Chapter 5: Application of Filters to Potential Field Gradient Tensor Data

Transcription

1 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.

2 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 B ij satisfy Laplace s equation. Bijxx + Bijyy + Bijzz = 0 (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.

3 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.

4 Chapter 5: Filters 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: iux ( + vy) F u v f x y e dxdy (, ) (, ) = (5-2) The inverse transform can therefore be defined as: 1 iux ( + vy) f ( x, y) = F ( u, v) e dudv 4π (5-3) 2 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 h u v e 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.

5 Chapter 5: Filters 102 Figure 5.1 shows the B xx 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 B xx 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.

6 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 (B xz ) 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 B xz 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 B xz 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 B xz 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.

7 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 B yz 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 1 st 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 ( ) 4 u v π +, where u and v are the frequency domain variables (Fuller, 1967). Figure 5.6 shows the second derivative of the B yz data introduced previously. In these data, the edges are better defined, but there is more noise.

8 Chapter 5: Filters 105 Figure 5.6. The application of a 2 nd 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 (θ m θ f ) -1, where: θ m mu ˆ ˆ x + mv y = mˆ z + i and k θ f ˆ ˆ ˆ Bxu+ Byv = Bz + i (5-4) k Here the vector ( ˆ ˆ ˆ x, y, z) m m m is the magnetic moment unit vector, and ( ˆ ˆ ˆ x, y, z) B 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 B xx exhibits a peak and a trough around the anomaly in Figure 5.7, Figure 5.8 shows a

9 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 B xz 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 B xx, B yy and B xy components of the magnetic gradient tensor when reduced to the pole shows a trough directly above the anomaly (for this scenario). The B zz component shows a peak, and the remaining components show a peak and trough either side of the anomaly.

10 Chapter 5: Filters 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 B xy 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 B xy component of the gradient tensor due to a magnetic dipole with noise added. The source is in the centre of the field area.

11 Chapter 5: Filters 108 Figure The B xy 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 The B xy 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.

12 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 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: F SIG = F SIG + F SIG + F SIG (5-5) tensor x y z 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 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.

13 Chapter 5: Filters 110 Figure The top row of images show six components of the gradient tensor due to a dipole with moment 1000Am 2 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 The top row of images show six components of the gradient tensor due to a dipole with moment 1000Am 2 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.

14 Chapter 5: Filters 111 Figure The top row of images show six components of the gradient tensor due to a dipole with moment 1000Am 2 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 B yy and B yz ), but they are easy to see on the single Tensor Analytic Signal. Figure Features not clearly discernable on all the gradient tensor images (left) are easy to perceive on the single Tensor Analytic Signal image (right).

15 Chapter 5: Filters 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 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 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.

16 Chapter 5: Filters 113 Figure 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

17 Chapter 5: Filters 114 the dipole oriented in the z direction, are there multiplicative combinations that produce a single peak above the anomaly (B xx B yy, B xx B zz, B yy B zz and B xx B yy B zz ). Figure The components of the gradient tensor with 10% noise contamination that have been used to produce images in the following four figures. Figure 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.

18 Chapter 5: Filters 115 Figure 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 Figure 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.

19 Chapter 5: Filters 116 Figure 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 Figure 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 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.

20 Chapter 5: Filters 117 Figure The multiplicative combinations of combinations of two gravity gradient tensor components of the regolith model reveal no further information from the data. Figure The multiplicative combinations of combinations of three gravity gradient tensor components of the regolith model reveal no further information from the data.

21 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: B B B xx xy xz det xy yy yz xx yy zz 2 xy yz xz yz xx xz yy xy zz B = B B B = B B B + B B B B B B B B B (5-6) B B B xz yz zz 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 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.

22 Chapter 5: Filters 119 Figure 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: ( xx yy zz ) ( xy yz xz ) ( yz xx ) ( xz yy ) ( xy zz ) det 2 mod B = 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.

23 Chapter 5: Filters 120 Figure 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: Bxx B inv xy B inv xz B inv xx Bxy B xz Bxy B inv yy B inv yz = inv Bxy Byy Byz B B xz B xz Byz Bzz inv yz B inv zzinv 1 (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.

24 Chapter 5: Filters 121 Figure 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 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.

25 Chapter 5: Filters 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 (B xx, B yy and B zz ) 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.