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Chapter 5: Filters 98
Chapter 5: Application of Filters to Potential Field
Gradient Tensor Data
5.1 Introduction
One of the objectives of this research is to investigate the use of spatial filters on potential
field tensor data. In the previous two chapters I have constructed numerous potential field
maps of a particular 3-D regolith model, and these synthetic data can be used to illustrate the
use of filters to further extract information for interpretation. As shown earlier, not all the
features defined in the regolith model were evident in the field map (e.g., the palaeochannel).
This chapter is concerned with the filtering of potential field data, so as to enhance certain
features.
A potential field filter is a numerical process that highlights different aspects of gravity or
magnetic field data (Bhattacharyya, 1972; Clement, 1973; Ku et al., 1971). Different types
and forms of filters highlight different features. Filters can emphasise boundaries between
geological units, highlight deeper or shallower structures, or show features from different
angles (Telford et al., 1996).
Filters are usually applied to potential field data after the data have been collected and
processed into some standard format (e.g., Bouguer gravity). The filtering often involves
placing some moving spatial window over the data and analysing the data in each window.
The data in the window are subject to some mathematical treatment and output into a new
data file. Alternatively, filters can be applied to the data set as a whole. Examples of each
will be shown in this chapter.
In order to demonstrate the use of filters on geophysical data sets, various simulated
geophysical responses have been used. The first is the noise-free gravitational gradient tensor
data set taken from Chapter 3. While it has been shown that these features are not
immediately recognisable on the forward models previously calculated, the application of
filters may still enhance features and therefore be useful for non-near-surface exploration.
The second example is the noise-free magnetic gradient tensor data set also taken from
Chapter 3. Recall that the surface features were visible, but the deeper information was not.
Chapter 5: Filters 99
The application of filters may help remedy this situation. The third data set is from Chapter 4;
it is the regolith model with noise added, and a magnetic dipole representing mineralisation.
The fourth data set is the magnetic gradient response of a dipole (with a high level of noise
added) which will be used to demonstrate the use of several filters, including “reduction to the
pole” filters for a gradient tensor response. Gradient tensor data are similar to total field data
in that an anomaly will appear to be positioned away from a source due to the inclination of
the inducing field.
The components of the magnetic gradient tensor satisfy the conditions for being potential
fields (with the exception that they are scalar fields, not vector fields). That is, the x, y and z
double derivatives of Bij satisfy Laplace’s equation.
0ijxx ijyy ijzzB B B+ + = (5-1)
In equation (5-1), the first two subscripts represent which gradient tensor component is being
examined, and the third and fourth subscript represent the double derivatives of the
component. Analytical equations for the terms with four subscripts will be derived in Chapter
7, as they are needed to calculate the magnetic field around certain complex source types.
5.2 Description and Application of Filters
Most filters fall into two categories: Filters relying on the Fourier transform of the field, and
convolution methods. That is, filtering can be undertaken in the spatial frequency domain
using the Fourier transform or in the space domain by convolution. Fourier transform
techniques involve converting the data set into the frequency domain, operating on this data
set in some way, and then returning it to the space domain. This allows different spatial
frequencies (wavelengths) of data to be highlighted or suppressed. Some of the techniques
include: Reduction to the Pole (where data are recalculated as if the inducing magnetic field
were vertical), Band Pass Filters (whereby selected wavelengths of data outside a specified
band are removed), Derivative Filters (a process for determining the first and second vertical
derivatives of the data) and Field Continuation (a process that calculates what a field should
look like if it was measured at a different height). Filters that attenuate short wavelength
features are referred to as low-pass filters, while filters that suppress long wavelength features
are called high-pass filters.
Chapter 5: Filters 100
Convolution methods involve convolving a filter impulse response h(x,y) with the data B(x,y).
The filter impulse response has to be defined by the user. A finite window is placed around
the data point in question, selecting adjacent points in each direction. The size of the window
determines how many samples are taken. The extracted data is then isolated to form a new
data set. Examples of convolution methods include: Averaging or Smoothing Filters (where
the entire data set is smoothed), Sunlight Filters (where the data set is highlighted from a
specific direction) and Edge-Detection Filters (where boundaries in a certain direction are
highlighted). If the convolutional model is valid (i.e., a linear operation) then each process
can be done in each domain, but Fourier transforms are global operators and can’t easily
accommodate local variations in medium properties.
Some processing techniques do not fall into the two categories (described above), as they
involve a direct calculation utilising more than one data set (not involving moving windows
or Fourier transforms). In Chapter 2, it was mentioned that there are three Analytic Signals of
the field that can be computed directly from the gradient tensor components. The three
equations of (2-12) can be applied directly to data already calculated to enhance features.
Since a source may not produce an anomaly in all of the components of the gradient tensor
(e.g., the landmines in Figures 3.18 to 3.23), all independent components of the gradient
tensor should be used simultaneously to best capture and enhance any anomalies that may be
present. Having said this, the majority of filters act on a single data set, and so I propose that
when running a filter on gradient tensor data, it should be run on all components. As all the
noise-free individual gradient tensor components (i.e., the data sets from Chapter 3) should
have (by definition) identical frequency information, I have run the same filters on all the
individual gradient tensor components. The gradient tensor components with added noise
(i.e., from Chapter 4) will behave differently and could be analysed by running filters with
different properties (e.g., larger or smaller window size).
Examples of the processed data are shown throughout the next sections. The program
“Encom Profile Analyst” has been used to filter the data presented in this chapter, and the
program “MATLAB®” used to compute the analytic signals. The number (and type) of
filters used was limited to those available in the program “Encom Profile Analyst.”
Chapter 5: Filters 101
5.2.1 Fourier Transform Filters
The Fourier transform of a function f(x,y) is an integral transform (Blakely, 1996), and in two
dimensions can be defined as follows:
( ) ( ) ( ), , i ux vyF u v f x y e dxdy∞ ∞
− +
−∞ −∞
= ∫ ∫ (5-2)
The inverse transform can therefore be defined as:
( ) ( ) ( )2
1, ,4
i ux vyf x y F u v e dudvπ
∞ ∞+
−∞ −∞
= ∫ ∫ (5-3)
Whilst data is in the Fourier domain, numerous operations can be carried out. The specific
operations are described in the following sections.
Upward and downward continuation
The process of transforming a data set so that it appears that it has been measured at a
different height is called continuation. The process is called upward continuation if the data
set is being moved further away from the source, and downward continuation if the data is
being moved toward the source (Clarke, 1969; Hansen and Miyazaki, 1984; Henderson, 1970;
Henderson and Zietz, 1949b; Jacobsen, 1987; Pawlowski, 1995). The benefits of upward
continuation include removing high spatial frequency noise and highlighting regional
features. Downward continuation can highlight subtle features, but at the expense of also
highlighting any noise present in the data.
The process of upward or downward continuation involves converting the data set into the
Fourier domain, and multiplying through by the term 2 22 h u ve π± + , where h is the new height,
and u and v are the frequency domain variables. The data are then converted back into the
space domain for visual analysis. If h is positive, the process is called upward continuation, if
h is negative, the process is called downward continuation.
Chapter 5: Filters 102
Figure 5.1 shows the Bxx component of the magnetic gradient tensor for the regolith model
introduced in Chapter 3. The data has been downward continued 1m. It is obvious that while
this process has introduced noise to the data, the boundaries between the regolith units are
better delineated. Similar images are obtained from the other components of the gradient
tensor.
Figure 5.1. Application of downward continuation. Note how the filtering has enhanced the regolith boundaries and the cadaver anomaly, but has also created much noise not previously present in the data.
Figure 5.2 shows the Bxx field response from the noisy regolith model of (including the added
dipole) after upward continuation by 4m. By comparison with the original, it is immediately
apparent that much of the short wavelength noise has been removed, and the mineralisation
feature is enhanced. Again, a similar result is obtained from the other components of the
gradient tensor.
Figure 5.2. Upward continuation has enhanced the mineralisation feature, and removed some of the low wavelength noise.
Chapter 5: Filters 103
Bandpass filters
When data is in the Fourier domain, it is possible to remove features corresponding to
different wavelengths. Bandpass filters are used specifically to do this, and some examples
follow.
Figures 5.3 and 5.4 show one of the off-diagonal components of the gradient tensor (Bxz) for
the regolith model in Chapter 4. Figure 5.3 has had the apparent wavelengths (in both x and y
directions) below 2m and above 5m removed, and as a result, there is much less noise in the
data (these values were chosen simply to illustrate the effect of the filter on the data). Figure
5.4 shows the Bxz component after retaining only wavelengths between 40m and 80m. Note
that the features now seem much broader than the original, and the response of the dipole is
significantly enhanced.
Figure 5.3. The Bxz component of the gradient tensor for the regolith model in Chapter 4, with noise and a dipole added. The bandpass filter has removed wavelengths below 2m and above 5m.
Figure 5.4. The Bxz component of the gradient tensor for the regolith model in Chapter 4, with noise and a dipole added. The bandpass filter has removed wavelengths below 40m and above 80m.
Chapter 5: Filters 104
Vertical derivative filters
First and second derivative potential field maps are often used to detect edges and geological
boundaries in data (Clarke, 1969; Henderson and Zietz, 1949a; McGrath, 1991). Such
derivatives can be viewed as a high pass filtering operation. Filters can be applied to
calculate the first and second derivatives of field data, and even fractional derivatives (Cooper
and Cowan, 2003). Such filters generally add numerical noise to field data. This is apparent
from Figures 5.5 and 5.6. Figure 5.5 shows the first vertical derivative of the Byz component
of the gradient tensor corresponding to the regolith model in Chapter 3 (Figure 3.17). The
boundaries in the original image, while visible, are a little scattered, and the filtering process
has enhanced them.
Figure 5.5. The application of a 1st vertical derivative filter has enhanced the boundaries between regolith units.
The second vertical derivative is found by multiplying the Fourier domain data with the term
( )2 2 24 u vπ + , where u and v are the frequency domain variables (Fuller, 1967). Figure 5.6
shows the second derivative of the Byz data introduced previously. In these data, the edges are
better defined, but there is more noise.
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Figure 5.6. The application of a 2nd vertical derivative filter has enhanced the boundaries between regolith units, and created extra noise.
Reduction to the pole
As previously mentioned, reduction to the pole (RTP) is a process whereby asymmetric
anomalies in a magnetic field (due to magnetic bodies) are reshaped so that the peaks occur
directly above the magnetic body (Baranov and Naudy, 1964; Hansen and Pawlowski, 1989;
Lu, 1998; Silva, 1986). The process (Blakely, 1996) involves transforming the field data into
the Fourier domain and multiplying through by the term (θmθf)-1, where:
ˆ ˆˆ x y
m z
m u m vm i
kθ
+= + and
ˆ ˆˆ x y
f z
B u B vB i
kθ
+= + (5-4)
Here the vector ( )ˆ ˆ ˆ, ,x y zm m m is the magnetic moment unit vector, and ( )ˆ ˆ ˆ, ,x y zB B B is the unit
magnetic vector giving the direction of the magnetic field. The symbol i here is the imaginary
number 1− . The RTP process therefore involves altering the orientation of the field
anomaly such that it would be equivalent to that measured at the magnetic pole.
Generally, Total Magnetic Intensity maps will produce a peak or a trough positioned around a
body, but the gradient tensor components will generally exhibit several peaks and troughs
around the body. Figure 5.7 shows the (noisy) components of the magnetic gradient tensor
around a dipole source, and these have been reduced to the pole in Figure 5.8. Note that
while Bxx exhibits a peak and a trough around the anomaly in Figure 5.7, Figure 5.8 shows a
Chapter 5: Filters 106
trough directly above the anomaly. Reduction to the pole will not always return a peak or
trough directly above the causative feature, as can be seen in Figure 5.8, where the Bxz
component of the same data set reveals a peak and trough either side of the anomaly.
Figure 5.7. Gradient tensor components due to a magnetic dipole, with a large amount of noise added.
Figure 5.8. The Bxx , Byy and Bxy components of the magnetic gradient tensor when reduced to the pole shows a trough directly above the anomaly (for this scenario). The Bzz component shows a peak, and the remaining components show a peak and trough either side of the anomaly.
Chapter 5: Filters 107
5.2.2 Convolution filters
Convolution filters work by convolving a filter (2-D impulse response) with a data set. For
potential field gradient tensor data, I have found that smoothing (or averaging) filters also
enhance anomalies to best locate a source position. They are low-pass filters and hence
remove short wavelength data, ideal for data sets with much high frequency noise. A
common filter used in actual magnetic data processing is a despiking filter. This is effectively
a smoothing filter and removes spikes from the data. As all my data are synthetic, a despiking
filter has not been tested.
Running Average (or Smoothing) Filters
By taking the running average of a point with neighbours (surrounding points), it is possible
to effectively smooth an image. An example of this process as applied to gradient tensor data
is shown here. Figure 5.9 contains the Bxy component of the gradient tensor response
(reduced to the pole) with noise added. The anomaly is not easy to pick, but is present in the
centre of the image. Figure 5.10 shows the same data after having been filtered by a 9 by 9
average filter. The anomaly is obvious here, and even more so in Figure 5.11, where a 31 by
31 point running average filter was used instead. A similar result is obtained from the
remaining gradient tensor components.
Figure 5.9. The Bxy component of the gradient tensor due to a magnetic dipole with noise added. The source is in the centre of the field area.
Chapter 5: Filters 108
Figure 5.10. The Bxy component of the gradient tensor due to a magnetic dipole with noise added. The map has been enhanced with a 9 by 9 average filter.
Figure 5.11. The Bxy component of the gradient tensor due to a magnetic dipole with noise added. The map has been enhanced with a 31 by 31 average filter. The centre of the prominent blue area represents the position of the body.
Other filters
There are many other types of filtering operations. For example, the Encom software package
“Encom Profile Analyst” frequently presents the data after application of a sunlight filter.
Chapter 5: Filters 109
Apart from giving the data a three-dimensional look, this can be used to highlight features
from a specific angle. I have not found these filters to be particularly useful in extracting
further information from the data sets presented in this thesis. As with most filters, the
sunlight filter is always applied to a single component of the field. An obvious question is:
what would happen if the same filter were implemented on all the gradient tensor
components, and then combined in some way to analyse the total results? The answer
depends entirely on how the components are combined, and to answer this question fully I
need to look at different ways of combining the components of the gradient tensor. Perhaps
the most common form of combining components of the gradient tensor is the Analytic
Signal.
5.2.3 Analytic Signals
Using equations (2-12) it is possible to create three analytical signals of any gradient tensor
data. This has been used in the past with standard potential field data to highlight and
enhance anomalies. I would like to introduce a further analytic signal, hitherto unused in the
literature. It is simply a combination of the three analytic signals of equations (2-12). For a
vector field F, it can be written:
2 2 2tensor x y zF SIG F SIG F SIG F SIG= + + (5-5)
I simply refer to this as the Tensor Analytic Signal of the field, as it computes a single data set
from all the components of the gradient tensor. It is possible to weight the gradient tensor
components, or to leave them in raw recorded form. Three examples of the use of this signal
are given in Figures 5.12 to 5.14. The figures illustrate the gradient tensor response of a
dipole (the dipole being in a different orientation for each image), along with the Analytic
Signals underneath. In Figure 5.14, note that the three Analytic Signals contain maxima and
minima around the source position, while in all cases a large peak is shown directly above the
anomaly in the Tensor Analytic Signal response.
Chapter 5: Filters 110
Figure 5.12. The top row of images show six components of the gradient tensor due to a dipole with moment 1000Am2 in the x direction. The second row contains the three Analytic Signals as computed via equation (2-12), and the final image at the bottom is the Tensor Analytic Signal, as computed from equation (5-5). The Tensor Analytic Signal shows a larger response than the three Analytic Signals.
Figure 5.13. The top row of images show six components of the gradient tensor due to a dipole with moment 1000Am2 in the y direction. The second row contains the three Analytic Signals as computed via equation (2-12), and the final image at the bottom is the Tensor Analytic Signal, as computed from equation (5-5). The Tensor Analytic Signal shows a peak where the three other Analytic Signals show a (relatively) flat line.
Chapter 5: Filters 111
Figure 5.14. The top row of images show six components of the gradient tensor due to a dipole with moment 1000Am2 in the z direction. The second row contains the three Analytic Signals as computed via equation (2-12), and the final image at the bottom is the Tensor Analytic Signal, as computed from equation (5-5). The Tensor Analytic Signal shows a single peak where the other three Analytic Signals show some variation in their response.
Another example of the Tensor Analytic Signal calculated from the synthetic data of Chapter
4 is shown here (Figure 5.15). It displays the six components of the gradient tensor for the
regolith model (with noise and dipole representing mineralisation) alongside the Tensor
Analytic Signal. The features may not be easily recognizable or discernable on all the
individual gradient tensor components (especially Byy and Byz), but they are easy to see on the
single Tensor Analytic Signal.
Figure 5.15. Features not clearly discernable on all the gradient tensor images (left) are easy to perceive on the single Tensor Analytic Signal image (right).
Chapter 5: Filters 112
5.2.4 Experimental Combinations
The gradient tensor contains five components that can be measured independently of each
other. Noise in one component (if random and isotropic) may not correspond to anything in
another component. In other words, there is no correlation of noise between components.
Therefore as protection against random noise it is best to work with all components together,
rather than one in isolation. Chapter 6 will also introduce the use of eigenvalues and
eigenvectors as ways of interpreting and displaying potential field gradient tensor data. The
eigenanalysis will also be used to examine other magnetic sources in Chapter 7. While these
can be treated as a form of filter, the theory behind them will be deferred until then.
Analytic Signals and Smoothing
A question that arises relates to the order in which filters should be applied. Not all processes
are commutative. In Figure 5.16, the raw gradient tensor data has been used to calculate the
Tensor Analytic Signal, which has then been smoothed with a 9 by 9 smoothing filter. If the
Tensor Analytic Signal had been computed from smoothed gradient tensor data, the result
would be the image shown in Figure 5.17. In this illustration, the mineralisation is still
prominent, but the channel feature is less evident. This suggests smoothing should be
undertaken as a “late-stage” filtering process, after processes such as calculating the Analytic
Signal(s) have been undertaken.
Figure 5.16. Features not clearly discernable on all the gradient tensor images (left) are easy to see on the Tensor Analytic Signal image (right). The Tensor Analytic Signal has been smoothed with a 9 by 9 averaging filter. The prominent blue “bars” on the image reflects an area of data that has been lost due to this process.
Chapter 5: Filters 113
Figure 5.17. The Tensor Analytic Signal here has been computed from smoothed gradient tensor data. The creek feature is less visible than in previous Figures. Again, the prominent blue “bars” on the image reflects an area of data that has been lost due to the smoothing process.
Multiplication of Gradient Tensor Components
In searching for ways of combining gradient tensor components, I have experimented with
simply multiplying the components together. Note that with six gradient tensor components
to manipulate, there are 15 combinations using 2 components, 20 combinations using 3
components, 15 combinations using 4 components, 6 combinations using 5 components, and
1 combination with all six components (note that if I were to use just the five independent
components of the gradient tensor, the combinations of multiplications would be a subset of
the above group). As there are many combinations, Figures 5.18 to 5.23 show them for a
single dipole of orientation in the z direction. Figure 5.18 shows the six individual
components of the gradient tensor due to a magnetic dipole that have been used to create the
combinations. The images have added random noise of a maximum strength of 10%.
Examining similar images (not shown here) for scenarios with the dipole in various other
orientations reveals that, generally, 2 or 4 peaks present themselves around the anomaly, and
the signal becomes more concentrated closer to the source position. Only in this case, with
Chapter 5: Filters 114
the dipole oriented in the z direction, are there multiplicative combinations that produce a
single peak above the anomaly (BxxByy, BxxBzz, ByyBzz and BxxByyBzz).
Figure 5.18. The components of the gradient tensor with 10% noise contamination that have been used to produce images in the following four figures.
Figure 5.19. The fifteen multiplicative combinations of the gradient tensor components, allowing two components to be multiplied together at time. Scales are the same as for Figure 5.18.
Chapter 5: Filters 115
Figure 5.20. The twenty multiplicative combinations of the gradient tensor components, allowing three components to be multiplied together at time. Scales are the same as for Figure 5.18.
Figure 5.21. The fifteen multiplicative combinations of the gradient tensor components, allowing four components to be multiplied together at time. Scales are the same as for Figure 5.18.
Chapter 5: Filters 116
Figure 5.22. The six multiplicative combinations of the gradient tensor components, allowing five components to be multiplied together at time. Scales are the same as for Figure 5.18.
Figure 5.23. The one multiplicative combination of the gradient tensor components, allowing all six components to be multiplied together at time. Scales are the same as for Figure 5.18.
From the above examples, it would seem that multiplying the diagonal components of the
field together produces actual peaks above the anomalies (although this is for a particular
model). Operating on the gravity gradient tensor data from Chapter 3 reveals that the more
multiplications that are carried out between the components, the sharper the response. Figure
5.24 shows the multiplicative combinations of two components at a time, and Figure 5.25
shows the multiplicative combinations of three combinations at a time. The original data set
can be seen in Figures 3.14 to 3.19 in Chapter 3. It is obvious that no further features are
being resolved, and the features that can be seen in the original data are not as clear.
Therefore, multiplication of gradient tensor components does not appear to reveal further
geological information from depth, although for a magnetic dipole a series of peaks tend to
form around the source.
Chapter 5: Filters 117
Figure 5.24. The multiplicative combinations of combinations of two gravity gradient tensor components of the regolith model reveal no further information from the data.
Figure 5.25. The multiplicative combinations of combinations of three gravity gradient tensor components of the regolith model reveal no further information from the data.
Chapter 5: Filters 118
Determinant of the 3 × 3 Gradient Tensor
The determinant of the gradient tensor may be useful for interpretation, as it provides a single
data set to work with, rather than five. The determinant of the gradient tensor is given by:
2 2 2det 2
xx xy xz
xy yy yz xx yy zz xy yz xz yz xx xz yy xy zz
xz yz zz
B B BB B B B B B B B B B B B B B B B
B B B= = + − − − (5-6)
This is effectively a composite of the combinatory multiplicative terms from the previous
section (although three terms are actually permutations). Images for the determinant are
therefore similar to the images given in the last section. For a dipole source, the determinant
may show a peak or trough directly above the source, but for the majority of dipole
orientations, both a peak and trough are obtained, either side of the source position. Figure
5.26 shows such an example (the determinant equalling zero above the source position), and
Figure 5.27 shows an example where a trough is obtained directly above the source position.
Figure 5.26. The six components of the gradient tensor for a dipole oriented in the x direction (left), and the corresponding determinant (right). The determinant here produces a peak and trough on either side of the source position.
Chapter 5: Filters 119
Figure 5.27. The six components of the gradient tensor for a dipole oriented in the z direction (left), and the corresponding determinant (right). The determinant here produces a trough directly above the source position.
The five terms in equation (5-6) can be altered mathematically, so as to create an increased
cumulative effect upon a data set. For example, by calculating these five terms (before adding
them together) for the gravity gradient tensor data from Chapter 3, I note that the second and
fifth term have a negative signal, and the third term has both positive and negative terms. By
calculating the square of each term and determining the square root of the added total (in a
similar fashion to calculating Analytic Signals) allows the addition of the data set to not
“cancel-out” any information, and increase the field response. This is referred to as phase
coherent addition, to avoid destructive interference. The equation governing this response
would therefore be:
( ) ( ) ( ) ( ) ( )mod
2 2 22 2 2 2 2det 2xx yy zz xy yz xz yz xx xz yy xy zzB B B B B B B B B B B B B= + + + + (5-7)
Figure 5.28 shows the determinant (in blue) calculated along a profile directly above a
gravitational point source, and the modified determinant superimposed (in red) for
comparison. The anomaly pattern for the modified determinant is tighter around the source
position, which is located 20m along the horizontal axis.
Chapter 5: Filters 120
Figure 5.28. The modified determinant exhibits a tighter fitting curve around a source than the determinant itself. The source here is positioned at 20 metres along the horizontal axis.
Inverse Matrix of the 3 × 3 Gradient Tensor
Where the determinant of the gradient tensor is not equal to zero, there exists an inverse
matrix. The notation used here is:
1
inv inv inv
inv inv inv
inv inv inv
xx xy xz xx xy xz
xy yy yz xy yy yz
xz yz zzxz yz zz
B B B B B BB B B B B B
B B BB B B
−⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥=⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎣ ⎦⎢ ⎥⎣ ⎦
(5-8)
When the determinant is equal to zero, no inverse matrix exists. Much care must be taken in
the interpretation of the components of the inverse matrix, as will become obvious. Note that
in Figure 5.27, the determinant is equal to zero directly above the dipole source. This is
highlighted in the inverse response (Figure 5.29) where a “spike” is seen above the anomaly.
This will not occur for all dipole orientations, as can be seen in Figure 5.30, where no
“spikes” can be used to locate the anomaly.
Chapter 5: Filters 121
Figure 5.29. Shown on the left are the gradient tensor responses of a magnetic dipole oriented in the x direction. On the right are the components of the inverse matrix. The spike seen in some of these components represents points where the determinant is equal to zero, and hence no inverse exists. In this scenario, the dipole is placed directly below this point.
Figure 5.30. Shown on the left are the gradient tensor responses of a magnetic dipole oriented in the y direction. On the right are the components of the inverse matrix. There are no “spikes” seen in any of these components.
Chapter 5: Filters 122
5.3 Discussion and Conclusions
I have presented some standard filters and shown how they can be applied to potential field
gradient tensor data. By treating the components of the gradient tensor as separate field maps,
filters can be applied to the individual components. By applying the same filter to all the
components, the resulting maps can be used in conjunction for interpretation. Some filters,
especially Reduction to the Pole, will not always produce an obvious “peak” directly above a
source position, and so the gradient tensor components should be combined in some way to
enhance the signal. Analytic Signals are especially useful in locating source position, and the
introduction of a “Tensor Analytic Signal” has allowed the three analytic signals to be
analysed in a single data set. Smoothing of data has allowed potential anomalies to be
observed in the data where previously not discernable. If smoothing is applied before
calculating the Tensor Analytic Signal, prominent features from that first smoothing process
will be highlighted further, possibly at the expense of diminishing the amplitude of other
features.
Several multiplicative combinations of gradient tensor components have been calculated.
Generally, the combinations of diagonal components of the gradient tensor (Bxx, Byy and Bzz)
produce images with peaks (or troughs) directly above the source position (for a dipole
source). For more complex geology, little is obtained from the multiplicative combinations.
The determinant of the gradient tensor can also be used for interpretation of gradient tensor
data, as it allows the data to be interpreted from a single data set. For a dipole source, a peak,
trough, or adjacent peak and trough are found above a source position. The inverse matrix of
the gradient tensor can also be used for interpretation, but care needs to be taken. If the
determinant is equal to zero, no inverse exists, producing a “spike” in the inverse data. This
does not necessarily mean that a source is present; it only means that the determinant is equal
to zero, which could suggest a source.
Further interpretation techniques which are not technically classed as filters (e.g., eigenvalues
and eigenvectors) will be introduced in the next chapter, as they relate directly to an inversion
routine.