasis Montaj from Geosoft Inc. is generally used for the visualisation of spatial data.

• GMSYS from Northwest Geophysical Associates Inc. (NGA) is used for modelling gravity

and magnetic data, but not gradiometry data.


• Geomodel from RockWare® is a freeware program that allows forward modelling of simple

geometric shapes. It does not forward model gradient tensor responses, just the more

common total field or vertical components of gravity and magnetics. It also has a basic

inversion feature.

My research has involved developing new algorithms to help examine potential field gradient

tensor data, and therefore I had to write my own codes to allow this. Two software

development platforms that I have used extensively in preparing this thesis are:

• Matlab™ which offers a high-level scientific programming language that can be used for

gridding, forward modelling, filtering, and inversion.

• Sliceomatic which is a freeware attachment to Matlab™ and is used to visualize three-

dimensional data in this thesis.

1.5 Thesis Objectives

Increased resolution of field data is possible by measuring the gradients of potential fields

(Hansen, 1999; Slack et al., 1967). These improvements in the field measurements should

therefore enhance our understanding of the shallow-subsurface, but by how much remains to

be quantified. This is one of the objectives of this thesis.

The measurement of the components of the gradient tensor is different than measuring a

single component of the field. If five gradients are measured, there is (by definition) more

than one data set that can be used for interpretation. The application of filters to a single data

set is commonplace, but is less common for combinations of data sets. Another objective of

this dissertation is therefore to examine the application of filters to multi-component gradient

data.

Inversion techniques are routinely used to create geological models from geophysical

measurements. There are many different types of inversion routines and as with filtering, it is

common practice to only use one data set for inversion. A third objective of this dissertation

is to select an appropriate inversion routine and determine some guidelines as to how multi-

component inversion can be efficiently conducted.


Some researchers have also highlighted the need for an understanding of some complex

magnetic sources that may exist in near-surface situations (Keith Leslie, personal

communication). These sources, known as multipoles, can be used to simulate areas where

more than one magnetic source is present. The final objectives of this dissertation are

therefore to develop mathematical relationships for multipoles, and to construct an automated

inversion technique (i.e., a process that can be implemented automatically with minimal input

from a user) for detecting and locating such sources.

The specific dissertation aims can be stated thus:

• 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 gradient 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 construct an automated

inversion technique for such sources.

The thesis is divided into 9 chapters and two appendices. In this, the first chapter, I have

provided background perspective and motivation for the study. I also gave an introduction to

regolith exploration in Australia. Chapter 2 contains much of the mathematics required to

understand the gravity and magnetic field responses of various models, and the inter-

relationships between the gradient tensor components. Chapter 3 is concerned with

calculating the theoretical response of a specific 3-D regolith model, and the implications for

actual field surveying. In Chapter 4 I look at more complex features (especially noise or

variability in the magnetic susceptibility value) that may arise in regolith situations and what

this means for geophysical exploration. Chapter 5 is about filter theory, and how information

can be extracted or enhanced from the data computed in Chapters 3 and 4. Chapter 6

develops several inversion techniques for extracting depth and direction information from the

gradient tensor data. These involve Genetic Algorithms (GAs) and Eigenanalysis. Chapter 7

extends the magnetic field theory to more complex source types known as multipoles, and


Chapter 8 introduces a new automated inversion routine that can locate such sources. Chapter

9 is a Discussion and Conclusions chapter. Appendix 1 contains some mathematical notes

regarding multipoles, and Appendix 2 contains a list of publications, conference presentations

and posters that have arisen from this thesis.

Chapter 2: Potential Field Theory 20

Chapter 2: Potential Field Theory, including Vector and

Gradient Tensor Mathematics

2.1 Introduction

In order to model gravitational and magnetic fields, an understanding of the mathematical

laws that govern these fields in vital. Both gravitational and magnetic fields are vector fields,

and therefore the study of vector calculus is crucial to their treatment. This chapter deals with

the mathematical modelling of these fields, showing examples where appropriate. There are

three main sections: some background mathematics, the basic gravity and magnetic formulae,

and extension of the magnetic formulae to cases where the materials produce their own

magnetic field.

2.1.1 Mathematical Treatment of the Potential Field

This first section contains the derivations needed to describe all the different components of a

potential field. As an example, in the previous chapter I stated that a difference exists

between the field quantities gz and gzz. These will be defined mathematically in this section.

A vector field, F, is a potential field if it has zero curl. One can then write F as the gradient of

a scalar potential ϕ , such that (Blakely, 1996; Heath et al., 2003):

F ϕ= ∇ (2-1)

where (in Cartesian coordinates):

i j kx y z

∂ ∂ ∂∇ = + +

∂ ∂ ∂ (2-2)

As F is a vector, it can be written as the vector sum of three Cartesian components:

F i j kx y zF F F= + + (2-3)


Using this notation, the three vector components of the field F and be defined in terms of the

potential as follows:

xFxϕ∂

=∂

, yFyϕ∂

=∂

, zFzϕ∂

=∂

(2-4)

Using the notation g for the gravitational field and B for the magnetic field, the most

commonly measured components of these fields for geophysical exploration are gz and Btmi

(Telford et al., 1996). Btmi is the total magnetic intensity and can be written:

2 2 2Btmi x y zB B B B= = + + (2-5)

As the Earth’s gravitational field at a point is effectively a vector pointing to the centre of

mass of the Earth, the largest component is the z, or vertical, component. This is why gz is

most commonly used, although other components may be used for exploration (Dyke et al.,

2002; Lee, 2001; Murphy, 2004) and will be discussed shortly.

The use of Btmi does not allow directional information to be taken into account for exploration,

as this quantity is just a scalar. Directional information from the vector (and the yet to be

discussed gradient tensor) components allows further interpretation of the field (Foss, 2004;

Rajagopalan and Christensen, 2000).

Less commonly used components of the fields are the three gradients of the field, defined as

the derivatives of the total field with respect to x, y and z (Heath, 2002). These can be written:

F i j kyx zFF F

x x x x∂∂ ∂∂

= + +∂ ∂ ∂ ∂

(2-6)

F i j kyx zFF F

y y y y∂∂ ∂∂

= + +∂ ∂ ∂ ∂

(2-7)

F i j kyx zFF F

z z z z∂∂ ∂∂

= + +∂ ∂ ∂ ∂

(2-8)


Note that these three “gradients” are vectors, and the components of these vectors can be

defined by the gradient tensor of the field (to be introduced in the next section). The

magnitude of each of these gradients can be written:

22 2F yx zFF F

x x x x∂⎛ ⎞∂ ∂∂ ⎛ ⎞ ⎛ ⎞= + +⎜ ⎟ ⎜ ⎟⎜ ⎟∂ ∂ ∂ ∂⎝ ⎠⎝ ⎠ ⎝ ⎠

(2-9)

22 2F yx z

FF Fy y y y

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

(2-10)

22 2F yx zFF F

z z z z∂⎛ ⎞∂ ∂∂ ⎛ ⎞ ⎛ ⎞= + +⎜ ⎟ ⎜ ⎟⎜ ⎟∂ ∂ ∂ ∂⎝ ⎠⎝ ⎠ ⎝ ⎠

(2-11)

These quantities are also referred to as Analytic Signals of the field (Rajagopalan and

Christensen, 2000). These are written:

FxF SIG

x∂

=∂

, FyF SIG

y∂

=∂

, FzF SIG

z∂

=∂

(2-12)

Finally, there exists another analytic signal which is derived from the Hilbert transform of the

Btmi scalar field (Debeglia and Corpel, 1997; Nabighian, 1972; Qin, 1994; Roest et al., 1992).

A simple notation is:

22 2tmi tmi tmi

tmiB B BB SIGx y z

⎛ ⎞∂ ∂ ∂⎛ ⎞ ⎛ ⎞= + +⎜ ⎟⎜ ⎟ ⎜ ⎟∂ ∂ ∂⎝ ⎠ ⎝ ⎠⎝ ⎠ (2-13)

This is useful in interpretation of magnetic field data (Hrvoic and Pozza, 2006; Hsu et al.,

1996; Zhang, 2001), and examples of Analytic Signals will be shown in Chapters 5 and 7.

The three components in equation (2-13) can be written analytically:

xx x xy y xz ztmi

tmi

B B B B B BBx B

+ +∂=

∂ (2-14)


xy x yy y yz ztmi

tmi

B B B B B BBy B

+ +∂=

∂ (2-15)

xz x yz y zz ztmi

tmi

B B B B B BBz B

+ +∂=

∂ (2-16)

The quantities with the double subscripts are spatial derivatives of a particular Cartesian

component of the field, as explained below.

2.1.2 The Gradient Tensor of a Potential Field

Just as a vector field F defines the existence of the three Cartesian components, Fx, Fy and Fz,

it also defines the gradient tensor of the field. The components of the gradient tensor are

found by taking the derivatives of the vector components with respect to x, y and z. Formally,

the definition of a tensor involves the matrix multiplication of two three-component vectors

(Bourne and Kendall, 1977). The gradient tensor of a vector field is therefore a matrix

defined by the gradient operator and the vector field F (Pedersen and Rasmussen, 1990):

xx xy xz

x y z yx yy yz

zx zy zz

x F F Fy F F F F F Fz F F F

⎡ ⎤∂ ∂⎡ ⎤⎢ ⎥⎢ ⎥ ⎡ ⎤∂ ∂ = ⎢ ⎥⎣ ⎦⎢ ⎥⎢ ⎥⎢ ⎥∂ ∂⎣ ⎦ ⎣ ⎦

(2-17)

For gravity and magnetic fields in particular, the following two relations hold:

0F∇ • = 0F∇× = (2-18)

By expanding these out in full, the further relationships for F are obtained:

xy yxF F= , yz zyF F= , xz zxF F= (2-19)

0xx yy zzF F F+ + = (2-20)


The gradient tensor is therefore symmetric and the last equation is Laplace’s equation, which

holds in source-free regions (Blakely, 1996). This leaves five independent components of the

gradient tensor instead of nine, and the gradient tensor can now be written:

xx xy xz xx xy xz

yx yy yz xy yy yz

zx zy zz xz yz xx yy

F F F F F FF F F F F FF F F F F F F

⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥=⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥− −⎣ ⎦ ⎣ ⎦

(2-21)

2.2 Potential Field Relations

Having introduced some of the basic terminology for the various field quantities and vectors

in this dissertation, I now look at specific analytical formulae for certain objects. While the

literature contains many mathematical derivations of potential field formulae (Barnett, 1976;

Bhattacharyya, 1964; Cady, 1980; Jain, 1991; Nagy, 1966; Plouff, 1976; Rao et al., 1981; Rao

and Babu, 1991; Sharma and Bose, 1977; Skeels and Watson, 1949), the following sections

serve to provide a consistent notation for this dissertation.

2.2.1 Gravitational Potential

From equation (2-1), the gravitational field in terms of a potential can be written as follows:

g ϕ= ∇ (2-22)

Alternatively, the potential can be written as an integral of the field:

( )0

g r drr

ϕ ′ ′= ⋅∫ (2-23)

The potential of a gravitational point source can be written (Blakely, 1996):

mGr

ϕ = − (2-24)


where G is the Universal Gravitational Constant (equal to 6.673 × 10-11Nm2/kg2), m is the

mass of the object and r is the distance to the object. The variable r can be written in terms of

x, y and z:

2 2 2r x y z= + + (2-25)

Note that:

2

mGr rϕ∂

=∂

(2-26)

which is a form of Newton’s law of gravitation. I use x, y and z to obtain the three Cartesian

components:

3xmxg G

x rϕ∂

= =∂

(2-27)

3ymyg G

y rϕ∂

= =∂

(2-28)

3zmzg G

z rϕ∂

= =∂

(2-29)

Therefore, the total field vector can be written:

( )3g i j kmG x y zr

= + + (2-30)

Gravitational Point Source

Assuming that the mass introduced previously is confined to a single point in space, it is

possible to assign a value to it and determine the gravitational field at all points around it.

The formulae for the total field vector and its three components have been derived already: it


remains to calculate the gradient tensor components. These are listed below. The diagonal

components of the matrix are:

2

3 5

1 3xx

xg Gmr r

⎛ ⎞= −⎜ ⎟

⎝ ⎠ (2-31)

2

3 5

1 3yy

yg Gmr r

⎛ ⎞= −⎜ ⎟

⎝ ⎠ (2-32)

2

3 5

1 3zz

zg Gmr r

⎛ ⎞= −⎜ ⎟

⎝ ⎠ (2-33)

(Note that 0xx yy zzg g g+ + = as required)

The off-diagonal components of the matrix are:

5

3xy

xyg Gmr

= − (2-34)

5

3yz

yzg Gmr

= − (2-35)

5

3xz

xzg Gmr

= − (2-36)

For illustrative purposes, if I take an area of measurement over the x-y plane for x and y

running from 1 to 100m, and place a mass of 100kg at a depth of 20m, then the following

plots are obtained. The measurements are taken at “surface” level, (z = 1m). The simulated

measurements of the Cartesian components are in Gals, and the gradient tensor components

are in Eö.


Figure 2.1. The gx response of a point source (left) as described by equation (2-27), followed by the gy response (middle, equation (2-28)), and the gz response (right, equation (2-29)). The source here is 20 metres deep and has a mass of 100kg.

Figure 2.2. The gxx response of a point source (top left) as described by equation (2-31). Next to this is the gyy response and the gzz response, as described by equations (2-32) and (2-33) respectively. The gxy response (bottom left) is described by equation (2-34), and the gyz and gxz responses (bottom middle and bottom right) are described by equations (2-35) and (2-36) respectively. The source here is 20 metres deep and has a mass of 100kg.

In Figure 2.1, the two horizontal Cartesian components gx and gy exhibit a peak and a trough

on either side of the source position, while the gz component shows a single peak above the

anomaly. The diagonal components of the gradient tensor exhibit either a peak or trough

above the source position. Two of the off-diagonal components (gyz and gxz) display a peak

and trough either side of the source position, and the gxy component map features a saddle

point above the source.


Noting that the mass of an object is equal to the product of its density and volume, the

following substitution can be made:

m Vρ= (2-37)

into the previous equations.

Assuming a spherical volume, the volume V in equation (2-37) can be replaced with a cubed

radius term as follows:

343

V aπ= (2-38)

where a is the radius of the sphere. By substituting (2-37) and (2-38) into the point source

equations (2-27) to (2-36), the equations for the gravitational response of a sphere are

obtained. Note that the only difference between the equations (and therefore the graphs) is in

magnitude. Thus a sphere can be represented by a point source of appropriate mass at the

centre of the sphere (Blakely, 1996). The only difference occurs inside the sphere, where

Laplace’s equation does not hold, and Poisson’s equation must be used instead.

Rectangular Prism Source

In order to obtain the formulae required to determine the gravitational response of complex

objects, it is necessary to integrate the results of formulae for a point source over some

volume. For a rectangular prism, the point source formulae must be integrated over x, y and z

to determine the gravitational response. The limits of this integration will determine the

boundaries of the prism.

Note that the general equation for the gravity case is:

( )3

2 3i i

i j k

r rK KRr r r

=+ +

(2-39)


where K represents all the constants, and the ri (etc…) are x, y and z (depending on which

formulae is being used). Taking this through a triple integration leads to the following steps:

( )22 2

32

1 1 12 2 2

jk i

k j i

rr ri

i j kr r r i j k

r drdr drr r r

=+ +

∫ ∫ ∫ (2-40)

2

22

1 11

2 2 2

1i

jk

k ji

rrr

j kr r i j k r

dr drr r r

⎡ ⎤−⎢ ⎥ =⎢ ⎥+ +⎣ ⎦

∫ ∫ (2-41)

( )22 2

11 1

2 2 2lnjk i

ik j

rr r

j i j k krr r

r r r r dr⎡ ⎤⎡ ⎤− + + + =⎢ ⎥⎢ ⎥⎣ ⎦⎣ ⎦∫ (2-42)

( ) ( )

22

2

11

1

1 1tan tan ln ln

kj

i

ij

k

rrr

j kkk i i k j j k

i i r r r

r rrr r r r r R r r Rr r R

− −

⎡ ⎤⎡ ⎤⎡ ⎤⎛ ⎞ ⎛ ⎞⎢ ⎥⎢ ⎥− + − + − +⎢ ⎥⎜ ⎟ ⎜ ⎟⎢ ⎥⎢ ⎥⎢ ⎥⎝ ⎠ ⎝ ⎠⎣ ⎦⎣ ⎦⎢ ⎥⎣ ⎦

(2-43)

The analytical result of such triple integration, after substituting in the various upper and

lower limits, is quite unwieldy and need not be written out in full here. A simplification used

in this dissertation is outlined below.

[ ][ ][ ] 2

1

2

1

2

1),,(*

w

w

v

v

uuwvuFF = (2-44)

Equation (2-44) is equal to:

),,(),,(),,(),,(),,(),,(),,(),,(

111112121122

211212221222

wvuFwvuFwvuFwvuFwvuFwvuFwvuFwvuF

−++−+−−

(2-45)

As an example, from equation (2-25):

( ) ( ) ( )222 wzvyuxR −+−+−= (2-46)

and letting:


X x uY y vZ z w

= −= −= −

and i i

i i

i i

X x uY y vZ z w

= −= −= −

(2-47)

the following is obtained:

21

21

21

21

21

22

21

22

21

21

22

22

22

21

21

22

21

22

22

22

21

22

22

22

*

ZYXZYXZYXZYX

ZYXZYXZYXZYXR

++−++++++++−

+++++−++−++=

(2-48)

Therefore, I can now write out the equations governing the gravitational response of a

rectangular prism. Using the above notations, I arrive at the following for the three Cartesian

components of the vector field:

( ) ( )*

1 1tan tan ln lnxZ YZg Gm Z X X Z Y R Y Z RX XR

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

(2-49)

( ) ( )*

1 1tan tan ln lnyZ XZg Gm Z Y Y Z X R X Z RY YR

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

(2-50)

( ) ( )*

1 1tan tan ln lnzX XYg Gm X Z Z X Y R Y X RZ ZR

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

(2-51)

The five independent components of the gradient tensor can therefore be written:

*

1tan ⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎠⎞

⎜⎝⎛= −

XRYZ

Gmg xx (2-52)

*

1tan ⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎠⎞

⎜⎝⎛= −

YRXZ

Gmg yy (2-53)


( )( )*ln RZGmg xy += (2-54)

( )( )*ln RXGmg yz += (2-55)

( )( )*ln RYGmg xz += (2-56)

Again (for illustrative purposes), examples of the fields corresponding to a rectangular prism

with dimensions 20 × 20 × 20 metres are shown in Figures 2.3 and 2.4. The prism is situated

in the middle of the mapped region, with the depth to the top of the body being 2 metres. The

source has a mass of 1.6 × 107kg (corresponding to the above volume and a density of 2g/cc),

and measurements are taken at a height of 1m above the surface.

The shapes of these anomalies are similar to those for a point source (i.e., the arrangement of

peaks and troughs), but the gradient tensor components reveal some further information

regarding the shape and position of the source. Note how the diagonal components of the

gradient tensor yield lineations around the edges of the source. The gxx component marks the

North and South boundaries and the gyy component marks the East and West boundaries. The

gzz component marks all four boundaries. The gyz and gxz components define the North-South

and East-West extents of the body and gxy is effectively illustrating the corners of the

rectangular prism. The Cartesian components (gx, gy and gz) exhibit similar properties, but

they are not nearly as prominent.

Figure 2.3. The gx response of a prism source (left) as described by equation (2-49), followed by the gy response (middle, equation (2-50)), and the gz response (right, equation (2-51)).


Figure 2.4. The gxx response of a prism source (top left) as described by equation (2-52). Next to this is the gyy response and the gzz response, as described by equations (2-53) and (2-20) respectively. The gxy response (bottom left) is described by equation (2-54), and the gyz and gxz responses (bottom middle and bottom right) are described by equations (2-55) and (2-56) respectively.

2.2.2 Poisson’s Relationship – Connection to Magnetics

It is only in the last century that the underlying quantum theory which explains how magnetic

fields arise from matter was developed. This theory is quite complex (Bransden and Joachain,

1992) and will not be discussed here. However, magnetic responses can be calculated from

gravity using Poisson’s Relationship, which connects gravity and magnetic fields (Blakely,

1996). This relationship assumes that changes in density and magnetic susceptibility (to be

defined here) are coincident. The relationship is:

gB dSdζ

= (2-57)

with:


eF kSGρ

= (2-58)

where Fe is the inducing magnetic field, k is the magnetic susceptibility of the source, G is the

Universal Gravitational constant, ρ is the density of the object and ζ is the direction of

magnetisation. Again, the magnetic field (Fe) is measured in Teslas.

In addition to the present (induced) component of the magnetic field, there is also (in the case

of ferromagnetic materials) a remanent component (or past component) acquired at the time

of formation or deposition when the Earth’s magnetic field may have been pointing in a

different direction. It is this total magnetisation (induced plus remanent) which is important.

To define the magnetic susceptibility, I first need to define the magnetic permeability of a

region (μ0 in a vacuum, μ for material media). The magnetic permeability of an object is

defined as the ratio of a resulting magnetic field (or magnetic induction, B) and the magnetic

intensity (H). The units for H are Amps/metre, and those for μ0 are Tesla metres/Amp

(Tm/A) or Henry/metre (H/m). The connecting relations are:

( ) ( )0 0 1B H H M Hkμ μ μ= = + = + (2-59)

where M is the material magnetisation, or dipole moment per unit volume. The magnetic

susceptibility of an object is defined as the ratio of the magnetic permeability of the medium

to the magnetic permeability of a vacuum (4π × 10-7Tm/A), minus one. Ferromagnetic

materials have a high value of susceptibility. Using the previous notation, the magnetic

susceptibility can be written:

0

1k μμ

= − (2-60)

The magnetic susceptibility is therefore a dimensionless quantity, but the “size” of the

quantity depends on which system of units is being used. Using the SI system for analysing

magnetic susceptibilities of material, near-surface (regolith) materials can be expected to have

susceptibility values of around 0 to 10-3 units. Using the “micro-cgs” system for analysing

magnetic susceptibility of material (Sherrif, 2002), this translates to susceptibilities 0 to 100


units. Susceptibility values can also be negative (diamagnetic material), such that the material

magnetisation is in the opposite direction to the inducing field. Ferromagnetic materials can

have susceptibilities 5 orders of magnitude higher (Telford et al., 1996).

Using notation from Heath et al. (2003), equation (2-56) is written in the following form:

( ) ( )B i j k i j kx y z x y zdB B B S g g g

dζ= + + = + + (2-61)

This formula is only useful in its present form if the direction of magnetisation is in the

direction of one of the coordinate axes. This is generally not true, except at the North and

South poles of the Earth (ζ = z) and at the equator (ζ = x). Even at the equator, if there is

some declination of the field, there will be a y component that must be taken into account. I

can therefore replace the directional derivative term in equation (2-58) with a set of weighted

partial derivatives:

( ) ( )i j k i j kx y z x y zB B B S g g gx y z

α β γ⎛ ⎞∂ ∂ ∂

+ + = + + + +⎜ ⎟∂ ∂ ∂⎝ ⎠ (2-62)

The weights α, β and γ can be determined from the inclination (I) and declination (D) of the

field at whatever region of the Earth is being examined. Figure 2.5 shows the relationship

between the quantities.

Figure 2.5. Relationship between the inclination and declination of a field, and three Cartesian components.


The three weights can be determined through the following derivation. Start with the

restriction:

1α β γ+ + = (2-63)

Note that:

tan Dβ α= (2-64)

and:

2 2 tan Iγ α β= + (2-65)

Substituting equations (2-64) and (2-65) into equation (2-63) yields the following:

2 2tan tan 1D Iα α α β+ + + = (2-66)

Now note that:

2 2

cos Dαα β+ = (2-67)

Substituting equation (2-67) into (2-66) yields:

tantan 1cos

IDD

α α α+ + = (2-68)

This yields the following formula for the first weight:

costan cos sin

DI D D

α =+ −

(2-69)

The other weights are trivial to derive and are given by:


tan Dβ α= (2-70)

1γ α β= − − (2-71)

Having determined the weights for the direction of the Earth’s field, I can continue the

determination of the magnetic field due to induction in susceptible media.

From equation (2-62), the Cartesian components of the field can be taken as separate

equations:

x xB S gx y z

α β γ⎛ ⎞∂ ∂ ∂

= + +⎜ ⎟∂ ∂ ∂⎝ ⎠ (2-72)

y yB S gx y z

α β γ⎛ ⎞∂ ∂ ∂

= + +⎜ ⎟∂ ∂ ∂⎝ ⎠ (2-73)

z zB S gx y z

α β γ⎛ ⎞∂ ∂ ∂

= + +⎜ ⎟∂ ∂ ∂⎝ ⎠ (2-74)

Rewriting equations (2-72) to (2-74) I arrive at the following:

( )x xx xy xzB S g g gα β γ= + + (2-75)

( )y xy yy yzB S g g gα β γ= + + (2-76)

( )z xz yz zzB S g g gα β γ= + + (2-77)

where again the second subscript on g stands for differentiation with respect to a particular

spatial co-ordinate. In other words, the three components of the total magnetic field can be

determined from the weighted sum of selected components of the gravity gradient tensor,

assuming Poisson’s relationship holds.


Similar relations exist for the magnetic gradient tensor components, except that derivatives of

the gravity gradient tensor components must be found. The formulae can be found by taking

the derivatives of equations (2-75) to (2-77) with respect to x, y and z. The following

formulae are obtained:

( )xx xxx xyx xzxB S g g gα β γ= + + (2-78)

( )yy xyy yyy yzyB S g g gα β γ= + + (2-79)

( )zz xzz yzz zzzB S g g gα β γ= + + (2-80)

( )xy xxy xyy xzyB S g g gα β γ= + + (2-81)

( )yz xyz yyz yzzB S g g gα β γ= + + (2-82)

( )xz xzx yzx zzxB S g g gα β γ= + + (2-83)

The above quantities all have units of T/m. By deriving the above formulae through different

means, the following relations can also be stated:

iij iji jiig g g= = (2-84)

ijk ikj jik kij jki kjig g g g g g= = = = = (2-85)

For simplicity, from now on in this thesis only the first term in the above relations will be

used if more than one is applicable (e.g., gxyz instead of gzxy). Note that the equations (2-78) to

(2-83) will therefore change notation slightly, although they are the same mathematically.


2.2.3 Magnetic Response of Susceptible Media

Having determined the relations between the various components for the gravity and magnetic

field, it is now possible to present the equations that will determine the response of various

models of magnetically susceptible media. For illustrative purposes in this section, the

inclination of the field is set at 60 degrees, and the declination at 10 degrees. The medium is

assigned a magnetic susceptibility of 0.00001 SI units, and the Earth’s magnetic field has

been assigned a magnitude of 50000nT. All other parameters (position, search area, mass)

will remain unchanged from the earlier gravity examples.

Point Source

Substituting equations (2-27) to (2-29) and equations (2-31) to (2-36) into equations (2-75) to

(2-83), the following equations are obtained for the magnetic field of a point source of

magnetically susceptible material. Equations (2-86) to (2-88) are the Cartesian components

of the vector field as defined by equation (2-3).

2

3 5 5 5

1 3 3 3x

x xy xzB Sr r r r

α β γ⎛ ⎞⎛ ⎞ − −⎛ ⎞ ⎛ ⎞= − + +⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎜ ⎟⎝ ⎠ ⎝ ⎠⎝ ⎠⎝ ⎠

(2-86)

2

5 3 5 5

3 1 3 3y

xy y yzB Sr r r r

α β γ⎛ ⎞⎛ ⎞− −⎛ ⎞ ⎛ ⎞= + − +⎜ ⎟⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟⎝ ⎠ ⎝ ⎠⎝ ⎠⎝ ⎠

(2-87)

2

5 5 3 5

3 3 1 3z

xz yz zB Sr r r r

α β γ⎛ ⎞⎛ ⎞− −⎛ ⎞ ⎛ ⎞= + + −⎜ ⎟⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎝ ⎠

(2-88)

Equations (2-89) to (2-91) below give the diagonal components of the 3 × 3 magnetic gradient

tensor due to a point source.

3 2 2

7 5 7 5 7 5

15 9 15 3 15 3xx

x x x y y x z zB Sr r r r r r

α β γ⎛ ⎞⎛ ⎞ ⎛ ⎞ ⎛ ⎞

= − + − + −⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎝ ⎠ (2-89)


2 3 2

7 5 7 5 7 5

15 3 15 9 15 3yy

xy x y y y z zB Sr r r r r r

α β γ⎛ ⎞⎛ ⎞ ⎛ ⎞ ⎛ ⎞

= − + − + −⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎝ ⎠ (2-90)

2 2 3

7 5 7 5 7 5

15 3 15 3 15 9zz

xz x yz y z zB Sr r r r r r

α β γ⎛ ⎞⎛ ⎞ ⎛ ⎞ ⎛ ⎞

= − + − + −⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎝ ⎠ (2-91)

The off-diagonal elements of the magnetic gradient tensor are given by equations (2-92) to (2-

94).

2 2

7 5 7 5 7

15 3 15 3 15xy

x y y xy x xyzB Sr r r r r

α β γ⎛ ⎞⎛ ⎞ ⎛ ⎞ ⎛ ⎞= − + − +⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠ ⎝ ⎠⎝ ⎠

(2-92)

2 2

7 7 5 7 5

15 15 3 15 3yz

xyz y z z yz yB Sr r r r r

α β γ⎛ ⎞⎛ ⎞ ⎛ ⎞⎛ ⎞= + − + −⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎝ ⎠

(2-93)

2 2

7 5 7 7 5

15 3 15 15 3xz

x z z xyz xz xB Sr r r r r

α β γ⎛ ⎞⎛ ⎞ ⎛ ⎞⎛ ⎞= − + + −⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠ ⎝ ⎠⎝ ⎠

(2-94)

Examples of all these fields are shown in Figures 2.6 and 2.7 (these examples are not related

to any known area of geology, they act simply as illustrations of how the magnetic field

behaves around these source types). The point is at a depth of 20 metres under the centre of

the plot. As with the plots corresponding to the Cartesian components of the gravitational

field due to a point source (Figures 2.1), the magnetic Cartesian components (Figure 2.6) also

exhibit a peak and trough around the body, with Bz showing a single peak. The slight “offset”

of these anomalies is due to the inclination and declination of the inducing field. The gradient

tensor components also share similar characteristics (compare Figures 2.2 and 2.7), with a

slight distortion due to the inducing field. Again, the diagonal components of the gradient

tensor feature a peak or trough above the body (only approximately above in this case), and

the Byz and Bxz display a peak and trough around the source. The Bxy component (as with gxy)

exhibits a saddle point.


Figure 2.6. The Bx response of a point source (left) is described by equation (2-86), followed by the By response (middle, equation (2-87)), and the Bz response (right, equation (2-88)). The inclination of the field is 60 degrees, and the declination of the field is 10 degrees.

Figure 2.7. The Bxx response of a point source (top left) is described by equation (2-89). Next to this are the Byy response and the Bzz response, as described by equations (2-90) and (2-91) respectively. The Bxy response (bottom left) is described by equation (2-92), and the Byz and Bxz responses (bottom middle and bottom right) are described by equations (2-93) and (2-94) respectively. The inclination of the field is 60 degrees, and the declination of the field is 10 degrees.

Rectangular Prism Source

The magnetic response of a rectangular prism follows from earlier equations. The three

Cartesian components of the vector field can be written:

( )( ) ( )( )*

1tan ln lnxYZB S Z R Y RXR

α β γ−⎛ ⎞⎛ ⎞⎛ ⎞= + + + +⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠⎝ ⎠

(2-95)


( )( ) ( )( )*

1ln tan lnyXZB S Z R X RYR

α β γ−⎛ ⎞⎛ ⎞⎛ ⎞= + + + +⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠⎝ ⎠

(2-96)

( )( ) ( )( )*

1ln ln tanzXYB S Y R X RZR

α β γ −⎛ ⎞⎛ ⎞⎛ ⎞= + + + +⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠⎝ ⎠

(2-97)

The three diagonal components of the magnetic gradient tensor can be expressed as follows:

*

2 2

2

1 11 1

1xx

YZX XR X RB S

R Z R R Y RYZXR

α β γ

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

(2-98)

*

2 2

2

1 11 1

1yy

XZY YR Y RB S

R Z R R X RXZYR

α β γ

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

(2-99)

*

2 2

2

1 11 1

1zz

XYZ Z R Z RB S

R Y R R X R XYZR

α β γ

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

(2-100)

The three independent off-diagonal components of the magnetic gradient tensor can be

written:

*

1 1 1xy

X YB SR Z R R Z R R

α β γ⎛ ⎞⎛ ⎞ ⎛ ⎞⎛ ⎞ ⎛ ⎞ ⎛ ⎞= + +⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟+ +⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎝ ⎠ ⎝ ⎠⎝ ⎠

(2-101)

*

1 1 1yz

Y ZB SR R X R R X R

α β γ⎛ ⎞⎛ ⎞ ⎛ ⎞⎛ ⎞ ⎛ ⎞ ⎛ ⎞= + +⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟+ +⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎝ ⎠ ⎝ ⎠⎝ ⎠

(2-102)


*

1 1 1xz

X ZB SR Y R R R Y R

α β γ⎛ ⎞⎛ ⎞ ⎛ ⎞⎛ ⎞ ⎛ ⎞ ⎛ ⎞= + +⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟+ +⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎝ ⎠ ⎝ ⎠⎝ ⎠

(2-103)

Examples of the previous nine equations are shown in Figures 2.8 and 2.9. The prism has

similar dimensions as before (20 × 20 × 20 metres volume), and is in the same position (depth

to top is 2 metres). Again, the field responses hint at the shape of the object, and this can be

seen here in the Cartesian components as well. The peaks and troughs visible in the

horizontal Cartesian components relate to the boundaries of the prism. The “square” shape is

apparent in the Bz component. The gradient tensor components also exhibit the sides of the

object in a similar fashion to Figure 2.4, but they are much better delineated here. This is

primarily due to the fact that the formulae governing these magnetic gradient tensor

components come from derivatives of the gravity gradient tensor equations and will therefore

be more compact.

Figure 2.8. The Bx response of a rectangular prism source (left) is described by equation (2-95), followed by the By response (middle, equation (2-96)), and the Bz response (right, equation (2-97)). The inclination of the field is 60 degrees, and the declination of the field is 10 degrees.


Figure 2.9. The Bxx response of a rectangular prism source (top left) is described by equation (2-98). Next to this are the Byy response and the Bzz response, as described by equations (2-99) and (2-100) respectively. The Bxy response (bottom left) is described by equation (2-101), and the Byz and Bxz responses (bottom middle and bottom right) are described by equations (2-102) and (2-103) respectively. The inclination of the field is 60 degrees, and the declination of the field is 10 degrees.

2.3 Magnetic Dipole Field

In order to describe the magnetic field around an object that is ferromagnetic in nature (i.e.,

materials that retain their own magnetisation after the inducing field is switched off, e.g.,

magnetite and bar magnets), it is necessary to derive new formulae. These are given in

standard physics texts (Cowan, 1968; Halliday et al., 1997) and so will not be repeated here. I

will start from the equations relating to a magnetic field around a dipole source.

Two vectors are needed to describe the field around a magnetic dipole. The first vector

describes the orientation and strength of the magnetic dipole, and is labelled m (units Am2).

The second vector, r, is the position vector from the source point to the field point, i.e., it

describes the direction and distance to each point surrounding the magnetic dipole. The

vectors m and r can be written in terms of Cartesian coordinates as:

kjim zyx mmm ++= (2-104)


kjir zyx ++= (2-105)

and the italicised r (the distance from the dipole to the field point) is:

222 zyxr ++= (2-106)

The relation between vectors m and r is illustrated in Figure 2.10.

Figure 2.10. Only two vectors, m and r, are needed to calculate the magnetic response of a magnetic dipole.

Magnetic field strength (more appropriately referred to as magnetic induction) is measured in

Teslas. Most of the examples in this paper are in nanoTeslas (nT), and as I will be looking at

first, second and third spatial derivatives of these, units also exist in nanoTeslas per metre

(nT/m), nanoTeslas per metre squared (nT/m2), and nanoTeslas per metre cubed (nT/m3).

The magnetic field generated by a static magnetic dipole can now be written:

( )⎟⎠⎞

⎜⎝⎛ −

•= 35

0 34 rrdipolefield

mrrmBπ

μ (2-107)

The μ0 represents magnetic permeability of free space and is equal to 4π × 10-7H/m. Equation

(2-107) is a vector, and can be written in terms of its Cartesian components using the unit

vectors (i, j and k):


kjiBdipoledipoledipoledipole zyxfield BBB ++= (2-108)

By expanding equation (2-107) and substituting into equation (2-108):

( )( ) ( )⎟⎟⎠

⎞⎜⎜⎝

⎛ ++−

++•=++ 35

0 34 r

mmmr

zyxBBB zyxzyx dipoledipoledipole

kjikjirmkjiπ

μ (2-109)

Isolating the three components i, j and k in equation (2-109), the following is obtained:

( ) ( )

( )⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

⎟⎠⎞

⎜⎝⎛ −

•

+⎟⎟⎠

⎞⎜⎜⎝

⎛−

•+⎟

⎠⎞

⎜⎝⎛ −

•

=++

krm

jrmirm

kji

35

35350

3

33

4

rm

rz

rm

ry

rm

rx

πμ

BBBz

yx

zyx dipoledipoledipole (2-110)

A generalised form of the three components of the total field vector can thus be written:

( )⎟⎠⎞

⎜⎝⎛ −

•= 35

0 34 r

mr

iB iidipole

rmπ

μ (2-111)

where i is either x, y or z (not to be confused with the unit vector i) Examples of these three

components of the field are shown in Figure 2.11. The dipole is oriented 45 degrees from

North (up) and has a dipole moment of 1Am2. There is no external field.

Figure 2.11. The Bx response of a dipole source (left) is described by equation (2-111) for i = x, followed by the By response (middle, for i = y), and the Bz response (right, for i = z). The dipole is oriented 45 degrees clockwise from North (up).


The most commonly used form of the magnetic field for geophysical exploration is the total

magnetic intensity (TMI), and is a combination of the three Cartesian components introduced

in the previous section. For a dipole, it can be written:

( ) ( ) ( )222dipoledipoledipoledipoledipole zyxfieldTMI BBBB ++== B (2-112)

An example of this field is shown in Figure 2.12.

Figure 2.12. The total magnetic intensity of a magnetic dipole, with dipole moment equal to 1Am2 in the vertical direction and at a depth of 5m.

Now, equation (2-111) can be extended to determine the equations which describe gradient

tensor responses of the magnetic field due to a static dipole.

If I differentiate equation (2-111) with respect to some spatial component i, I obtain:

( ) ( )⎟⎟⎠

⎞⎜⎜⎝

⎛ •−

•+= 7

2

550 1536

4 ri

rrim

B iiidipole

rmrmπ

μ (2-113)

If I differentiate (2-111) with respect to either of the other spatial variables, j, then I obtain the

following:

( )⎟⎟⎠

⎞⎜⎜⎝

⎛ •−+= 755

0 15334 r

ijr

imr

jmB ji

ijdipole

rmπ

μ (2-114)


Examples of these gradient tensor components are shown in Figure 2.13. The physical

properties of the dipole used are the same as for the three Cartesian components shown in

Figure 2.11.

Figure 2.13. The Bxx response of a dipole source (top left) is described by equation (2-113). Next to this are the Byy response and the Bzz response. The Bxy response (bottom left), Byz and Bxz responses (bottom middle and bottom right) are described by equation (2-114). The dipole is oriented 45 degrees clockwise from North (up).

2.3.1 Line of Dipoles

The solution for a single dipole can be used to represent a spherical source. The formulae

governing the magnetic field response of a magnetic dipole can be extended to create a line of

dipoles (Jain, 1991). A line of dipoles or a sheet of dipoles can be used to approximate other

geological bodies like pipes and sills. This is achieved by integrating the formulae for a

single dipole over x, y or z, depending on the azimuth of the line. Some of the equations have

already been derived. Note that the integral of Bxx with respect to x will yield Bx.

As a line of dipoles must have some direction in space, the equations must be integrated with

respect to that direction. Generally however, it is only possible to integrate in the x, y and z


directions. So some form of co-ordinate rotation must be undertaken in order to create a line

in an axis direction.

Here I present the equations of Bi, Bii and Bij integrated with respect to i, j and k, where i, j and

k can be any of x, y and z. The integration process is reasonably extensive, and a full

derivation will not be presented here. The equations are:

( )( ) 22

11

2 20

2 3

34

ii

i j k ii

i i

m c i m j m k r i m iB dic r cr

μπ

⎡ ⎤− + + −⎢ ⎥= −⎢ ⎥⎣ ⎦

∫ (2-115)

( )( )( ) 22

11

2 3

02 3

3 2

4

jj

j i k ii

jj

i m a m i m k a j m jB dja r ar

μπ

⎡ ⎤− + + +⎢ ⎥= −⎢ ⎥⎣ ⎦

∫ (2-116)

( )( )( ) 22

11

2 3

02 3

3 2

4

kk

k i j ii

kk

i m b m i m j b k m kB dkb r br

μπ

⎡ ⎤− + + +⎢ ⎥= −⎢ ⎥⎣ ⎦

∫ (2-117)

[ ]2

2

11

ii

ii i ii

B di B=∫ (2-118)

( )( )( )

( )( )( )

2

2

1

1

2 2 2

2 30

2 3 2 2 4

3 5

3

4 3 15 20 8

m rj

i jj

iij j i k

j

j m i r j a ma rB dj

i m a j m i m k a aj j

a r

μπ

⎡ ⎤+ • − −⎢ ⎥+⎢ ⎥= ⎢ ⎥− + + +⎢ ⎥⎢ ⎥⎣ ⎦

∫ (2-119)

( )( )( )

( )( )( )

2

2

1

1

2 2 2

2 30

2 3 2 2 4

3 5

3

4 3 15 20 8

m rk

i kk

iik k i j

k

k m i r k b mb rB dk

i m b k m i m j b bk k

b r

μπ

⎡ ⎤+ • − −⎢ ⎥+⎢ ⎥= ⎢ ⎥− + + +⎢ ⎥⎢ ⎥⎣ ⎦

∫ (2-120)

2 2

2

11 1

i ii

ij ji j ii i

B di B di B⎡ ⎤= = ⎣ ⎦∫ ∫ (2-121)


[ ]2

2

11

jj

ij i jj

B dj B=∫ (2-122)

( )( )

( )( )( )

2

2

1

1

2 2

2 30

3 2 2 4

3 5

3

4 3 15 20 8

k

i jk

ijk k i j

k

k m j m i r kb rB dk

ij m b k m i m j b bk k

b r

μπ

⎡ ⎤+ −⎢ ⎥+⎢ ⎥= ⎢ ⎥− + + +⎢ ⎥⎢ ⎥⎣ ⎦

∫ (2-123)

where:

2 2a i k= + , 2 2b i j= + , 2 2c j k= + (2-124)

The usage of the terms a, b and c is to simplify equations (2-115) to (2-123). To apply these

equations, one must first select which component(s) of the field is to be modelled (say Bxy),

and the direction in which the line of dipoles is oriented (say in the z direction). The formulae

governing the response are therefore equations (2-123) and (2-124) for i=x, j=y and k=z.

Figures 2.14 and 2.15 show some numerical examples of equations (2-115) to (2-124) to

calculate the field response of a line of dipoles. The dipole is oriented such that it is

horizontal (parallel to the x-y plane) at a depth of 10 metres (a suitable depth for near-surface

exploration), and has a dipole moment in the north direction of 10Am2. The line runs north

south, and is 5 metres long. The simulated field “measurements” are taken every 1 metre on

(effectively) the Earth’s surface. The grid is 20 by 20 metres and the middle of the source

projects (in the upwards direction) to the centre of this grid.

Note that the signal is strongest around the end points of the line. This line is equivalent to a

long bar magnet, so no field response is expected from the middle. The exception is in Bx, but

this is expected, as Bx is the magnetic field strength in the x direction and will remain constant

along the length of the line.


Figure 2.14. The magnetic response of a line of dipoles shows maximum response at the ends of the dipole, except in Bx, where the whole line is obvious (because the line is oriented north-south).

Figure 2.15. The magnetic gradient tensor response of a line of dipoles shows maximum response at the ends of the dipole.

2.3.2 Sheet of Dipoles and a Prismatic Dipole Source

An obvious extension of this work is to use integration to create a two-dimensional sheet of

dipoles and a three-dimensional prism of dipoles. It should be obvious that these formulae

will be quite long, and care must be taken in both their derivation and their usage. Note that I

have so far presented 9 equations for a line of dipoles. After substituting x, y and z into these

equations, there are 27 equations. As each of these formulae must be integrated with respect


to x, y and z for a sheet of dipoles, this would suggest that there would be 81 equations for the

sheet, and 243 for the prismatic dipole. There is no need to write these in explicit form here,

as the next section will outline a simpler method of calculating the magnetic response of a

prismatic dipole.

2.3.3 Comparison of a Dipole and a Polarised Source

In order to forward model magnetic sources it is necessary to choose between a dipole source

and a magnetically susceptible body that is “polarised” due to the inducing magnetic field.

Both techniques are equally valid and forward modelling a point source and a dipole source

(equations (2-86) to (2-94) and (2-111) to (2-114) respectively), yield similar shaped field

anomalies. The only difference is in the scale (or magnitude) of the anomalies, as for a point

source a magnetic susceptibility value is required as input, and for a dipole source a dipole

moment is used. In fact, by equating the point and dipole source equations for Bx, the

following can be derived:

point dipolex xB B= (2-125)

( )20

3 5 5 5 5 3

31 3 3 34

m re xxF k mx xy xzG r r r r r r

μα β γρ π

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

(2-126)

2 2

03 5 5 5 3 5 5 5

1 3 3 3 1 3 3 34

ex y z

F k x xy xz x xy xzm m mG r r r r r r r r

μα β γρ π

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

(2-127)

This final equation suggests that the dipole moment is effectively equivalent to the three

weighted variables introduced in section 2.2.2, except that the dipole moment has units Am2

and the weights are dimensionless. Assuming an inducing field of 90 degrees (γ = 1), the

above equation can be simplified to:

0

4e

zF k mG

μρ π

−= (2-128)


By substituting values into equation (2-128) for the inducing field (Fe) and the density of the

source (ρ), it is possible to find relations between magnetic susceptibility (k) and the dipole

moment component (mz).

Returning to the prismatic dipole problem (i.e., the large number of equations required to

determine the magnetic response of a prismatic dipole), equation (2-127) shows that the point

and dipole sources are effectively equivalent. As formulae for the magnetic response of a

magnetically polarised rectangular prism have already been derived from the point source

equations (equations (2-95) to (2-103)), these should be equivalent to the prismatic dipole

case.

2.4 Interrelationships Between Various Field Components

It is often possible to calculate various components of the field from other components, the

simplest of which is from Laplace’s equation. Other methods include calculating the vertical

gradient from the horizontal gradients, or the entire gradient tensor from the total field data

(Agarwal and Lal, 1972; Nelson, 1988; Paine, 1986; Pedersen, 1989; Pedersen, 1991).

Another simple example is to use a “first vertical derivative filter” (to be discussed further in

Chapter 5) to calculate gzz from gz. This is basically a map of the differences between points

in a data set. This technique has been used to compare airborne Falcon™ data with measured

ground gravity data and is illustrated in Figure 2.16. Figure 2.16 shows two images, the

image on the right being the Falcon™ data. The image on the left shows processed ground

data, upward continued from the 1st Vertical Derivative values. A large black area which

corresponds to the low gradient is present in both the images. This area represents a

Kimberlite pipe, an igneous intrusion where diamonds may be found. The upper parts of

Kimberlite pipes often have a lower density than the rest of the pipe and many other rocks

(depending on depth of weathering) and therefore produce a gravitational low (for more

information, see http://www.gravitydiamonds.com.au/BHPBFalconBrochure.pdf).


Figure 2.16. Gravity gradient images from processed ground data (left) and the FALCON™ survey (right). The large black area corresponds to a Kimberlite pipe. A scale is shown in the middle.

Nelson (1988) demonstrates that any one component of the gradient tensor needs to be

measured to obtain all five components of the gradient tensor. The remaining four

components can be calculated through use of Fourier transforms. Denoting the Fourier

transform of a function f(x,y) as Φ(f(x,y)) = F(u,v), and the spatial frequency domain variables

as u and v (with k2 = u2 + v2), relations (2-129) to (2-133) hold. Equations (2-129) and (2-

130) show the relationship between gzz and the two components gxz and gyz:

( )( ) ( ) ( ), , ,xz xz zziug x y G u v G u vk

= =F (2-129)

( )( ) ( ) ( ), , ,yz yz zzivg x y G u v G u vk

= =F (2-130)

Equation (2-131) demonstrates the relationship between gxx and gxz, and equation (2-132)

shows the relationship between gyy and gyz.

( )( ) ( ) ( ), , ,xx xx xziug x y G u v G u vk

= =F (2-131)


( )( ) ( ) ( ), , ,yy yy yzivg x y G u v G u vk

= =F (2-132)

The remaining gradient tensor component, gxy, can also be computed from gyz:

( )( ) ( ) ( ), , ,xy xy yziug x y G u v G u vk

= =F (2-133)

In these equations, 1i = − . It is therefore possible to calculate all the components of the

gradient tensor from one component (e.g., from gzz). But this means that any noise present in

measurements of the single component will be transferred into the computed components. If

however the gradient tensor components are measured separately, they are independent of

each other, so they do not contain noise from other components. This will become important

in comparing inversion techniques, later in this thesis.

It is also possible to calculate gx, gy and gz (and therefore g) from gzz through Fourier

transform relations (see Nelson (1988) for mathematics). It is therefore possible to calculate

the entire gradient tensor, the vector field (g) and (by definition) the three Cartesian

components of the field from a single map of any one gradient tensor component (e.g., gzz).

Many scientists therefore question whether it is actually necessary to measure all five

independent components of the gradient tensor, when only one is needed to calculate the rest.

One reason for measuring all five components is given above; noise is not repeated in the

data. If noise is repeated in the tensor components then it may appear as a false target for

geophysical exploration. It will be shown later (Chapter 4) that gravity and magnetic gradient

tensor data taken close to the surface can contain a lot of what appears to be noise, but is

actual geological information.

A second reason for measuring (rather than calculating) the gradient tensor components

separately leads on from this and relates to inversion routines. If repeated noise is present in

the gradient tensor components an inversion routine will want to assign a geological anomaly

to it. However, if the five components of the gradient tensor contain their own noise (and the

inversion routine is trying to fit a geological model to all the components simultaneously),

any geological anomalies produced through the inversion algorithm that correspond to noise

in a single component will be discarded, as the noise is not present in the other components.


2.5 Conclusions

This chapter has developed the formulae (from potential field theory) required for the forward

modelling of gravity and magnetic Cartesian and gradient tensor responses for idealised

structures. The formulae have been presented in general cases for point sources and

rectangular prisms.

The use of Poisson’s relationship for deriving the magnetic equations (from the gravity

equations) has been extended to incorporate an inducing field in any direction. This has been

undertaken using weighed partial derivatives. Poisson’s relationship also illustrates how

some of the gravity gradient equations can be used to calculate directly the magnetic response

(e.g., gxx, gxy and gxz to calculate Bx).

Equations governing the magnetic response of ferromagnetic materials (i.e., materials that

retain their own field) have been derived. The equations governing the magnetic response of

susceptible media have been also been derived through the application of Poisson’s

relationship. I have shown that these two sets of equations are effectively equivalent. As an

example, the magnetic response of a prismatic dipole can also be expressed through the

equations for a prism within an inducing field.

I have also shown that there exists a relationship between the gradient tensor and Cartesian

components through the use of Fourier transforms. This process requires an already complete

data set to operate on; it cannot be used on a single measurement of a field. Calculating field

components through Fourier transforms will also repeat any noise present in the original data.

Finally, the formulae presented in this chapter provide a consistent system of notation for the

mathematics in this thesis.

Chapter 3: Forward Modelling 56

Chapter 3: Modelling Potential Field Gradient Tensor

Responses of Three-Dimensional Regolith Structures

3.1 Introduction

The regolith includes materials such as (but not restricted to) sands, clays, silts and gravels; it

may also contain structures such as palaeochannels, mineral deposits, weathered saprolite and

anomalous debris. Knowledge of the location and properties of these features is beneficial to

land users, including farmers, civil engineers, explorers and miners.

The changes in the physical properties of regolith features can often be quite subtle over small

distances (Clark, 1997; Emerson et al., 2000; Tracey and Direen, 2002), and hence if

exploration within the regolith is to be successful, analysis of this high spatial frequency data

needs to be undertaken. The geophysical responses of these features are generally only found

in the noise window of many current gravity and magnetics exploration data.

This chapter examines the effectiveness of gravity and magnetic tensor data in exploration

through and within a simulated regolith cover. A simulated regolith model is created, from

which the theoretical gravity and magnetic tensor responses are calculated. The range of

output values are compared to published case history responses of current systems (Doll et al.,

2001; Dransfield et al., 2001; Hammond and Murphy, 2003).

3.2 Preliminary Work

By using equations for simple objects such as spheres and prisms derived in the previous

chapter, it is possible to create some simple geological structures to evaluate the behaviour of

the components of the gradient tensor behave. Take the geological example of a fault or

vertical step. A fault is a plane (approximately) of fracture in a geological unit caused by

brittle failure and along which some displacement has occurred (Allaby and Allaby, 1999).

This can be modelled on a computer as two rectangular prisms, one being offset from the

other. The gravity or magnetic responses of rectangular prisms are known so the gravity and

magnetic responses can be determined from equations given in the previous chapter. Figure


3.1 shows a simple dip-slip fault with the TMI component of the magnetic field shown above

it. The inducing field is vertical, the geological material has magnetic susceptibility of 0.0001

SI and the fault throw is 5 metres. Figure 3.2 shows the same model, but this time exhibiting

the Bxx component of the magnetic gradient tensor. Both field maps show a distinct anomaly

around the fault position, with the Bxx component of the gradient tensor being more compact.

Other components of the gradient tensor (Bxy, Bxz, Bzz, Byz) show similar anomalies (this fault

remains constant in the y direction for a large distance, and hence Byy shows little response).

Another common geological scenario is a sill. A sill is a concordant intrusion of igneous

material that does not reach the surface of the Earth. Sills can also be modelled via computer

using equations for rectangular prisms. An example of the Bxx component of the gradient

tensor response of a simple sill is shown in Figure 3.3. The upper part of the sill is 10 metres

long and 2 metres thick (with a depth to top of 10 metres), and the lower part of the sill is 2

metres thick in the x and y direction, and extends to a depth of 100m. The corresponding field

data (compared to the previous example) has a long spatial wavelength, and because the

source is closer to the surface, the magnitude of the response is increased. The remaining

components of the gradient tensor show similar long wavelength responses.

Figure 3.1. The TMI response of a geological fault with susceptibility 0.0001SI and vertical inducing field.


Figure 3.2. The Bxx component of the magnetic gradient tensor due to a geological fault with susceptibility 0.0001SI and vertical inducing field.

Figure 3.3. The Bxx component of the magnetic gradient tensor due to a sill as represented with rectangular prisms.


Just as the magnetic response of a dipole can be integrated over a line to produce a line of

dipoles (see section 2.3.1), the equations for a simple gravity point source can be integrated

over a line to produce a line source of anomalous density. The equations for a gravitational

point source can also be integrated to produce the gravitational response of a ring. The ring

can be integrated a second time to produce a disc, and finally the disc integrated to yield a

cylinder. Such elongate sources can be used to simulate a tunnel (low or zero density), or a

vein of mineralisation (high density). Whilst veins are not simple cylinders in nature, such an

approximation is adequate for the purposes of illustration. An example of a tunnel and the gyy

component of the gravity gradient tensor response is shown in Figure 3.4. The gyy component

is chosen as it produces the response with the largest magnitude. In this case, the selected

gradient tensor response exhibits a peak above the source.

Figure 3.4. The gyy component of the gravity gradient tensor due to a tunnel, as represented by a horizontal cylinder. The peak is approximately 24Eö, which is the highest amplitude response compared to the remaining components of the gradient tensor for this scenario.


Figures 3.5 and 3.6 show two further examples of geological sources that are detectable by

gravity gradients. Figure 3.5 shows a normal fault represented by two slabs, which is

relatively larger than a spherical source in the same area. As the fault has a large gravitational

(gzz) anomaly associated with it, the anomaly caused by the sphere is barely detectable. This

illustrates the difficulty of locating small sources where larger sources are present. These data

were calculated by taking samples of the field at 1-metre intervals. If these data were sampled

at a larger spacing, the small anomaly above the sphere could easily be lost. Figure 3.6

illustrates the gravitational attraction of a dyke, illustrated by the gxx component of the

gradient tensor. The dyke is at a depth of 14 metres and has a density contrast of 2g/cc and is

easily detectable at the surface. In the last two examples, I have selected the gzz and gxx

components to illustrate the field responses. All the components of the gradient tensor show

similarly sized wavelength features, although the diagonal components do tend to have the

greatest signal.

Figure 3.5. Combinations of geologic features illustrates how small, near-surface sources (i.e., the sphere in this example) may not be apparent on a gradient tensor map due to larger sources.


Figure 3.6. Gravitational response (gxx) of a dyke source. The dyke is at a depth of 14m, and has a density of 2g/cc. The field response shows that this theoretical anomaly would be detectable by current acquisition systems.

Each of these geological models are relatively near to the surface of the Earth and produce

responses that should be detectable with modern instruments. However, defining an exact

boundary of what is (and what isn’t) detectable is difficult due to the many variables involved.

Some simple examples are therefore included here. Figure 3.7 shows four plots of depth vs.

density contrast of a spherical gravitational source; the individual plots corresponding to a

different size source. The sphere ranges in density contrast from 0.1 to 5g/cc, in depth from 1

to 10m and the radius is given the values 0.5, 1, 2 and 4m (separate plots). The colour at each

combination of these values dictates how many of the gradient tensor components can detect

the source. Dark blue indicates that the source is detectable by all the components of the

gradient tensor, and red indicates that the source is undetectable. The plot shows a distinct

curve of values that define the boundaries of being detectable for this source type. The source

is almost always detectable at the surface, but not at a depth of 13m.


Figure 3.7. This plot shows the “detectability” of a gravitational spherical source. The sphere has been given a radius of 0.5m (top left), 1m (top right), 2m (lower left) and 4m (lower right). The blue represents combinations of depth, density and radius such that the sphere is detectable with all the gradient components, and red represents combinations that are not detectable by any components.

Figure 3.8 demonstrates the results for a similar experiment, but this time for a cubic source.

The volume of the cube is first set at 1m3 and the density allowed to vary from 1 to 5g/cc and

the depth from 1 to 20m. The experiment is repeated using a larger volume of 10m3. Such a

body is expected to produce stronger gradients than for a sphere of comparable size, because

of the sharp edges. The shapes of the detectability curves given in Figure 3.8, are slightly

different in each case, but essentially show the same result, viz., detectability increases as

density contrast increases, depth of burial decreases and size of body increases.

Small sources, even close to the surface, may still be undetectable. As mentioned in Chapter

1, the regolith often comprises many different materials, which can exhibit similar physical

properties to one another, i.e., there will be less contrast between the physical parameters of

different regolith units. Further forward modelling is therefore undertaken to: (a) take into

account these small variations and (b) illustrate the types of maps that may be expected from

gradient tensor data.


Figure 3.8. This plot shows the “detectability” of a gravitational cubic source. The sphere has been given a 1m3 volume in the left illustration, and a 10m3 volume in the right. The bark blue represents combinations of depth, density and volume such that the source is detectable with all the gradient components, and red represents combinations that are not detectable by any components. 3.3 Features to Include in a Regolith Model

A simulated model has been built up from of many cubic metre blocks (voxels) stacked

together to create the effect of a three-dimensional volume of regolith. Each voxel is assigned

a value for its density and magnetic susceptibility. This creates a three-dimensional model of

regolith with realistic features. This model is not meant to represent any particular known

area of geology, but to simply create a simulation from which numerically calculated gravity

and magnetic data can be determined. Similarly, this model does not contain all possible

regolith features (e.g., ferruginous duricrust, sand dunes, soil profiles, etc…), as this would be

impractical.

Figures 3.9 and 3.10 show selected depth slices of the model, and Table 3.1 shows the density

and magnetic susceptibility values assigned to the separate regolith units. These values have

been extracted from the literature (Clark, 1997; Telford et al., 1996; Tracey and Direen, 2002;

Weast, 1971). The three-dimensional simulated regolith model is divided up into 10 different

geological units, labelled A to J. The model includes soils of different type, a creek running

from west to east through the “northern” part of the model, a palaeochannel feature, buried

mineralisation, land mines and other anomalies. Each unit is assigned a uniform value of

density and magnetic susceptibility in accord with regolith “type” as shown in Table 3.1. The

gravity and magnetic tensor responses are then each simply the sum of the responses from the

individual voxels that make up the model, located at different source positions from the

observation point.


The freeware program “Sliceomatic” provides excellent visualisation for three-dimensional

information and has been used in the past for geophysical modelling (Carey, 2003). As it is

not possible to capture 3-D rotatable displays adequately on paper, the following Figures have

been provided to illustrate the model.

Figure 3.9. Eight selected depth slices of the regolith model.

Figure 3.10. Four selected depth slices of the model using the visualisation package “Sliceomatic”, including three types of alluvium, an acidic soil, a creek, and the uppermost parts of a palaeochannel. Each depth slice is fully labelled.


Table 3.1. Some values extracted from the literature that describe the physical properties of common material found in the regolith.

The state of the art instrumentation that measures components of the tensor, and against

which I need to compare values, are the BellGeospace system (gravity), the FALCON system

(gravity), the GETMAG project (CSIRO) and the Oak Ridge National Laboratory (ORNL)

unexploded ordnance (UXO) detection system (magnetics), which were reviewed in Chapter

1.

From the Bell FTG system specifications (Hammond and Murphy, 2003), it appears possible

to measure gravity vector gradients above 10Eö, with a resolution of around 1Eö. The

FALCON system (Dransfield et al., 2001) is able to measure gravity gradients of –30 to 30Eö,

with (again) a resolution of around 1Eö. Published data from of the ORNL magnetic system

show measured values from 0 to 10nT/m (Doll et al., 2001), and data from the GETMAG

project range from -600 to 600nT/m (Schmidt et al., 2004).

These ranges are based on data that have been averaged (i.e., adjacent readings have been

averaged) in order to increase the signal to noise ratio. This process removes shorter

wavelength variations and therefore degrades lateral resolution.


The ORNL system measures the magnetic gradient components by measuring the difference

between two scalar magnetometers (Doll et al., 2001), while the gravity systems of Bell and

FALCON use accelerometers to determine the vector gradients (Hammond and Murphy,

2003; Lee, 2001).

3.4 Forward Modelling Method

The Cartesian reference frame used to specify source and field points is illustrated in Figure

3.11. The x-y plane is taken as horizontal and the z-axis as vertical, the increasing values of z

representing increasing height. The positions u1 and u2 on the x-axis represent the limits of

the rectangular prism in the x direction, and similarly v1 and v2, and w1 and w2 represent the

bounds in the y and z direction. The measurement of the gravity or magnetic gradient tensor

is made at the field point P(x,y,z).

Figure 3.11. The variables required to create gravity and magnetic forward models of a rectangular prism. Many rectangular prism elements are used to build up a complex model.


Equations (2-52) to (2-56) and (2-98) to (2-103) are all that are required to calculate the

components of the gravity and magnetic gradient tensors for a rectangular prism. Note that a

voxel in this study is simply a prism for which the side lengths are equal (i.e., a cube).

The five gravity and five magnetic tensor components are calculated for the specified three-

dimensional regolith model described previously by summing the individual responses. The

resulting range of values should indicate if current gravity and magnetic tensor technology is

capable of detecting and discriminating regolith features.

Care must be taken in order to obtain results with the correct units. The formulae often

require the mass of each voxel rather than the density. With the chosen voxel size of one

cubic metre, the density values in g/cc need just be multiplied by 1000 to convert to

kilograms. All distances must be in metres, the magnetic susceptibilities in SI units and the

corresponding units for the Universal gravitational constant (6.673 × 10-11Nm2/kg2). Finally,

it is assumed that the inducing magnetic field is equal to 50,000nT, which is an average value

for the magnitude of the Earth’s magnetic field (Telford et al., 1996). As this is simply a

constant multiplying factor in the equations, the results can be simply scaled (divided through

by this amount and multiplied by another amount) to obtain the anomaly pattern for a

different magnitude of the Earth’s field.

A simplified version of the regolith model consisting of large prisms was imported into the

program “Potent” (version 3.09) to gauge roughly the expected field responses. Figure 3.12

shows the gz response and Figure 3.13 shows the Btmi response. Potent could not be used for

the complex model introduced previously. Matlab™ was used for the complex modelling as

this provided a suitable interface with the model data.


Figure 3.12. The gz response of a simplified regolith model with a peak value of about 1mGal.

Figure 3.13. The Btmi response of the simplified regolith model with a maximum value of around 3nT, and a minimum of around –1nT.


3.5 Results and Discussion

This section shows the simulated gradient tensor responses for the 3-D model of Figures 3.9

and 3.10. The values were computed for ground level (z = 0). Samples of the field were

taken every 1m in both the x and y direction. This is undertaken to simulate a detailed ground

survey. Each diagram shows a separate gradient tensor response, with a scale to illustrate the

range of the response. This range will be used to help determine if any of the gradient tensor

responses are sensitive enough to detect features present in Figures 3.9 and 3.10.

It is immediately apparent on each figure that numerous regolith and landform features are

visible. Almost all of the responses show contrasts in gradient values that can be used to

define boundaries between such units, although some of the gradient tensor maps appear

rather smooth (e.g., the off-diagonal components of the gravity gradient tensor), and the

boundaries between the units are less obvious.

Figure 3.14. The gxx component of the gravity tensor as measured at the surface. Values range from –0.1 to 0.03Eö.


Figure 3.15. The gyy component of the gravity tensor as measured at the surface. Values range from –0.1 to 0.01Eö.

Figure 3.16. The gzz component of the gravity gradient tensor as measured at the surface. Values range from –0.01 to 0.2Eö.


Figure 3.17. The gxy component of the gravity gradient tensor as measured at the surface. Values range from –0.06 to 0.04Eö.

Figure 3.18. The gyz component of the gravity gradient tensor as measured at the surface. Values range from –0.08 to 0.08Eö.


Figure 3.19. The gxz component of the gravity gradient tensor as measured at the surface. Values range from –0.08 to 0.08Eö.

Figure 3.20. The Bxx component of the magnetic gradient tensor as measured at the surface. Values range from –800 to 600nT/m.


Figure 3.21. The Byy component of the magnetic gradient tensor as measured at the surface. Values range from –300 to 500nT/m.

Figure 3.22. The Bzz component of the magnetic gradient tensor as measured at the surface. Values range from –800 to 800nT/m.


Figure 3.23. The Bxy component of the magnetic gradient tensor as measured at the surface. Values range from –150 to 150nT/m.

Figure 3.24. The Byz component of the magnetic gradient tensor as measured at the surface. Values range from –1000 to 600nT/m.


Figure 3.25. The Bxz component of the magnetic gradient tensor as measured at the surface. Values range from –800 to 800nT/m.

Figure 3.26. The Bxx component of the magnetic gradient tensor as measured from a height of 10m. Values

range from –0.5 to 1nT/m.


Figure 3.27. The Bzz component of the magnetic gradient tensor as measured from a height of 10m. Values

range from –0.2 to 1.6nT/m.

Figure 3.28. The Bxz component of the magnetic gradient tensor as measured from a height of 10m. Values

range from –1 to 1.5nT/m.


Figure 3.29. The Bxx component of the magnetic gradient tensor as measured from a height of 80m. Values range from –0.15 to –0.02nT/m.

In all of the simulations, samples of the field were computed at 1m intervals in the horizontal

plane, whether “measured” at ground level or at a flight height of 10m or 80m. In reality,

such detailed surveying is expensive, and less detailed surveying commonly takes place, e.g.,

the 300m sample interval for the FALCON® AGG system mentioned in Chapter 1. Also, the

complex simulations are based on a model which is only 20m deep, so longer wavelength

features from deeper sources are not examined for this detailed regolith scenario. The simple

models in section 3.2 also reflect near-surface geological scenarios, and show similar

wavelength features.

The responses of airborne systems are wavelength-dependent. That is, they measure gradients

from separated accelerometers. Therefore any spatial wavelengths shorter than twice the

distance between the accelerometers will not be resolved. This is not such a problem for

high-altitude surveying, but is an important consideration for near-surface measurements.

This is evident in the simulations, as there are regolith features that produce wavelengths as

small as a few metres.


From each of the figures that correspond to simulated measurements at the ground surface, it

is possible to detect the boundaries of some of the regolith units (units A, C, and D). The

creek is obvious to pick, because it has a significant density and magnetic susceptibility

contrast with the surrounding media and occurs at ground level.

The other near-surface features, the ordnance, are also recognised as the large peaks on the

map. In reality, land mines are not one metre cubic blocks which “outcrop”, so the simulated

anomalies in the gravity and magnetic field caused by these ordnance are unrealistic. They

would normally be buried and much smaller. However, in the case of the magnetic gradient

tensor responses, these “land mines” produce reasonably large responses, indicating that a

very detailed survey may indeed succeed in detecting small unexploded ordnances.

In the interests of forensic geophysics (Powell, 2004), a simulated cadaver has been included

in the model as a buried near-surface target. Like the land mines, while not being a perfectly

simulated feature, it is simply added to see if any response is detectable. The cadaver was

represented by two adjacent voxels, with a slightly decreased density, and slightly higher

magnetic susceptibility than the surrounding media. It is not particularly noticeable on any of

the responses (Figures 3.14 to 3.25), although on careful inspection, a small peak is visible at

approximately the 80-metre mark on the west-east axis, and at the 60-metre mark on the

north-south axis on some of the plots.

The boundaries between the major regolith units were made quite irregular, in order to

simulate realistic scenarios. Often soil units intermingle or show gradational rather than sharp

contacts. It is remarkable that this level of detail has been recovered in the potential field

images (albeit noise-free), at least for the boundaries close to the surface.

The large palaeochannel (unit G), which occurs at mid-depth in the model, is not visible on

any of the tensor component images. This suggests that regolith features more than a few

metres below the surface may not be detectable via surface measurements of this type. The

boundaries between the deeper regolith units are also not detectable, but these features (and

others) may become visible with potential field continuation (Telford et al., 1996). Upward

continuation is a process used to enhance low frequency data, and hence accentuate deeper

structures. Such low-pass filters may prove useful in further processing of the data computed

from this study. Downward continuation may also enhance deeper features, at the expense of


amplifying noise (see Chapter 5). From all the simulated data, it appears that only very near-

surface features are easily detectable on the unprocessed images.

It is worth noting some of the similarities and differences between the plots of the

components of each (i.e., gravity and magnetic) gradient tensor. For example, the amplitude

of the model is less in the off-diagonal components gxy, gyz and gxz (Figures 3.17 to 3.19),

compared to the diagonal components of the tensor (Figures 3.14 to 3.16). Some edge effects

are noticeable on the off-diagonal components gyz and gxz, and dominate the maps. Also

noticeable in some of the maps of the magnetic gradient tensor is the lack of response of the

simulated land mines. They are only noticeable on the Byz and Bxz responses (Figures 3.24

and 3.25). This adds further argument for the measurement (rather than calculation from

total-field measurements) of the independent components of the gradient tensor (Pedersen and

Rasmussen, 1990), as these responses appear to be hidden in all the other components

(Figures 3.20 to 3.23).

The range of data is highly important in order to ascertain which gravity and magnetic

gradient tensor measurements may be useful for exploration within the regolith. All of the

simulated gravity responses in this study are in the range of -0.2 to +0.2Eö, while the actual

values measured during field surveys undertaken to date show values ranging from –30 to

+30Eö, with a resolution of around 1Eö. This suggests that current acquisition systems for

measuring the components of the gravity gradient tensor are not sensitive enough to detect

features within the regolith. In apparent contradiction to this are the results shown in Figures

3.4 and 3.5 (the tunnel and the fault), suggesting that some near-surface features should be

detectable. However, Figures 3.7 and 3.8 give some guidelines as to how detectable these

sources should be, and the physical parameters of the tunnel and the fault plot in the blue

areas of these figures (suggesting that they are detectable). The regolith model shown in

Figures 3.9 and 3.10 contains changes in density (at the surface) of the order of 0.2 to 0.5g/cc.

Such contrasts for cubic bodies may be undetectable from Figure 3.8 (i.e., they plot in the

red). There is therefore a fine line as to what is and isn’t detectable from gravity gradiometry

for near-surface exploration, with Figures 3.7 and 3.8 illustrating the boundary for some

different source types. Also note that factors such as field conditions and instrument

reliability play a vital part in what will ultimately be detectable and undetectable.

The numerical simulations are built up from numerous assumptions, including the fact that the

regolith model is constructed from cubes and that there is no underlying geology to produce


an added response. Long-wavelength underlying geology will contribute little to these

simulations, as adding an almost constant scalar value to each data point will not affect the

relative response, or resolution.

The situation for the magnetic gradient tensor components is quite different from the gravity

case, with the simulated values being larger than some actual measurements. The modelled

values range from –1200 to +600nT/m, while the experiments run by Doll et al. (2001)

suggest much lower values, in the order of 0 to +10nT/m. The magnetic gradient tensor

components are measured by determining the differences between two scalar magnetometers

(Doll et al., 2001), so these large gradients should (theoretically) be detectable. However,

work by (Schmidt and Clark, 2006), suggests that these simulated values are close to what

should be expected. The GETMAG project has recorded values in the order of –600 to

+600nT/m (Schmidt et al., 2004), values similar to those calculated in this work.

Gradient tensor responses have also been calculated at different flight heights. Figures 3.26 to

3.28 illustrate the Bxx, Bxz and Bzz components at the height of 10m. These figures

demonstrate how the gradient field has weakened as the detector has moved away from the

source. For a flight height of 10m, it is possible to detect some of the boundaries between

regolith units, although these are quite blurred. The range of values suggests that these

boundaries may be detectable, only if the recording instrument had a resolution of less than

1nT/m. As is apparent in Figure 3.29, no regolith features are discernable at a flight height of

80m. The long-wavelength features visible on this map are due to the prism model as a

whole.

In addition, forward modelling the gradient tensor response of the simplified regolith model

yields similar ranges of values for the magnetic gradient tensor components. Figure 3.30

shows the Bxx response of this model, with a minimum of (approximately) -150nT/m and

maximum of 150 nT/m. These values are somewhat less in magnitude to the forward

modelling shown in Figures 3.20 to 3.25. There are further similarities in Figures 3.20 and

3.30, including the fact that the signal is mostly seen as spikes around the boundaries of the

regolith units.


Figure 3.30. The Bxx component of the magnetic gradient tensor due to the simplified regolith model yields values comparable to the complex model.

3.6 Conclusions

The range of values obtained from the gravity gradient tensor simulations suggest that current

acquisition tools for the gravity gradient tensor may not succeed in measuring useful data for

very-near-surface regolith exploration (i.e., the complex regolith scenario shown in Figures

3.9 and 3.10), but would succeed if the source has a large density contrast with the

surrounding media (e.g., the tunnel in Figure 3.4). Figures 3.7 and 3.8 give some guidelines

as to how detectable a gravitational source is from measurements of the gradient tensor,

depending on its shape, depth and density.

The magnetic gradient tensor, however, yields values similar to those which have been

actually measured to date, and so it should be possible to measure these gradient tensor

components for successful use in near-surface regolith exploration. Simulations suggest that

measurements should be useful for delineating boundaries between regolith units.

Also, not all the features in the regolith model were present in all the maps of the components

of the gradient tensor (i.e., the landmines in the magnetic case), adding weight to the


argument to measuring all the independent components of the gradient tensor for

interpretation. My regolith model excluded ferruginous duricrust, a situation quite common

in parts of Australia (Emerson et al., 2000). Such duricrust has high magnetic susceptibility

and density and could dominate the geophysical signatures. Other “sources” at depth, such as

those illustrated in Figure 3.9, would be more difficult to locate beneath a duricrust cover. An

analogous situation occurs in electromagnetics where conductive overburden can mask or

screen the responses from subsurface ore bodies.

Chapter 4: Incorporating Uncertainty 83

Chapter 4: Incorporating Physical Property Variability in

Potential Field Tensor Forward Modelling of Near-Surface

Regolith Structures

4.1 Introduction

Regolith units and landforms contain Earth materials having different physical properties.

The first part of this chapter examines magnetic susceptibility variations of the near-surface

material at Fowlers Gap, to see if there is any correlation between susceptibility and regolith

units and landforms. The modelling carried out in Chapter 3 was idealised in that magnetic

susceptibility was held constant in each rock or soil unit. Natural variability may mask some

of the boundaries and features being sought (Puranen et al., 1968; Stevens, 1999). I will focus

in this chapter on magnetic modelling incorporating susceptibility variability. Gravity

modelling will not be pursued because it was shown previously that the regolith signatures are

really quite small.

Fowlers Gap is located approximately 100km north of Broken Hill in New South Wales,

Australia (see Figure 4.1). It is an arid zone research station for which many detailed regolith

landform maps have been constructed. Therefore, for the purposes of this study, I will only

be looking at the surface part of the regolith, and determining what relationship (if any) exists

between surface magnetic susceptibility and regolith materials. Primarily, I want to see how

the regolith distorts the magnetic field of the Earth. For this, the magnetic susceptibility of

regolith materials must be examined. There are other magnetic properties of the regolith,

such as remanence and direction of magnetisation, that could be taken into consideration

(Emerson et al., 2000; Emerson and Macnae, 2001), but for the purposes of this chapter, I

want to primarily look at changes in magnetic susceptibility.

Figure 4.1. Fowler’s Gap is located approximately 100km North of Broken Hill in NSW, Australia.


4.2 Fowlers Gap Measurements Method

In order to collect as much data as possible, I simply walked along various traverses of

regolith features (channel deposits, drainage systems, drainage depressions to name a few),

taking readings every few (between 2 and 10) metres. Each data point consisted of an easting,

a northing and a single magnetic susceptibility reading. The magnetic susceptibility reading

was taken from an analogue hand-held magnetic susceptibility meter (Geoinstruments

susceptibility meter JH-8 no. 089, made in Finland). The precision and accuracy of such a

device is not known, so placing error bars on the data is not possible. Some authors (Emerson

et al., 2000) recommend taking numerous measurements at each location but this was not

attempted in this study. The meter was designed for hand samples, so placing it against the

ground should pose no difficulty (Susceptibility values can range into negative numbers, but

all the readings taken at Fowlers Gap were positive). The readings represent the dominant

surface material (almost always a soil). The data were entered into a computer and plotted as

colour-coded data points. This was superimposed over the aerial photograph and a regolith

map of the area. A magnetic susceptibility histogram was also created, outlining the

distribution of data values.

The geology of Fowlers Gap is composed mostly of Adelaidean metasediments (Hill and

Roach, 2005). These include quartzites, shales, dolomitic shales and some dolomite. Some

of the bedrock is ferruginous (enriched in iron oxyhydroxides), suggesting some magnetic

susceptibility values in this area may be relatively high. The majority of regolith material is

related to the bedrock, or in the case of alluvium, from sources not found in Fowlers Gap. It

is also related to weathering, as weathering can drastically alter the susceptibility of magnetic

materials.

4.2.1 Results

In total, 338 data points were acquired. Figure 4.2 shows the collected data overlain on an

aerial photograph of the area. Figure 4.3 is a histogram that shows the range of the data.

Using a logarithmic scale, the mean magnetic susceptibility is 1.7395, and the standard

deviation is 0.2509. The recorded values give a mean magnetic susceptibility of


approximately 79 (× 10-5 SI), and the median value of 50. The standard deviation is about 33.

Most of the magnetic susceptibility values are quite low; the distribution tapers off above 100.

In Figure 4.2 red corresponds to high values, and blue to low values. Figure 4.4 shows the

data overlain on a regolith map of the area. Various sections of the data have been

highlighted that will be discussed in a later section. Magnetic susceptibility distributions are

typically presented after taking the logarithm (base 10) of the data (Irving et al., 1966) and

this has been undertaken for the histograms here.

4.2.2 Discussion of Features Depicted in the Map

Area 1 is within an alluvial channel (regolith code ACah). The values for magnetic

susceptibilities here are relatively high, and have a large range of values. Susceptibility

readings vary from 95-230 (× 10-5 SI). Alluvial channels contain sediments from many

sources, and may therefore explain the large range of readings.

Figure 4.2. An aerial photograph of Fowlers Gap mapping area, with grid size 1 kilometre squared. The collected data has been overlain on the map. Blue points represent low magnetic susceptibility and red points represent high magnetic susceptibility.


Figure 4.3. This histogram shows the variation of data in the sampled area. There are 338 data points, mean susceptibility 79 and median 50.

Figure 4.4. The data has been placed over a regolith map of the area (Hill and Roach, 2005). The numbered ovals are areas that are discussed here. The coded symbols represent different regolith types and features (see Table 4.1).


The readings start to level out after leaving the Alluvial Channel. Area 2 is slightly weathered

bedrock, on an erosional rise. With bedrock present, it is reasonable to suggest that the

readings obtained in this area are representative of the bedrock. A road (Fm) is present in this

area, and similar readings are obtained on either side of it.

Area 3 is quite small, but contains a lot of variation in magnetic susceptibility, regolith type

and landforms. The regolith types are alluvial sediments, channel deposits and highly

weathered bedrock. The primary landforms are depositional plain, erosional plain and

drainage depression. Magnetic susceptibility units range from 50 to 145 (× 10-5 SI).

Area 4 is part of a sheet flow deposit on a depositional plain. The unit is large (approximately

100 metres across), and displays constant range of susceptibility data. There is some variation

towards the middle of this area which suggests (at least in this case) that magnetic

susceptibility measurements will not suffice on their own as a method of defining regolith

types and landforms.

From area 5 and to the East, the magnetic susceptibility values tend to remain at lower levels

(generally less than 80). The reason for this change is not obvious. The regolith type and

landform type change gradually from colluvial sediments on an erosional rise to colluvial

sediments on an erosional plain. Technically, the only change here is in the slope of the

surface, but the source for the sediments deposited here must also be taken into account. Note

that west of area 5, there is a system that relates to the deposits from the channel at area 1

(this channel extends above the map and to the east). To the east of 5, the erosional rise may

act as a barrier against further sediments from the east. This may be the cause for the change

in the readings here.

Area 6 is on a slope, and exhibits combinations of alluvial sediments in drainage depressions

and colluvial sediments on a low hill. The magnetic susceptibility readings in this area are the

lowest in the area (down to 10 (× 10-5 SI) units) and are all below 60 (× 10-5 SI). This does

not mean that all Aed and Cel units everywhere will always have low magnetic susceptibility

units; this is only a reflection of the sediments in this area.

Area 7 represents a cross-section through the alluvial sediments in drainage depressions and

the colluvial sediments on a low hill. This was undertaken to see if there was a way of


picking the boundaries of such units. This does not seem to be the case, as there does not

appear to be any correlation here. The changes in magnetic susceptibility here seem not to be

dependent on the regolith type and landform, although the values remain reasonably constant

in the area.

Area 8 contains the highest value obtained in the mapping area. One single value was as high

as 500 (× 10-5 SI). The regolith type and landform is SSer, slightly weathered bedrock on an

erosional rise. This may be related to the ferruginous saprolite reported in the area,

characterised by micaceous shale with minor quartz veins, sandstones and dolomite.

Area 9 is a section through an alluvial drainage depression. This drainage system contains a

fairly constant set of data, with some high values occurring (up to 110 (× 10-5 SI)). The

drainage system contains run-off from the low hills (i.e., from the same source), so the fairly

constant set of values is understandable.

Finally, area 10 contains numerous features. There is some bedrock to the east of the area

that has small values of magnetic susceptibility associated with it. This suggests that the

bedrock changes over the total mapping area. The main discussion point about this area is

that the changes in magnetic susceptibility do not appear to be related to the change in

regolith type and landform. It should be kept in mind that the area in question comprises

mature metasediments. The results would be quite different for volcanic regions having

higher levels of regolith magnetisation. A straightforward relationship (between magnetic

susceptibility and regolith units) is not immediately obvious at Fowler’s Gap. Further

statistical analysis is given in the next section, assigning mean susceptibility and standard

deviations to each regolith unit.

4.2.3 Statistical analysis of regolith units and landforms

In the Folwers Gap area, I sampled nearly every type of regolith unit and landform present.

Table 4.1 provides an explanation of the regolith code and lists the various features, the

number of samples taken, the average magnetic susceptibility value and standard deviation for

each unit.


Table 4.1. The table shows the regolith units/landforms seen at Fowlers Gap, and their corresponding average magnetic susceptibility values and standard deviations, determined from the number of samples shown.

The obvious question arising from this analysis is whether these values can be used to define

characteristic histograms for regolith units? This may be possible, as the histograms of these

data sets show. Figures 4.5 to 4.7 show the histograms for the three of the regolith

units/landforms. These plots have been created from taking the logarithm of the magnetic

susceptibility values (Irving et al., 1966).


Figure 4.5. Histogram corresponding to magnetic susceptibility values taken of alluvial sediments that are in drainage depressions.

Figure 4.6. Histogram corresponding to magnetic susceptibility values taken of colluvial sediments on an erosional low hill.


Figure 4.7. Histogram corresponding to magnetic susceptibility values taken of Sheetflow sediments on an erosional plane.

As can be seen in Figures 4.5 to 4.7, these regolith units/landforms have characteristic

frequency polygons (or histograms) of magnetic susceptibility, initially suggesting it may be

possible to identify a similar regolith unit/landform from an equivalent data set. However, the

range of values often overlap (as can certainly be seen in the overall histogram in Figure 4.3),

so this may become problematic. Also, if not enough samples are taken (such as SSer), or if

the histogram does not represent a standard distribution (such as SSep), it will be harder to

associate a regolith unit/landform to the plot.

4.2.4 Conclusions from Fowlers Gap Susceptibility Mapping

A hand-held magnetic susceptibility meter was taken to Fowlers Gap in New South Wales

and near-surface measurements taken at close spacing along two profiles. The data suggests

that magnetic susceptibility measurements can be used to describe (or characterise) the area in

a qualitative sense, although a direct quantitative relationship between regolith type,

landforms and susceptibility is not apparent.


4.3 Extended Forward Modelling

Having determined “real” magnetic susceptibility values (and hence noise from the standard

deviation calculations) from Fowlers Gap, it is now possible to extend the forward modelling

scenarios from Chapter 3 to more realistic simulations of regolith geology, in which physical

property variability is incorporated. This is done by assigning each of the regolith units in the

model from the last chapter with properties of the regolith units described above.

Table 4.2 lists the regolith units A to J (as introduced in Chapter 3) and their new descriptors,

which will be used for further forward modelling.

Table 4.2. Table outlining which regolith landforms are to be used for further modelling.

The forward modelling algorithm assigns the average magnetic susceptibility value for each

regolith unit to each voxel in the model. Added to each voxel is the standard deviation of

each unit, multiplied by a random number, taken from a Gaussian distribution. This allows

the total regolith landform units to contain realistic noise, in the form of random physical

property variation in each unit. Figures 4.8 and 4.9 show selected forward model responses.

The remaining components of the gradient tensor are similar in nature to these and have not

been reproduced here.


Figure 4.8. The Bxx component of the gradient tensor response of the regolith scenario with noise added.

Figure 4.9. The Bxy component of the gradient tensor response of the regolith scenario with noise added.

From these figures it is apparent that only the alluvial channel feature is visible using the data

generated for the noisy model. All of the other features, e.g., the landmines, the different soil

types, are not discernable on any of the maps; they are lost within the noise “speckle”.


However, also note that the responses have increased in magnitude. For Bxx, instead of values

ranging from –800 to 600nT/m (as in the previous chapter), values ranging from –1000 to

1500nT/m (for Bxx) were obtained. The signal is also increased for Bxy, and the rest of the

gradient tensor components. This is primarily due to the new magnetic susceptibility values

for the regolith types, which are often two or more orders of magnitude higher than the

previously used values. This does not affect the results from Chapter 3, as these magnetic

responses are still detectable.

It is also important to note here that many current magnetic gradiometers have a target

resolution of 0.01nT/m (Schmidt et al., 2004). If noise of this magnitude were added to the

figures above, very little change would occur in the plots.

As well as adding geological noise (uncertainty) to the data, it is possible to include the effect

of ferromagnetic material (for which formulae were derived in Chapter 2) to the simulation, to

see how it adds to the overall response. In the following simulation, a dipole source has been

added in the position where the mineralisation occurs (depth 12 metres, x (south-north)

position 57 meters and y (west-east) position 80 metres). The mineralisation has been given a

dipole moment of 10Am2 in the x direction. Figures 4.10 and 4.11 show selected components

of the gradient tensor (Bxz and Byz) of this scenario. The mineralisation is apparent in these

figures as the slightly elevated peaks and troughs apparent around the coordinates mentioned.

Figure 4.10. The Bxz component of the gradient tensor response of the regolith scenario with noise and a dipole representing the mineralisation.


Figure 4.11. The Byz component of the gradient tensor response of the regolith scenario with noise and a dipole representing the mineralisation.

The Bxz map here locates the dipole best, as a peak occurs directly above the simulated

mineralisation. The application of filters in the next chapter will attempt to extract more

information on source depth.

Finally, representing the landmines as randomly oriented dipoles has a surprising effect.

Even with very small dipole moments (the magnitude of m in the order of 0.1Am2), the

magnetic response for the landmines dominates the gradient tensor maps. Figure 4.12 shows

the Bxy response.

Note also that the magnitudes of the responses have increased again. They are now on the

order of 104nT/m. This relates to the close proximity of the field measurements to the

sources, as seen from the formulae for the responses of a dipole and prismatic susceptible

media. Equations (2-113) and (2-114) have r-5 and r-7 terms, meaning that the response for a

dipole will be greater than for the surrounding susceptible media, as the samples of the field

are taken on the “surface” of the anomaly (i.e., for r less than 1m). The susceptible media

contribution, given by equations (2-98) to (2-103), involves r-2 and r-1 terms, producing far

weaker signals (for small r) than for the dipoles. Note that, for larger values of r (i.e., r

greater than 1m), the r-2 and r-1 terms will dominate. Also, the lateral extents of the responses


due to the landmines are quite small (a wavelength of a few metres), and therefore it would be

harder to determine the exact position.

Figure 4.12. The Bxy component of the gradient tensor response of the regolith scenario with noise, a dipole representing the mineralisation, and dipoles representing the landmines. 4.4 Conclusions

Measurements of surface magnetic susceptibility values from Fowler’s Gap suggest that no

consistent relationship exists between magnetic susceptibility and regolith or landform types.

This suggests that using the magnetic susceptibility of surface materials to map regolith or

landscape features is unfeasible. However the magnetic susceptibility of materials is still an

important part of near-surface exploration, as mineral deposits may not relate to regolith or

landscape features, but instead to unseen contrasts of magnetic susceptibility beneath the

surface.

The magnetic susceptibility measurements have allowed detailed magnetic forward modelling

of the regolith scenario introduced in the previous chapter. The forward models have

illustrated that the magnitude of the field responses are greatly increased, especially as

measurements are taken close to the source. The models also show that small features can


have a large potential field response (i.e., the land mines), although they would be difficult to

accurately locate due to their small size.

While the noise added to the forward models is characteristic of the Folwers Gap area, and

does not reflect all regolith scenarios in Australia, the modelling results indicate that magnetic

gradiometry as measured near the surface should be useful for regolith exploration,

supporting the results of Chapter 3.

Documents

Chapter 1: Introduction to Gravity, Magnetics and Regolith ... · PDF fileChapter 1: Introduction 1 Chapter 1: Introduction to Gravity, Magnetics and Regolith Exploration 1.1 Introduction