QUALITY CONTROL OF GRIDDED AEROMAGNETIC DATA 1 INTRODUCTION

Transcription

1 QUALITY CONTROL OF GRIDDED AEROMAGNETIC DATA Stephen Billings* and Dave Richards Rio Tinto Exploration, Australia Region * Now at Earth and Ocean Sciences, University of British Columbia, Vancouver, BC, V6T1Z4 1 INTRODUCTION Airborne geophysics, particularly aeromagnetics has been a cornerstone of the mineral exploration industry for over 50 years and has been a significant component of Rio Tinto s project generation and area selection activities. In 1997 Rio Tinto Australia Region spent approximately 10% of its exploration budget on geophysics, of which two thirds was on airborne surveys (~4% magnetic and ~2% EM). The amount of data that we have available continues to increase, and requires that we can process and interpret it well. Typically the product that we use for interpretation is the gridded aeromagnetic image. Many gridded images of aeromagnetic data include artefacts and unlikely correlations of anomalies across flight lines. This distorts the anomaly shapes and patterns used to interpret the data, as well as introducing (or removing) power at frequencies which may be important for further processing. Some of the features may be due to poor data quality, inappropriate choice of survey parameters, poor gridding algorithms or inappropriate choice of the parameters used by the gridding algorithm. The initial aim of this project was to begin to establish some guidelines to help ensure the best possible quality of gridded aeromagnetic data, and to investigate and document the possible use of the two-dimensional FFT as a tool for diagnosing problems in gridded datasets. During the course of the project, it became clear that there were also issues with the accuracy of the Fourier domain representation of the gridded dataset, and that these would affect the outcomes of any operations undertaken in the Fourier domain (these commonly include reduction to the pole, upward and downward continuation, vertical and horizontal gradients, and analytic signal).

2 2 FOURIER TRANSFORMS OF GRIDDED DATA The mathematical definition of the Fourier transform is given in the glossary. The Fast Fourier Transform (FFT) is a method for decomposing an image into a series of sine and cosine functions of different frequencies (or wavelengths). The lowest frequency is zero (DC) and represents the average value of the image. The highest frequency is determined by the image cell size via the Nyquist relationship. This highest frequency oscillates through a full cycle N times along the horizontal axis and M times along the vertical axis of the image. There are then a further N/2-1 frequencies along the horizontal axis and a further M/2-1 frequencies along the vertical axis. Each of these intermediate frequencies undergoes an exact integer number of oscillations between the image edges. If the entire image was composed of just a single sinusoidal wave of a particular frequency, then the FFT of the image would be zero everywhere except at the frequency corresponding to the wave. Real images are more complicated than this and the FFT will usually have nonzero components at all spatial frequencies. The low frequency (long wavelength) components oscillate only a few times over the extent of the image and will mostly reflect the regional magnetic field. On the other hand the high frequency (short wavelength) components will oscillate many times through the image and reflect the magnetic field variations of shallow (near-surface) origin. 2.1 HOW OBJECTIVE IS THE 2-D FFT? In this section we consider the relationship between the discrete Fourier transform (DFT) of practice and the continuous Fourier transform of theory. In the Geophysics literature Cordell and Grauch (1982) have also considered this issue. Interpolation: The interpolation algorithm will have a substantial influence on the calculated Fourier transform, particularly in the across-line direction. There, all parts of the spectrum with frequencies higher than the Nyquist cut-off (ie with wavelengths shorter than twice the line spacing) will be almost entirely a function of the interpolation algorithm. Computation: The FFT algorithm works fast by exploiting symmetries that are present for images of certain dimensions. The simplest FFT algorithms are only efficient when both image dimensions are a power of two. The algorithm implemented by Geosoft in Oasis montaj is the Winograd FFT which works fast when the image dimensions are either 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 24, 28, 30, 36, 40, 42, 48, 56, 60, 70, 72, 80, 84, 90, 112, 120, 126, 140, 144, 168, 180, 210, 240, 252, 280, 315, 336, 360, 420, 504, 560, 630, 720, 840, 1008, 1260, 1280 or For images dimensions larger than 2520, Geosoft uses a standard power of two FFT (sizes 4096, 8192, 16384, 32768, etc.). This restriction on FFT input usually means that the image dimensions must be increased and the extended areas filled with synthetic data. Additionally, the FFT requires that the image fills an entire rectangle, a condition not met by many geophysical surveys. Therefore, any survey gaps or irregular edges will also need to be filled with synthetic data. Regular sampling and finite domain: Even if the image dimensions are acceptable to the Winograd FFT algorithm there is an additional factor to be considered. The finite size of the image combined with the regular spacing between adjacent data-points means that there is an implicit assumption that the data sequence is periodic. Therefore, points on the Eastern edge of an image are neighbours to points on the Western edge and similarly for points on the North and South boundaries. Fourier transforms of images whose opposite edges don t match result in contamination of the spectrum in the form of a St George s Cross in the wave number domain representation of the Fourier transform (high intensities concentrated along the axes because of the high frequency power introduced by the steps at the edges of 2

3 the image). To avoid this contamination the image is usually expanded (by about 10%) with the image values chosen so that opposite edges of the survey match smoothly. Geosoft expands this grid to have dimensions acceptable to the Winograd FFT. The actual edge matching and filling of survey gaps is achieved by the Maximum Entropy algorithm of Burg (1975) or by an inverse distance squared method. Ricard and Blakely (1988) also describe a method for minimising edge effects but it is not practical for large geophysical surveys. Sampling rate: The maximum frequencies represented in the Fourier domain are determined by the spatial grid cell size through the Nyquist relationship, u 1 v max = 1 2 y = 2 x max and. Any frequencies higher than these will be aliassed (folded) back into the lower parts of the spectrum. The actual maximum frequencies resolvable by the data are determined by the along-line and across-line sampling rates of the survey. For magnetics the data are generally under-sampled across-lines (50 to 400 m) and over-sampled alonglines (5 to 10 m). When the data are gridded the along and across-line cell sizes are usually chosen to be one-quarter to one-fifth of the line spacing. This results in an increase in data points across lines and a decrease along lines (see Figure 1). Along-lines the central consideration is whether the cell-size is fine enough to prevent aliassing of signal in the interpolated image. Across-lines the situation is somewhat different; the key issue is that all power in the Fourier domain at frequencies higher than the line-spacing Nyquist is entirely due to the gridding algorithm. Therefore, selection of the grid cell size is critical and involves a delicate compromise between faithfully representing signal along-lines and preventing too many artefacts across-lines. 2.2 CONCEPTUALISING 2-D FOURIER TRANSFORMS Fourier transforms are usually displayed as 2-D power spectra with a logarithmic stretch applied to the data. The power spectrum reflects the strength of the sine and cosine components at each frequency. The reason it is displayed with a logarithmic stretch is that there is a huge variation in the amplitudes of the different spectral components. Most aeromagnetic datasets are dominated by the lower frequency Fourier components (i.e. magnetic intensity is smooth at airborne altitude). If they weren't an aeromagnetic image would be a composed of very rapid oscillations in magnetic intensity that would make interpretation extremely difficult. To clarify the different components of a power spectrum we will run through an example (Figure 2). The FFT estimates Fourier components between zero frequency and the Nyquist limit imposed by the grid cell-size. In the example used here, the maximum frequency is m -1 (spatial wavelength 200 m) corresponding to a pixel size of 100m. The black box in the centre of the spectrum is the line Nyquist at a frequency of m -1 (spatial wavelength 600 m) and corresponds to the maximum frequency resolved by the data acrosslines. The along-line Nyquist limit almost always exceeds the grid Nyquist and is therefore not shown. The power spectrum has been logarithmically stretched using the colour table shown in Figure 3. Red colours correspond to regions of high power, grading to yellow, green and then to blue, which represents the lowest power. The very centre of the frequency spectrum represents the average value in the image, while those frequencies immediately adjacent correspond to low frequency variation. Frequencies gradually increase as we move away from the centre of the frequency spectrum. This particular aeromagnetic image (Figure 2) is very smooth so that there is a rapid decrease in power with increasing frequency. The survey lines are oriented North-South and have obvious levelling problems. Therefore, if we were to move horizontally across the image we would encounter a very rough surface. This means that high frequency Fourier components are required to fit the image variation in this 3

4 direction, a characteristic that manifests itself as a concentration of power along the horizontal frequency axis. Notice that the power spectrum is elongated in the vertical direction which corresponds to the direction of the survey lines. This indicates that there is a lot of power and hence a lot of structure in the image in the vertical direction and reflects the dense along-line sampling. The other feature to note in the power spectrum is the Southwest to Northeast trending concentrations of power. These correspond to the Northwest to Southeast trending geological features which are most obvious in the Northwest corner of the image. When displaying power spectra it is necessary to select a colour table and a colour stretch method. We have used the default colour-table supplied in the Matlab package (Figure 3). This colour table can be found in here, and can be copied to the source Geosoft directory (as say matlab.tbl) from where it can be selected in just the same way as any other colour table. Note that power spectra calculated in Geosoft have already been logarithmically stretched. Therefore, to get the same colour stretch as in this report the Colour Method option should be set to linear when displaying a grid. There is an inherent symmetry in Fourier transforms of image data which means that only half of the frequency range is required to completely characterise the power spectrum (Figure 4). Geosoft exploits this property by saving only the right-hand half of the power spectra. However, this creates difficulties in visualising any concentrations of power along the vertical frequency axis. Therefore, we recommend that an option be included in Geosoft for saving (or displaying) the entire frequency range (ie including the left-hand half of Figure 4). This entire representation would not always be required but would assist in quality control applications. 2.3 QUALITY CONTROL USING THE 2-D FFT In the context of this project there are two different reasons for calculating the Fourier transform of an image: 1. To apply one of many enhancement operations that are efficiently achieved in the Fourier domain, such as upward and downward continuation, vertical and horizontal first and second derivatives, reduction to pole, analytic signal, etc. 2. To examine the frequency content of an image so that potential problems in the data or gridding method can be identified. It turns out that it is better to use different methods to achieve each of these goals, as we will now explain starting with the first application. Our discussion in the last section established that if the image is not periodic then contamination of the Fourier spectrum, in the form of a St George's Cross will occur. When high pass filtering operations, such as downward continuation, derivative calculations and analytic signal are applied this contamination will manifest itself as ripples (Gibb's effects) extending through the processed image. To prevent these artefacts it is essential to pad the image with synthetic data such that the extended image is periodic. The notion is to construct the synthetic data in such a way that opposite edges of the padded image join smoothly without steps or discontinuities. Figure 5 compares the results of downward continuing the original image with the downward continuation of its padded equivalent (the Fourier transforms of the two images are given in Figure 61). The original image suffers severe contamination from Gibb's effects but these are absent from the padded image. Thus we see that Fourier domain filtering operations require a good method for padding the image with synthetic data. 4

5 We turn now to the second problem quality control using the Fourier transform. Although the same padding technique can be used, we will show that this can mask potential problems in the gridded data set. An alternative method for calculating the power spectrum is to apply a window function to the image first. Window functions are generally zero at the image edges and increase to a maximum value of one at the centre of the image. They are useful because they largely overcome problems related to the finite size of the image and edge mismatch. In this report we will use the so-called Kaiser-Bessel window (Harris, 1978), which is shown in Figure 6 compared with a triangular window. To illustrate the benefits of using a window function for quality control, we will use the Menzies survey, which was flown at 50m line spacing (full specifications are here). We first interpolated the data to a 25m grid using the Geosoft BIGRID interpolation method, and then removed the best fitting linear trend (flat sloping plane) from the image. We then used five different methods of modifying the image prior to Fourier transformation (Figure 7). The first two are available in Geosoft and are Maximum Entropy and Inverse Distance Squared, the third is the Plug Padding routine implemented in Intrepid, while the fourth routine is the Matlab thin-plate spline which we programmed specially for this project. All four of these routines involve padding the edges of the image with synthetic data. The fifth and last method involved the application of the Kaiser-Bessel window function to the image. The resulting power spectra are shown in Figure 8. Fourier transformation of the original image results in severe contamination in the form of a St George s Cross. Both Geosoft methods are an improvement over the original but still contain significant contamination. The Intrepid Plug routine is a further improvement but also suffers from some edge contamination. In particular note the concentration of power along the horizontal frequency axis. The Matlab thin-plate spline method has some very minor edge artefacts, while the Kaiser-Bessel window appears to have no obvious contamination. The relative merits of the different methods are further emphasised by a plot of their radially averaged power spectra (Figure 9). At low frequencies the results are similar, while at high frequencies the methods differ considerably. Fourier transformation of the original image gives a power spectrum with a very slow rate of decay. The maximum entropy method initially decays even slower than this (inverse distance squared is very similar), before steepening its rate of decay at higher frequencies. The Intrepid Plug routine decays relatively rapidly at low frequencies but at higher frequencies the spectrum flattens out. The Matlab thin-plate spline and Kaiser-Bessel methods predict similar radially averaged power spectra, with both decaying much more rapidly at high frequencies than any of the other methods. The example presented above indicates that effective quality control can best be achieved by applying a window function to the image prior to Fourier transformation. Note that the windowing method still requires that the image fill an entire rectangle so that a good gapfilling algorithm is a necessity. The Matlab thin-plate spline method was the best of the four gap-filling methods used, although the Intrepid PLUG routine was also relatively effective. 5

6 3 GRIDDING ALGORITHMS In this report we consider five different gridding algorithms: Geosoft BIGRID: which uses bi-directional splines Geosoft RANGRID: minimum curvature with tension Exact Minimum Curvature: using the so-called thin-plate spline (CRAE) Image Processing System: a nearest neighbour algorithm with a smoothing filter Natural Neighbours: a triangulation based gridding algorithm The first two algorithms are included because they form the backbone of gridding in Geosoft. The thin-plate spline is representative of some emerging gridding techniques which have significant advantages when it comes to Fourier transformation (eg. Billings and Fitzgerald, 1998). The IPS algorithm is a quick and simple method which is still being used in Rio Tinto and we wished to determine how the quality of the output ranked with other techniques. The fifth algorithm is probably the most sophisticated implementation of gridding algorithms based on triangulations. These have become increasingly popular and have been implemented in many Image Processing and GIS packages (such as TNT-MIPS, IDL, and ER-Mapper). We do not consider any other interpolation methods, such as kriging, as we wanted to keep the project to a manageable size. Note that kriging is available in the Geosoft package. However, to use this method requires careful attention to the construction of a variogram and requires long run times even on moderately sized problems. The performance of the algorithms was analysed on a 50m elevation, 50m line spacing survey flown and processed by Kevron Geophysics in November 1994 at Menzies (full specifications are here). The flight-line direction was 66 0 so the data were rotated by 24 0 to make the survey lines east-west (using XYROTATE with x 0 = , y 0 = and angle=24 0 ). This is a non-essential step, but it simplifies the visualisation of line-levelling problems in the Fourier domain as they will occur either in the vertical or horizontal directions and not at some diagonal angle. Only a rectangular section of the survey in the (original) NW corner was used for the comparisons. This was to ensure that the gap-filling algorithm did not influence calculation of the power spectrum by the window method. 3.1 Geosoft BIGRID (bi-directional splines) This method rapidly interpolates data which has been collected on roughly parallel lines. The gridding process is carried out in two steps. First, each line is interpolated along the original survey line to yield data values at the intersection of each required grid line with the survey line. The intersected points from each line are then interpolated in the across-line grid direction to produce a value at each required grid point. Normally, surveys are designed so that flight line direction is as closely perpendicular to regional strike as possible. Invariably, however, there will be magnetic trends which are not perpendicular to the flight line direction. (To take account of this, Geosoft have recently released a trend enhancement option (not tested here) which operates independently of the two grid directions. It is mostly automatic but manual intervention is also possible). BIGRID allows the method of interpolation to be selected independently for the down-line and acrossline directions. The interpolations available are linear, cubic spline (minimum curvature) or Akima spline. 6

7 Filtering of the line data before interpolation is also possible. BIGRID can design and apply non-linear and/or linear numerical filters to the original line data. The use of the non-linear filter is a very effective way to remove data spikes (undesired high-amplitude shortwavelength features) from the original data. These same filters can also be applied to the line data independently of any gridding process. This is our preferred option as the effect of the filter can be visually assessed prior to the gridding Interpolation parameters Maximum line separation: Areas between lines that are separated by more than this distance are set to null after the gridding. Maximum point separation: Areas along lines that are separated by more than this distance are set to null after the gridding. Cells to extend down line: The number of cells to extend beyond the edges of the data. By default, the number of cells is set to 1, which ensures that the grid will always extend at least a fraction of a cell past the ends of valid data. Spline down line: Spline to use down the lines. Should have little influence, as the data sampling along lines is usually much finer than the grid cell size. Spline across line: Spline to use across lines. Cubic splines can be prone to overshoot when the magnetic field changes rapidly. The Akima spline (Akima, 1970) was designed to overcome these problems and should normally be used in preference to any of the other options. Low- and/or High-pass filter wavelengths: The short and long-wavelength cut-offs are specified in the same distance units as the grid. The default is to apply no filters. To apply a low-pass filter, specify only the short wavelength cut-off. To apply a high-pass filter, specify only the long wavelength cut-off. A word of caution: if the data from the contractor was within specification, there should be no need to filter, and it may do more harm than good if applied indiscriminately. If filtering is felt to be necessary we recommend doing it prior to the gridding process so that its effect on the line data can be assessed visually. Non-linear filter tolerance: The non-linear filter is a low-pass filter that removes signal based on logic and has two important advantages over linear filtering alone: 1. It ignores signal power and can therefore cleanly remove very strong signal without the familiar long-wavelength residual left by linear filters. 2. When data does not contain signal to be removed it is not changed at all and therefore contains all the original information. The decision to remove signal is based on wavelength only. If the data is considered part of the short wavelength information, and if it exceeds a specified tolerance, it is replaced by a smoothly interpolated value based on the neighbouring values. Because of this, the nonlinear filter is ideal for removing data spikes. Normally our final delivered located data from contractors is expected to have had all the spikes removed but not to have been smoothed. Pre-filter sample increment: If the distance between each data point on a line differs by more than a certain tolerance (expressed in %), the line is first re-sampled at a specified interval using the down-line spline. The data is then filtered and splined again to the grid cell 7

8 size. By default, resample interval is set to the smaller of the grid cell size or the smallest data interval on the first survey line, and the tolerance is 2% Effect of interpolation parameters For BIGRID, the down-line spline method has very little influence on the interpolated grid or its power spectrum due to the dense sampling along lines. The spline across lines can have a much more significant influence Across-line spline method: Power spectra for the Menzies survey interpolated to a 12.5m grid cell using Akima and Cubic splines are shown in Figure 10 (the actual images are very similar and hence are not shown). The power spectra are very similar in the along-line direction, while in the diagonal and across-line directions the Akima spline has much more power. This occurs because Akima splines can have greater curvature around grid points than the Cubic spline. The increased curvature allows the Akima spline to overcome commonly accepted problems with overshoot experienced with Cubic splines. The more rapid fall-off in power of the Cubic spline with increasing frequency is further emphasised by the radially averaged power spectrum (Figure 11). We believe that the Akima spline is a superior interpolation method because it is less prone to overshoot. Therefore, we would recommend using the Akima spline exclusively for interpolation Cell size: We investigated the effect of cell-size on the output of BIGRID by interpolating the Menzies survey to pixel grids of 10m (1/5 line spacing), 12.5m (1/4 line spacing), 16.7m, (1/3 line spacing) and 25m (1/2 line spacing). As expected the main effect of increased cell-size is a smoothing of detail in the interpolated image (Figure 12). The power spectra are perhaps more revealing with the main effect of a smaller cell size being an increase in the Grid Nyquist cut-off (Figure 13). The spectra all look similar within the region of overlap (ie. the cell size doesn t adversely effect the output). The elongated form of the power spectrum certainly appears to imply that the BIGRID method takes advantage of the rapid sampling along lines (this issue is investigated elsewhere in this report). Across-lines the high frequency content of the power spectrum is essentially due to spectral extrapolation (ie. the Across Line Nyquist is less than the Grid Nyquist). Therefore, it really becomes a matter of choice between loss of sampled information along-lines and inference of structure across-lines. We recommend that a good compromise is 1/3 to 1/4 of the line spacing. We believe that 1/5 of the line spacing is probably too fine, while ½ is too coarse. Note that the line spacing is critical when it comes to performing image enhancements in the Fourier domain. As an example, downward continuation involves multiplying the Fourier spectrum of the image by an exponential function. This exponential increases rapidly with frequency and if the cell size is very fine, the very highest frequencies will be substantially amplified (Figure 14). These high frequencies will generally be a mixture of noise (in the along-line direction) and interpolation effects (in the across-line direction), therefore, their amplification is undesirable. For any Fourier filtering that involves amplification of high frequencies we urge caution and recommend a cell size no finer than 1/3 of the line spacing. 3.2 Geosoft RANGRID (minimum curvature with tension) The RANGRID algorithm is similar to the minimum curvature algorithm of Briggs (1974) and Swain (1976) and allows a variable tension parameter to overcome problems with overshoot (Smith and Wessel, 1990). 8

9 RANGRID first estimates grid values at the nodes of a coarse grid (usually 8 times the final grid cell size). This estimate is based upon the inverse distance average of the actual data within a specified search radius (weighted average where weight coefficients are calculated from 1/distance from node). If there is no data within that radius, the average of all data points in the grid is used. An iterative minimum curvature method is then employed to adjust the grid to fit the actual data points nearest the coarse grid nodes. After an acceptable fit is achieved, the coarse cell size is halved. The same process is then repeated using the coarse grid as the starting surface. This procedure is iterated until the minimum curvature surface is fitted at the final grid cell size. A very important parameter in the RANGRID process is the number of iterations used to fit the surface at each step. The greater the number of iterations, the closer the final surface will be to a true minimum curvature surface. However, the processing time is proportional to the number of iterations. RANGRID stops iterating when: It reaches a specified maximum number of iterations, or A certain percentage of the actual data-points are within a limiting tolerance of the minimum curvature surface. By default these limits are 100 iterations and 99% of points within 1% of the data range Interpolation parameters Low Pass desampling factor: The de-sampling factor is a function of the grid cell size. This factor effectively acts as a low-pass filter by averaging all the X, Y points into the nearest cell defined by this factor. The default is 1, which produces no pre-filtering other than dealiassing at the size of the grid cell. Blanking distance: All grids cells further than the blanking distance from a valid point will be set to a null value. Tolerance: The tolerance to which the minimum curvature surface must match at the real data points. Default is 1.0% of the Z range of the data. Some care must be exercised when choosing this parameter if there are relatively flat areas in the data set. For example, with a data range of 2000 nt, a 1% tolerance would give a 20 nt error. This error would be more obvious in a flat area than in an area of high gradient. An isolated high amplitude anomaly in an otherwise flat survey might need adjustment of this tolerance. Percentage pass tolerance: Percentage of points that must lie within the tolerance limit about the actual data points, before the current step in the processing sequence is terminated. Default is 99%. Maximum iterations: Maximum number of iterations allowed to reach a solution. This number limits the computational time but should be increased if the solution is not reached in the required number of iterations. Tension: The degree of internal tension placed upon the minimum curvature surface (between 0.0 and 1.0). The default is to apply no tension (0.0), which produces a true minimum curvature grid. By increasing internal tension, overshooting problems in unconstrained, sparse areas can be avoided. This will also result in an increase in curvature (and introduction of higher frequencies) in the vicinity of the real data points. 9

10 3.2.2 Effect of interpolation parameters Tension: Tension causes increased curvature in the vicinity of data points and is intended to overcome possible problems with overshoot and undershoot which are associated with minimum curvature surfaces. The Menzies survey was interpolated to a 12.5m grid cell using tensions of 0, 0.2, 0.4, 0.6, 0.8 and 1.0 (Figure 15). For this particular survey there is little obvious difference in the interpolated images, although the high tension image does have a slight rippling parallel to the survey lines. The power spectra are also similar, with each being elongated in the direction perpendicular to the survey lines (Figure 16). The differences in the power spectra are more apparent when radial averages are calculated (Figure 17). The spectra are similar down to about m -1 (spatial wavelength ~70 m) at which point the surfaces with increased tension decay slower than those with lower tension. This occurs because tension increases the curvature around the data points and hence increases the amount of high frequency power. Our preference is to use zero tension (ie. minimum curvature), because then there is no extra parameter to tune. Also, with adequately sampled data, tension should have little influence on anomaly shapes, unless the anomalies are small in comparison to the line spacing. We recommend that non-zero tension be used only when the line spacing is large and there are rapid changes in gradient in the across-line direction. In that case the minimum curvature output should be compared to the output with moderate tension (say 0.2) Cell size: We investigated the effect of RANGRID output by interpolating the Menzies survey to pixel grids of 10m (1/5 line spacing), 12.5m (1/4 line spacing), 16.7m, (1/3 line spacing) and 25m (1/2 line spacing). As expected the main effect of increased cell size is increased smoothing (Figure 18). The power spectra for the different images reveal that RANGRID causes an elongation in power perpendicular to the line direction (Figure 19). This is in complete contrast to BIGRID where the power spectrum was elongated along lines (Figure 13). This seems to indicate that RANGRID is unable to exploit the fast sampling along lines (this point is investigated further elsewhere in this report). As with BIGRID, in the areas of overlap the power spectra are very similar. Given the apparent inability of RANGRID to exploit the dense sampling along lines, there is little point in a grid-cell finer than 1/3 of the line spacing (also see the comments made on choosing the cell size for BIGRID). 3.3 EXACT MINIMUM CURVATURE (THIN-PLATE SPLINES) Implements minimum curvature exactly by using the so-called thin-plate spline (Duchon, 1976). This consists of a sum of a set of weights times translates of a fixed basis function (the thin-plate spline) supplemented by a linear polynomial. The thin-plate spline has been popular in the terrain modelling and geostatistical literature (eg. Hutchinson and Gessler, 1994), but applications to geophysics have been limited due to a very poor computational scaling. Solving the equations by simple methods results in at least an eight-fold increase in computer time and four-fold increase in memory requirements, when the number of points is doubled. As explained in Billings and Fitzgerald (1998) this poor time and memory scaling can be avoided by implementing a three-step process. Very recently, Rick Beatson from the University of Canterbury in New Zealand has refined this process and is able to interpolate large geophysical surveys very quickly (for example a 100,000 point magnetic survey required only five minutes). Further improvements to his code are being implemented and he soon believes he could do a 20 million point survey in 30 minutes or less. Without regular access to Beatson s code, I implemented a simple segmentation routine, which breaks the survey area up into small overlapping segments. The thin-plate spline equations are solved within each segment with the values in the overlapping regions linearly 10

11 weighted with distance from the respective segment edges. I investigated the error between this segmentation routine and an exact thin-plate spline solution provided by Beatson. There was very little difference between the two methods, with a maximum relative error of 0.4%. The advantage of the thin-plate spline method is that it specifies a surface that is defined over all of space. This surface is independent of the grid-cell size used for the interpolated image (in contrast to RANGRID for example, where the choice of cell size directly influences the solution). Further, it is possible to Fourier transform this surface directly, rather than to generate just the discretised version which any grid based method will produce (Billings, 1998). This overcomes many of the problems with FFTs discussed above (such as cell-size, edge mismatch, survey gaps etc.) and results in a power spectrum free of edge artefacts (Figure 20). The exact method does raise its own set of computational considerations so we will not consider this route to Fourier transformation further in this report. We will, however, note it as a promising technique for the future. 3.4 INTERPOLATION USING THE IMAGE PROCESSING SYSTEM The interpolation algorithm used by Rio Tinto s (formerly CRAE s) Image Processing System, IPS (and also by the image analysis package Dimple), consists of nearest neighbour interpolation, followed by successive applications of a smoothing filter which reduces its window size after each iteration Interpolation parameters Apart from the usual parameters (such as grid cell size; method for selecting multiple observations in a given grid cell etc.) the three main parameters which control the output of the IPS gridding algorithm; are search radius, smoothing iterations; and smoothing radius. Search radius: The search radius is half the distance, in pixels, over which interpolation is to be performed. A zero radius sets one pixel at each observation. If the radius is too small the image will have gaps. A larger radius may remove gaps but will take longer to process. The maximum radius you can specify is 20 pixels. Number of iterations: After interpolation, filtering can be applied to smooth the image. The filtered value for a pixel is based on a weighted average of pixels within a calculated radius. After each iteration the radius reduces, producing a spiral effect. If you specify zero iterations no filtering is performed and a blob the size of the search radius is generated around each observation. A good guide is to keep the numerical value of this parameter the same as the search radius. The more iterations you specify the longer the processing will take. The number of iterations multiplied by the search radius must be less than 226. Smoothing radius: If your data has areas with few observations and other areas with comparatively many observations, the frequency information in the image may not be distributed uniformly. A smoothing radius used in conjunction with filtering can help to reduce this effect. The smoothing radius, in pixels, must be between 0 and the search radius. Ideally it should be at or just below the average spacing between observations in your data (non-ideal for magnetics where along- and across-line sampling rates differ so markedly) Effect of interpolation parameters The search radius for IPS gridding does not influence the final image as long as it is large enough to ensure that there are no blank values between survey lines. The number of 11

12 smoothing iterations and the smoothing radius on the other hand, do influence the quality of the interpolated image. To investigate their effect the 200m line spacing Mortlock survey with a mean terrain clearance of 60m and a 50m pixel size was used (full specifications are here). The survey lines were flown at a bearing of 65 o and XYROTATE was used to align the survey-lines North-South. This survey was used in place of Menzies as it contains many faults and we wished to determine the effect of the smoothing parameters on abrupt geological features. Six different sets of parameters were trialled: zero smoothing iterations; one smoothing iteration, smoothing radius of one; three smoothing iterations, smoothing radius of one; six smoothing iterations, smoothing radius of one; six smoothing iterations, smoothing radius of two; six smoothing iterations, smoothing radius of three. As expected, increasing either the number of smoothing iterations or the smoothing radius causes the interpolated image to be smoother (Figure 21). With too few smoothing iterations the image is very blocky, while with too large a smoothing radius the image becomes blurred. The increased blurring with number of smoothing iterations and smoothing radius is further evident from the power spectra (Figure 22). For six smoothing iterations and a radius of two, some artefacts begin to creep into the power spectrum. These become even more evident when the smoothing radius is increased to three. As with RANGRID the power spectrum is elongated perpendicular to the survey lines, indicating that the method cannot exploit the dense sampling along-lines. IPS gridding is a very rough method for interpolation. It is particularly unsuited to the interpolation of line-based data because of the circular smoothing kernel. We recommend that it not be used for gridding of aeromagnetics 3.5 NATURAL NEIGHBOUR INTERPOLATION This is an interpolation method which is based on a Delaunay tessellation of the survey data (Watson, 1992; Sambridge et al., 1995). The Delaunay tessellation gives the least long and thin triangles that join nearby points in the survey (Figure 23). Around each survey point a Voronoi cell is then created by bisection of each line in the Delaunay tessellation. For any location within a given Voronoi cell, all the points with adjoining Voronoi cells are the natural neighbours of that point (Figure 24). To interpolate the field at an arbitrary point, a new Delaunay tessellation and set of Voronoi cells are calculated with the new point included. The amount of overlap of the new Voronoi cell with each original Voronoi cell determines the weight that is applied to each point in calculating the new field value (Figure 25). The natural neighbour method has the following useful properties: 1. the original function values are recovered exactly at the reference points; 2. the interpolation is entirely local (every point is only influenced by its natural neighbour nodes); and 12

13 3. the derivatives of the interpolated function are continuous everywhere except at the reference points. The dense sampling along-lines but large separation between survey lines causes the triangles in the Delaunay tessellation to be very long and thin. See for example in Figure 26 which shows part of the Delaunay tessellation for a 50m line spacing survey with 7 metres between the along-line samples. As the line spacing increases the triangles become longer and longer, but maintain approximately the same width. This highly skewed nature of the Delaunay triangles, and consequently the Voronoi cells, means that the method is probably not suited to the interpolation of aeromagnetic data Interpolation parameters The only interpolation parameters are a choice of several different methods for calculating the weights of each of the natural neighbour points. We have used the recursive method recommended by Sambridge et al. (1995). One feature of the natural neighbour method is that there are no tuneable parameters Application to the Menzies surveys When the natural neighbour method was applied to the six Menzies surveys the time required for interpolation actually decreased as the number of points increased. This behaviour was completely counter to intuition and was investigated further by Malcolm Sambridge (the author of the software). He found that the time required to create the Delaunay tessellation increased with the number of survey points as one would expect (Table 1). However, the time required to then interpolate the data to the same grid decreased with the number of survey points. The reason for this behaviour is that the smaller data set results in a much larger number of natural neighbours of each interpolation point and the time for interpolation is a direct function of this parameter. This computational scaling would not hold in general (say if the survey points were randomly distributed) and only occurs because of the line based nature of the data collection. Survey # Survey points # Delaunay triangles Time for Delaunay (s) Total time (s) 50m line spacing 300m line spacing Table 1: Timings on a Sun-Sparc 4 for interpolation of the 50 and 300m line-spacing Menzies surveys. Both surveys were interpolated to the same 262 by 417 grid. 3.6 COMPARISON OF INTERPOLATION METHODS The Menzies 50m line spacing survey was again used for the purpose of evaluating the different gridding methods. From this base 50m line spacing survey five other surveys were created by leaving out acquisition lines. These new surveys were at 100, 150, 200, 250 and 300m line spacing and were used to analyse how under-sampling affected the performance of the different algorithms. All surveys were interpolated to a common 25m pixel grid. 13

14 3.6.1 Geosoft BIGRID The BIGRID images are shown in Figure 27. A visual inspection indicates that, at the scale shown, there are only minor differences between the 50 and 100m line spacing surveys. The difference becomes more obvious at 150m line spacing, at which point the bi-directional nature of the interpolation is apparent. At 200m line spacing, anomalies that were continuous at 50 m, break up into several discrete anomalies. At 300m some anomalies completely disappear (eg. red anomaly in the southeast corner). At 50m line spacing the power spectrum is almost radially symmetric Figure 28, with a slightly elongated shape parallel to the survey lines. At 100m line spacing the loss in acrossline structure is evident with the power spectrum becoming compressed in that direction. This compression becomes even more pronounced as the line spacing increases. Even at 300m line spacing, there is a concentration of power in the along-line direction, indicating that BIGRID exploits the fast sampling along lines. Perhaps most striking about the power spectra are the horizontal concentrations of power which are separated by a vertical frequency of one over the line spacing. This corresponds to features with a spatial wavelength equal to the line spacing Geosoft RANGRID Images interpolated using RANGRID are given in Figure 29. There are only minor differences between results for 50 and 100m but they become much more obvious at 150m. By 200m the distinct linear feature running through the centre of the image (at a bearing of ~325 0 ) is broken into a series of individual anomalies. At 300m line spacing only the main long wavelength features are preserved. Close inspection of the wider line spacing images reveals a number of bull s eye artefacts (isolated highs or lows). These correspond to anomalies that have been sampled by only a single survey line. In these situations BIGRID had a tendency to stretch these anomalous features in the across-line direction (Figure 27). This may be a reason to prefer RANGRID when the field is under-sampled. RANGRID causes the power spectrum to be oriented perpendicular to the survey lines (Figure 30). The changes in the power spectrum with line spacing are relatively subdued. The main effect of the transition from 50 to 100m is a loss in the high frequency content across lines. At 150 m, the power spectrum begins to lose even more power perpendicular to the survey lines but does appear to retain it in the off-diagonal directions. Power spectra for 200, 250 and 300m line spacing surveys are very similar Exact thin-plate spline Images interpolated using the exact thin-plate spline are shown in Figure 31. They are very similar to RANGRID for line-spacings of 150m or less. By 200m there are some differences with the RANGRID equivalents. For example, the linear feature trending at appears to be better represented by the thin-plate spline than by RANGRID. The power spectrum for the thin-plate spline with a 50m line spacing (Figure 32) is almost identical to the RANGRID equivalent at low frequencies (Figure 30). The main difference is the absence of the high frequency power in the vertical direction for the thin-plate compared to RANGRID. At all other line spacings the RANGRID and thin-plate spline spectra are very similar except at the highest frequencies. Notice that at 200m line spacing, there is a concentration of power along the vertical axis. We believe this is due to an artefact introduced by the segmentation routine that we used to solve the thin-plate spline equations. 14

15 3.6.4 IPS Gridding Images for IPS gridding using six smoothing iterations and a smoothing radius of two are shown in Figure 33. Again there are only minor differences between 50 and 100m line spacing. At 200m line spacing the linear feature at 315 o bearing has been decomposed into several individual anomalies. Notice that anomalies in the wider line-spacing images are almost circular. This results from the circular smoothing kernel used by IPS gridding and is our main objection to the method. The power spectra are shown in Figure 34. At 50m line spacing the spectrum is elongated in the across-line direction. As with RANGRID this indicates that the method cannot exploit the fast sampling along-lines. As the line spacing increases the spectra flatten in the along line direction. The artefacts in the power spectrum are particularly noticeable at m line spacing. As with BIGRID, these appear to have a spatial wavelength equal to the line spacing Natural neighbours Images for natural neighbour interpolation are shown in Figure 35 and appear to be reasonable up to a line-spacing of 150m. At 200m spacing, isolated anomalies (bull s eye artefacts) become noticeable and by 300m are a dominant feature of the image. These occur because the interpolant is constrained to pass through all the data points. This condition can be relaxed as discussed by Watson (1992), but is not considered here. The bull s eye anomalies cause the vertical concentrations of power in the frequency domain plots (Figure 36). The troughs in these vertical features correspond to a spatial wavelength approximately equal to the line spacing. As with RANGRID and IPS gridding the power spectra are elongated perpendicular to the survey lines indicating that the method cannot exploit the dense along-line sampling. Natural neighbour interpolation is slow for surveys with wide line spacing and causes noticeable bull s eye artefacts. As it is the most sophisticated implementation of interpolation based on triangulation we can conclude that these types of methods are unsuitable for aeromagnetic surveys Comparison of interpolation methods Radially averaged power spectra for all five interpolation methods at 100, 200 and 300m line spacing are shown in Figure 37. For the 100m line-spacing survey the radially averaged spectra are very similar at low frequencies and begin to differ only at about m -1 (spatial wavelength of 130m). As expected, RANGRID and the exact thin-plate spline have very similar radially averaged spectra at all frequencies. The BIGRID spectrum decays the fastest, with RANGRID decaying second fastest but is overtaken by IPS gridding at higher frequencies. Natural neighbour interpolation decays the slowest and has a distinct spectral flattening at a frequency of about 0.012m -1 (spatial wavelength ~80m). This indicates that the natural neighbour grid is texturally the roughest of the five images. At 200m line-spacing, the radially averaged power spectra for BIGRID, RANGRID and the exact thin-plate spline are almost identical down to about 0.015m -1 (spatial wavelength of 66m) from where the BIGRID spectrum decays faster. The smoothest image is for IPS gridding which has a distinct set of maxima and minima at high frequency. These artefacts lie outside the grid Nyquist corresponding to a pixel size of one quarter of the line spacing (frequency 0.01m -1 or spatial wavelength 100m). Again natural neighbour interpolation has the slowest spectral decay. At 300m line-spacing the power spectrum for natural neighbour interpolation falls off fastest down to a frequency of about m -1 (spatial wavelength 23m) before being overtaken by 15

16 IPS gridding. BIGRID initially decays the slowest before dipping under the other methods. At very high frequencies BIGRID has a slight increase in power but the spectrum contains little power in this region of the frequency domain. 3.7 Is there a benefit in fast sampling along lines? We already suspect that BIGRID is sensitive to the fast sampling along lines, while RANGRID is not. To determine whether this is indeed the case we successively eliminated points from the Menzies survey. The original survey had an along-line spacing of 7m and from it we formed datasets with every second point (14m spacing), every fourth (28m spacing), every eighth (56m spacing), every sixteenth (112m spacing) and every thirtysecond point (224m spacing) BIGRID Images for BIGRID with every point, every second, fourth, eighth, sixteenth and thirty-second point included are shown in Figure 40. There is little obvious difference between the images at the scale shown until only every sixteenth point is included. At that level of coarse sampling the bi-directional nature of the interpolation is evident and becomes even more obvious when only every thirty-second point is included. The difference become more obvious when the power spectra are compared (Figure 41). There is virtually no difference between the power spectra for full sampling and with every second point included. When only every fourth point is used, the power spectrum in the along-line direction falls-off faster. This corresponds to a sampling rate of 28m compared to a grid-cell size of 12.5m. Horizontal striations develop in the power spectrum, and become more noticeable as more points are eliminated. By eight points (56m spacing), the power spectrum has obvious highs and lows in the along-line direction and these become even more pronounced by sixteen points. At sixteen points the power spectrum becomes elongated in the across line direction. This occurs when the along-line sampling (112m spacing) is further apart than the line spacing (50 m). For thirty-two points (224m spacing) the power spectrum is seriously corrupted and reduces to concentrations of power along the two frequency axes. The radially averaged power spectra also tell an interesting story (Figure 44). Results from full sampling down to every eighth point are almost identical until a frequency of 0.02 m -1 (features 25 m apart). The mid-to-high frequency structure of all the images differs considerably, while at the highest frequencies the spectra are identical for full sampling, for every second and for every fourth point. In conclusion, BIGRID output appears to be sensitive to sampling rates at about the level of the pixel used in the gridding (12.5m in this example). Below that level there is little difference in the interpolated image or its power spectrum (cf full sampling and every second point included). When the sampling rate exceeds that level there is a loss of information in the along-line direction RANGRID Images for RANGRID with every point, every second, fourth, eighth, sixteenth and thirtysecond point included are shown in Figure 42. There is little obvious difference between the image for full sampling and those for every second, fourth and eighth points. Even with every thirty-second point included there are only some minor differences from the result for full sampling. The power spectra support this observation (Figure 43). Spectra for full 16

17 sampling and for every second point included are almost identical, with only subtle differences in the high frequency regions from the spectra for every fourth and eighth point. With every sixteenth point the lower frequency parts of the spectrum change slightly. This change becomes more evident for every thirty-second point. The radially averaged power spectra for all six images are almost identical (Figure 44). The analysis indicates that RANGRID is unable to exploit the fast sampling along-lines, something we had already inferred from the elongated form of RANGRID power spectra. 3.8 Conclusions In summary: BIGRID: Exploits the fast sampling along-lines but distorts anomaly shapes when the line spacing is too wide. This is our preferred front-line method especially when the survey area has been adequately sampled. The method is very fast so that if the resulting image is unsuitable little time is wasted in the calculation and evaluation. RANGRID: Is unable to exploit the fast sampling along-lines but results in less distortion of anomalies at wide line spacing. This is our preferred method for under-sampled data or where the results for BIGRID are unacceptable. Exact thin-plate spline: As for RANGRID, except that it does appear to represent linear anomalies better at wide line spacing. This method would be preferred to RANGRID if it were ever implemented in Geosoft. IPS Gridding: Causes anomalies to be circular and also has some artefacts in the Fourier domain. We recommend that this method not be used for gridding of aeromagnetic data. Natural neighbour interpolation: Is slow to run on widely spaced line data and causes bull s eye artefacts. We recommend that methods based on triangulations are not used for aeromagnetic data. We believe the method may be suitable for interpolation of geochemical or gravity data with variable sample spacing. 17

18 4 QUALITY CONTROL WITH THE FFT As discussed in the section on Fourier transforms application of a window function is critical to the calculation of the FFT, otherwise edge mismatch may mask real problems in the image. Some means of filling survey gaps is also required but must not contribute a significant amount of power at high frequencies. SB implemented a thin-plate spline method in Matlab because the Maximum Entropy and Inverse Distance Squared methods in Geosoft were unsatisfactory. Either SB s Matlab method or a method similar to the Intrepid PLUG routine be included in Geosoft. The application of the Matlab method to the Gunanya aeromagnetic survey is illustrated in Figure 45. There can be benefits in partitioning the survey to allow any changes in survey or image quality to be determined. The decision of where to partition the image should be based on changes in survey characteristics - different aircraft or instrumentation, different line direction, significant geological changes etc. However, where possible the segments should fill an entire rectangle so that there is no need to use a gap-filling algorithm. Additionally, the dimensions of the subset should be acceptable to the Winograd FFT algorithm. Including such a subsection tool in Geosoft would be relatively straightforward. In the section which follows, the power spectra for seven different surveys are studied. Several have line-levelling problems, others have distinct geological features, and four are heavily filtered grids supplied by contractors. 4.1 Block F: levelling errors This 300m line spacing survey was flown in 1978 by Geometrics at a mean terrain clearance of 80m. The survey lines were N-S and the magnetometer sample rate was once per second (~ 60m). Full survey specifications can be found here. The survey is dominated by long wavelength variation in magnetic intensity with some regions of higher frequency variation (Figure 46). The vertical striations show that the survey has not been levelled correctly. When interpolated to a 75m grid cell with BIGRID the power spectrum shows the elongated form characteristic of bi-directional splines (Figure 46). The most obvious feature in the spectrum is the concentration of power along the horizontal frequency axis. This arises from the line levelling problems in the survey. Also noticeable in the spectrum are some power concentrations at 30 o anti-clockwise from the vertical axis. These are due to the N60E trending geological feature (fault?) in the original image. This example illustrates a very important concept in the interpretation of power spectra. When linear features are orientated in a particular direction (N60E), the spectrum contains a concentration of power in the perpendicular direction (30 o anti-clockwise from the vertical axis). 4.2 Block C: levelling errors with angled survey lines This 300m line spacing survey was flown in 1978 by Geometrics at a mean terrain clearance of 80m. The survey lines were NE-SW and the magnetometer sample rate was once per second (~ 60m). Full survey specifications can be found here. The survey was interpolated to a 75m grid using BIGRID and demonstrates the effect of line levelling errors (Figure 47) when the line direction is not parallel to an image edge. By default, Geosoft creates grids aligned with the coordinate system, which in this example causes the line levelling errors to be manifest as a diagonal concentration of power. As with the last example this region of high power is oriented perpendicular to the direction of the 18

19 survey lines. Even though the survey lines were at 45 0 to the image axis, there is still a relatively large concentration of power along the horizontal axis (compared to the vertical axis). This may occur because BIGRID first interpolated values along the horizontal direction (E-W). There are some obvious structural features in the power spectrum but these are difficult to relate to image objects because of the highly skewed nature of the power spectrum. I would therefore recommend that (where ever possible) for quality control the survey lines are explicitly rotated (using XYROTATE) so that they are either North-South or East-West. 4.3 Argyle: Analysis of a contractor s grid This 300m line spacing survey was flown by Geometrics in 1980 at a mean terrain clearance of 100m. The magnetometer sample rate was 1.2 seconds, which approximately corresponds to a 50m along line sample. Survey lines were orientated NE-SW, and full survey specifications are available here. The survey was interpolated to a 100m pixel using RANGRID, with a contractor s grid at the same spacing also available. A close inspection of both images reveals that the grid supplied by the contractor is much smoother than the RANGRID image (Figure 48). This is confirmed by the power spectra for the two images (Figure 49). The fall-off with frequency is also much faster than that obtained with either BIGRID or IPS (the latter with search radius 8, 6 smoothing iterations and a smoothing radius of two pixels, Figure 50). Both power spectra have linear features orientated at ~60 o anti-clockwise from vertical, which corresponds to the numerous features trending at about N30E (several are marked with A s in Figure 48). The horizontal striations are probably due to the textured region in the far North of the image (marked with a B). The levelling errors which are obvious in the original image are not at all obvious in the power spectrum. 4.4 MIDDALYA: A PROBLEM IN THE CONTRACTOR S GRID Aerodata flew the survey at Middalya in 1978 at 300m line spacing and a mean terrain clearance of 80m. The magnetometer sample rate was once per second, or approximately 60m along N-S survey lines. More detailed survey specifications are available here. The data were gridded to a 100m pixel using BIGRID, with a contractor s grid at the same spacing also available (Figure 51). The survey has a relatively long wavelength variation in magnetic intensity and some very obvious levelling errors which cause the horizontal banding in both power spectra (Figure 52). Also noticeable are diagonal features in the power spectrum oriented at 45 o clockwise from vertical, which are due to some NW striking geological features. The contractor s grid is much smoother and has a power spectrum that is more symmetric than the BIGRID version. The radially averaged power spectra for both grids are almost identical down to ~ m -1 (spatial wavelength 1800m; Figure 53). Most obvious in the power spectrum for the contractor s grid is a feature at ~ to m -1, which corresponds to a spatial wavelength equal to the line spacing, or three times the grid cell size. This feature may be due to the radius of some type of smoothing operation (explicit or implicit) imposed by the contractor s gridding algorithm. 19

20 4.5 Duketon: A high quality survey Duketon is a recent, high quality survey flown by Kevron in 1994 at 50m line spacing and a mean terrain clearance of 50m. The magnetometer sample rate was 0.1 seconds, or approximately 7m along lines. Survey lines weren45e, and prior to analysis were rotated so they were horizontal (using XYROTATE with x o =417300, y o = and θ=45 0 ). Full survey specifications can be found here. The image was interpolated to a 12.5m pixel using BIGRID (Figure 54), with the power spectrum displaying the characteristic elongate form of bi-directional splines. The only obvious feature in the power spectrum is a North-East trending concentration of power corresponding to several NW trending structures in the original image (marked with A s). This power spectrum reveals no obvious problems. However, the slight broadening of the power spectrum in the vertical direction compared with the diagonal may be due to imperfect line levelling. This broadening of the spectrum appears to be characteristic of almost all aeromagnetic surveys studied to date. 4.6 Warriedar: A piecewise analysis At the time of writing the survey specifications were not available, nor was the line-data. The contractor-supplied image had a 50m pixel indicating a probable line spacing of 200m. This example illustrates that there can be substantial benefits in partitioning the survey prior to analysis. The survey was divided into three different pieces (Figure 55). Region 1 contains NE trending geological features, region 2 consists of folded sequences, while region 3 is mostly composed of deeper geological features with several obvious lineaments. The irregular edges in regions 1 and 2 were filled with the Matlab gap filling method. The power spectra for all three regions fall off at approximately the same rate in all directions (Figure 56) probably indicating that the original grids have been smoothed considerably. Immediately obvious for region 1 is the skewed nature of the power spectrum which is due to the NE geological strike. The relatively sharp cut-off in the power spectrum along the diagonal line A-B may indicate a limit on the scale of variation in the direction parallel to the geological strike. Region 2 has a power spectrum that is close to being radially symmetric, and reflects the geological variation. The power spectrum for region 3 contains numerous linear features, which correlate with faults at different angles in the original image. For example, the geological feature trending from C-to-D (Figure 55) causes the linear feature marked as C-D in the power spectrum (Figure 56). The lineament A-B causes the various features trending in the direction A-B in the power spectrum. 4.7 Gunanya: Another piecewise analysis The Gunanya survey was flown at a 300m line spacing along N-S survey lines. Additional details of the survey can be found here. A contractor s grid with a 75m grid cell was available and was split into four segments for the purposes of analysis (Figure 57). Area 2 had several gaps, which were filled using the Matlab algorithm (see Figure 45). In area 1 the most obvious features in the power spectrum are the NE trending faults which cause a region of high power trending 45 o counterclockwise from the vertical axis (Figure 58). Less obvious is the horizontal banding due to a few minor N-S trending features. In area 2 the main features are some circular structures and several NW trending faults. These faults cause some minor concentrations of power in the frequency domain. There is also 20

21 some power along the horizontal frequency axis and this probably indicates minor levelling errors. Area 3 contains numerous faults and contacts at various orientations, which manifest themselves as a series of linear features trending at various angles (most obvious with the line Nyquist limits). The most noticeable features in area 4 are the ENE trending faults. Small variations in their orientations result in several slightly rotated features in the power spectrum. Less distinct are the structures trending perpendicular to the faults. Lastly, the radially averaged power spectra for areas 1, 3 and 4 are relatively similar (Figure 59). Area 2 has a more rapid fall-off in power, reflecting its smoother variation. Notice that each radially averaged power spectrum has a distinct change in the decay rate at about m -1, which corresponds to a sampling rate just exceeding 100m. As with the Middalya survey this may be due to some type of smoothing in the gridding algorithm used by the contractor. 5 DISCUSSION AND RECOMMENDATIONS 5.1 Fourier transforms of 2-D images The Fast Fourier Transform requires that data fill an entire rectangle. Given the possibility of survey gaps and the desirability of a periodic image, a gap filling algorithm is essential. In this report we found that the two Geosoft methods (maximum entropy and inverse distance squared) are unsuitable and some alternative technique is required. So that quality control can be achieved within Geosoft we recommend the implementation of either the Matlab thinplate spline, or the Intrepid PLUG routine. Using the Matlab thin-plate spline method, the FFT was successfully used as a tool for quality control. This involved multiplying the original image by a window function which gives a better estimation of the power spectrum than a direct FFT of the original image. Implementation of this window function within Geosoft would be straightforward and is recommended. A slight modification of the Window Grid GX is also required to allow only subsections with dimensions acceptable to the Winograd FFT. 5.2 Gridding algorithms Of the five interpolation methods studied in this report the BIGRID method is preferred when the data is adequately sampled. It exploits the dense sampling along lines and is very fast to calculate. In addition, with Geosoft's new trend gridding module, it should be able to preserve linear features angled with respect to the survey lines. The Geosoft defaults appear to give the best results. In particular, we recommend a cell-size of one-third to onequarter of the line spacing and the use of the Akima splines for the across-line interpolation. The main problem with BIGRID is the distortion of anomaly shapes at wide line spacings, although this is a criticism of the survey design rather than the algorithm. For undersampled data RANGRID is recommended. Again the Geosoft defaults are adequate although care is needed in specifying the tolerance when flat regions are expected. A cell size of no less than one-third the line spacing appears suitable, as the method is unable to exploit the fast along line sampling. The method is, however able to handle smaller cell sizes (one-quarter the line spacing) without introducing artefacts in the Fourier domain. 21

22 The exact thin-plate spline would be preferred over RANGRID if a fast method were implemented within Geosoft. This thin-plate spline would be particularly suitable for Fourier domain operations as it is possible to Fourier transform a spline surface exactly. Neither IPS gridding nor natural neighbours is regarded as suitable for aeromagnetic data. The former because it is a very rough method for interpolation and causes anomalies to be circular at wide line spacing. The latter because of its peculiar tendency to slow down when the line spacing is increased, and its introduction of bull's eye artefacts at wide line spacing. We would, however, recommend that natural neighbours be trialed on irregularly spaced geochemical and gravity data. 5.3 Quality control with the 2-D FFT An atlas of power spectra has been started and we recommend that this be extended as new surveys are analysed. Some characteristic features of gridded data sets were identified: Bi-directional splines have power spectra that are elongated parallel to the survey lines. Minimum curvature power spectra are elongated perpendicular to survey lines. Levelling problems are manifest as concentrations of power perpendicular to the survey lines. Several of the contractors grids were smoothed and had artefacts in the Fourier domain (the radially averaged power spectrum was a useful tool in this context). Geological features such as faults and contacts give very distinct linear concentrations of power in the frequency domain. At all times we recommend rotating the survey lines (using XYROTATE) such that they are orientated either N-S or E-W. Preliminary trails of the quality control tools on some radiometric data-sets indicated that: Some forms of levelling applied to radiometric surveys are too heavy handed in that they severely suppress high frequency power perpendicular to the survey lines. The minimum curvature output of one commercial package preferentially smooths data in one direction. This gives a power spectrum that is noticeably skewed in the perpendicular direction. 5.4 Issues not addressed in this report Some issues identified in the original research proposal but not able to be addressed in this study were: The effect of different gridding algorithms on map products derived from grids. These include first and second vertical and horizontal derivatives, iterative inversion to simple models, analytic signal, upward and downward continuation, reduction to the pole, Euler and Werner deconvolution. Whether there is any advantage in working with profile data rather than grids to produce any of these derived products. 22

23 Whether a horizontal gradiometer actually confers any advantage. 6 LITERATURE SUMMARY Akima (1970) A new method for interpolation and smooth curve fitting based on local procedures : The original paper on Akima splines, which are piece-wise polynomials of degree three. The spline passes through all the data points and appears smooth and natural (in the sense that it will be closer to a hand-drawn curve than other methods such as cubic splines). Each piece-wise polynomial extends between two data-points with the polynomial coefficients determined by reference to the value and slope of the two data-points. Briggs (1974) Machine contouring using minimum curvature : The original paper on interpolation using the difference equations for minimum curvature. Superseded by Smith and Wessel (1990). Cordell and Grauch (1982) Reconciliation of the discrete and integral Fourier transforms : Investigates the relationship between the discrete and continuous Fourier transforms by using a simple 1-D example. A method for making the discrete and integral transforms equivalent is presented. It involves shifting the data sequence by integer multiples of the data interval and summing. This requires that the signal be extrapolated beyond the limits of the data interval. In the paper the authors use the equivalent source method (Cordell, 1992) for the extrapolation but note that it is currently impractical for large data sets. The reasons for this (excessive computational effort) are essentially the same as those that inhibit direct Fourier transformation of thin-plate spline surfaces. Harris (1978) On the use of windows for discrete harmonic analysis with the Discrete Fourier Transform : Provides a review of 23 different functions used for data windowing. The paper is principally concerned with signal processing applications (detection of harmonic signals in broad band noise) but the results are also pertinent for image processing. There is a good discussion on the necessity of data windowing and its effects on various figures of merit. Harris concludes that the Kaiser-Bessel window is the best performer of the 23 different functions. Reid (1980) Aeromagnetic survey design : Attempts to determine sensible survey parameters (height, line spacing etc.) for aeromagnetic surveys. It uses the percentage of power aliassed at a given sampling rate over an ensemble of magnetised blocks or over a single dipole anomaly. In the first case the critical sampling rate (line spacing) is ~2h (where h is the height above magnetisation), while in the latter it is ~h. Ricard and Blakely (1988) A method to minimise edge effects in two-dimensional discrete Fourier transforms : This paper presents a method for minimising edge effects based on rotation of the imaged data. The general idea is that as the grid is rotated, regridded, Fourier transformed and rotated backwards that the edge artefacts appear at different orientations in the Fourier domain and can be eliminated by averaging. Real signal, on the other hand will always appear at the same orientation in the Fourier domain. The main disadvantage of the method is that it is slow as the image must be re-gridded and Fourier transformed many times (the examples presented in the paper used 30 rotations). 23

24 Smith and Wessel (1990) Gridding with continuous curvature splines in tension : Shows how to modify the original minimum curvature equations to include a tension parameter. This results in only a very slight modification of the finite difference equations of Briggs (1974). This is a very well written paper and probably the best reference on minimum curvature interpolation by the finite difference equations. Some improvements to the original Briggs algorithm are implemented and are as follows: Removal of the regional field (linear surface only) by a least squares fit Use of successive over-relaxation to speed convergence Nesting of the interpolation in a multi-grid type approach Swain (1976) A Fortan IV program for minimum curvature : Fortran implementation of Briggs (1974) minimum curvature algorithm. Largely superseded by the Smith and Wessel (1990) paper. 7 REFERENCES Akima, H., A new method for interpolation and smooth curve fitting based on local procedures: J. Association Computing Machinery: 17, Briggs, I. C., Machine contouring using minimum curvature: Geophysics: 39, Billings, S. D., Geophysical aspects of soil mapping using airborne gamma-ray spectrometry: Unpublished doctoral dissertation, The University of Sydney. Billings, S. D. and Fitzgerald, D., An integrated framework for interpolating airborne geophysical data with special reference to radiometrics: ASEG Conference, Hobart. Burg, J. P., 1975, Maximum Entropy Spectral Analysis. Unpublished doctoral dissertation. Stanford University. Duchon, J., Interpolation des fonctions de deux variables suivant le principe de la flexion des plaques minces: R. A. I. R. O. Anal. Num., 10, Harris, F. J., On the use of windows for harmonic analysis with the discrete Fourier transform: Proceedings IEEE, 66, Hutchinson, M. F. and Gessler, P. E., Splines more than just a smooth interpolator: Geoderma, 62, Reid, A. B., Aeromagnetic survey design: Geophysics: 45, Sambridge, M., Braun, J and McQueen, H., Geophysical parameterisation and interpolation of irregular data using natural neighbours: Geophysical J. International, 122, Smith, W. H. F. and Wessel, P., Gridding with continuous curvature splines in tension: Geophysics: 55, Spector, A. and Grant, F. S., Statistical models for interpreting aeromagnetic data: Geophysics, 35,

25 Stanley, J. M., Sertsrivanit, S. and Clark, P. J., Magnetic exploration beneath a nearsurface magnetic noise source: Exploration Geophysics, 23, Swain, C. J., A Fortran IV program for interpolating irregularly spaced data using the difference equations for minimum curvature: Computers and Geosciences, 1, Watson, D. F., Contouring: a guide to the analysis and display of spatial data: Pergamon Press. 8 GLOSSARY Aliassing: The maximum frequency that can be resolved by regularly sampled data is determined by the Nyquist cut-off. Any higher frequencies present in the data will appear at lower frequencies and effectively contaminate the spectrum. This phenomenon is known as aliassing and is an inevitable consequence of discrete sampling when the sampling rate is not sufficiently rapid. For example, with a line spacing of 50 m, the maximum frequency that can be resolved is 0.01 cycles per metre. If there is signal, at say cycles per metre, then it will contaminate the spectrum at cycles per metre. The high frequency power is folded back into the lower frequency parts of the spectrum. Fourier transforms: The Fourier transform, F(u,v), of a continuous function, f(x,y), defined over 2-D space is given by [ ] ( x, y) exp 2π i( ux + ) F( u, v) = du dvf vy If (x,y) are spatial co-ordinates in units of metres, then (u,v) are frequency co-ordinates in units of cycles per metre. In real world applications it is not possible to measure the signal of interest continuously, nor is it possible to measure it indefinitely. Rather there are a finite number of samples, N, obtained over a finite domain, which means that the Discrete Fourier Transform (DFT) must be used. The DFT can be defined on an arbitrary set of points, (x n,y n ), and is usually evaluated at N frequency co-ordinates (u k,v k ), (1) N F( u, v ) = w f k k n n = 1 ( x, y ) exp 2π i( u x + v y ) n n k n k n (2) The w n are a set of quadrature weights to ensure that the DFT approximation of equation (2) agrees with the continuous form of equation (1) as closely as possible. Fourier transformation of irregularly sampled geophysical surveys by the DFT of equation (2) is theoretically possible. However, there are two difficult problems that are encountered when attempting this approach; 1. Calculation of the optimal quadrature weights is difficult and depends on the characteristics of the function being approximated. 2. When the data locations, the (x n,y n ), are not distributed on a regular grid, the DFT calculation requires O(N 2 ) operations. When N is large the computation takes a long time, especially when compared to the O(NlogN) cost of a Fast Fourier Transform algorithm (FFT) on the same sized dataset (see Table 2). 25

26 N Direct (secs) FFT (secs) , Table 2: Comparison of the cost of a direct calculation of a DFT on irregularly sampled data to an equivalent sized FFT on regularly sampled data. Calculations were made assuming that a point FFT takes one second (approximately correct on the computer used for the comparisons). To overcome these difficulties, in geophysics, we usually interpolate our data to a regular grid and then calculate the resulting DFT by a Fast Fourier Transform algorithm. With an N M spatial grid with cell spacings of x and y in the x- and y-directions respectively, the DFT is calculated on an N M frequency grid with spacings of u = 1 and v = 1. N x M y The maximum frequencies in the grid are determined by the Nyquist relationship and are given by u 1 v = 1. The 2-D DFT on regular data is then: = 2 x max and max 2 y F jk N M = x y f n = 1m = 1 nm jn km exp 2π i + (3) N M where F = ( j, k jk F ) N x M y and f f ( n x m y ) nm =,. Notice, that the quadrature weights, w n, that occurred in the DFT formula of Equation (2) have been set to a constant value of w n = x y. Inverse distance squared: A grid filling algorithm that starts by interpolating the blank areas of the grid by: 1. Replacing dummies within each grid row so that grid lines are periodic 2. Replacing dummies within each grid column so that grid columns are periodic 3. Averaging the results from the row and column filling The algorithm uses inverse distance weighting to the nearest data within each row and column. This procedure, which has been implemented in Geosoft, fills holes in the data and accounts for irregular edges of the grid. Kaiser-Bessel Window: Window functions are applied to images to avoid problems associated with finite size and edge mismatch. These allow the true power spectrum underlying the image to be more accurately estimated. If the original image has a pixel value of f nm at the n-th row and m-th column, then a new image, g nm, is formed by multiplying by the window function; i.e. 26

27 g = w nm n w Harris (1978) studied 23 different windows (for one-dimensional data) and concluded that the Kaiser-Bessel window was as good as any other window function. The Kaiser-Bessel window function is given by m f nm w n = I πα I 0 ( 2n N ) ( πα ) 0 1 / 2 where I o () is a zero-th order modified Bessel-function of the first kind, N is the number of rows in the image and α is a positive constant which controls the fall off of the window. The window function along the columns, w m, is defined analogously with m in place of n and M (the number of columns) in place of N. Following Harris (1978) this project used α=3.5. The Kaiser-Bessel window is shown compared to a triangular window function in Figure 6. Matlab thin-plate spline: This method for gap filling and edge matching was implemented specifically for this project in the Matlab software package. The basic idea of the method is to use a certain number of columns and rows around the edge of the image (minimum of 2 columns and 2 rows) and enforce continuity in image values plus various derivatives (up to 2 nd order) by applying a thin-plate spline interpolation to these edge pixels. The wrap around at opposite edges is arranged by assuming the image is periodically repeated. If the image is extended Eastwards and Northwards, the East and North most pixels are left in their current positions (segments AH and HG in Figure 60). However, the Western and Southern pixels are moved to the positions they would fill after periodic extension (to segments EF and BC respectively in Figure 60). Before these edge values are used to interpolate values within the extended parts of the image, pixels around the four corners of the image (C, D, E and H), are first used to interpolate synthetic data to the segments DE and CD. These values are then copied to the segments AB and GF, respectively, by using the assumed periodicity of the image. At the end of this process there are then image values all the way along the segment ABCDEFGHA. The thin-plate spline equations to these edge pixels (as well as a least the next column and row) are then used to interpolate values to the extended parts of the image. The method is also able to handle survey gaps by using pixels on the gap edges. Maximum Entropy: Maximum Entropy prediction is a grid filling method which samples the original data near the grid edges to determine its spectral content. It then predicts a data function that will have the same spectral signature as the original data. This means that if the original data is smooth, the predicted data will be smooth, and if the original data is noisy, the predicted data will be noisy. As a result, the predicted data will not significantly alter the energy spectrum of the original data alone. Also, this method allows noisy data on one edge of a grid to be gradually interpolated into smooth data that may be present on the opposite edge of a grid. The method is based on results given in the Ph.D. thesis of Burg (1975). Minimum curvature: The minimum curvature surface has the least curvature of all twicedifferentiable surfaces that interpolate the data. Briggs (1974) showed that this surface satisfies the differential equation, 2 2 ( z ) = f iδ ( x xi, y yi ) i 27

28 where (x i, y i, f i ) are the constraining data and δ () is the Dirac delta function. The differential equation satisfies the so-called free-edge boundary conditions, i.e. 2 z = 0 2 n n z = 2 and ( ) 0 along the edges (where n is normal to the boundary) and 2 z x y = 0 at the image corners. Briggs (1974) solved this differential equation by expressing it as a difference equation on the grid nodes. Smith and Wessel (1990) showed how to incorporate tension into the differential equation, and this results in a slight modification of the Briggs difference equation. Nyquist cut-off frequency: The maximum frequency that can be resolved with regularly sampled data. With a spatial sampling increment of x the Nyquist frequency is u = 1 max 2. x With gridded aeromagnetic data there are several different Nyquist cut-offs to be considered. The flight-line spacing and along-line sample increment give two different Nyquist cut-offs in the across-line and along-line directions (Figure 1). The data are usually interpolated to a regular grid with square pixels, which then imposes a Nyquist cut-off related to the grid cell size (Figure 1). If the grid cell size is smaller than the actual spatial sampling (as occurs across-lines), any power between the sampling and grid Nyquist frequencies is entirely due to the characteristics of the gridding algorithm. Note that there is no reason why an interpolated image has to have square pixels. Rectangular pixels will have two different grid Nyquist cut-offs in the two perpendicular directions. Plug Padding: Is a method for filling of survey gaps and edge matching that has been implemented in Intrepid. The method originated from a Fortran program written by Maurice Craig, and the only documentation appears to be an unpublished manuscript written by Maurice and Andy Green at CSIRO Exploration and Mining. Blank values in the image are assigned the average value of the four adjacent pixels (those directly above and below, and to either side, in the form of a cross). If one or more of these pixels are also blank they are ignored. The wrap around (of the East to the West edge and the North to the South edge), is easily arranged by treating points on opposite edges of the image as neighbours. Rather than a brute force approach at the finest level, the method uses a multi-grid technique. The image dimensions are always chosen to be divisible by a power of two (eg an image of 650 x 950 might be enlarged to 768 x 1152 = x ). The first iteration works on the sub-lattice of points that comprises the top-left point and all points separated from it by multiples of 2 N rows and columns (where N is the maximum power of two by which both image dimensions are divisible; in the example above N=6). The process is next repeated on the finer grid with separation distance 2 N-1, and so on, down to the finest grid with a separation of one cell. St George s Cross: A St George s Cross is a concentration of power along both axes in the frequency domain, and arises when opposite edges of an image do not join smoothly together (Figure 61). It occurs because the FFT algorithm implicitly assumes that the image is periodic and when it is not, the opposite edges are effectively step functions. Fourier transforms of step functions have large amounts of power at high frequencies. The vertical 28

29 features in the spectrum are due to mismatch in the horizontal edges of the image while the horizontal features are from mismatch in the vertical edges of the image (Figure 61). Tension: The concept of tension relates to an analogy with elastic plates, where the datapoints represent point forces applied to the plate. The total stored elastic strain energy in the flexed plate is approximately proportional to the curvature. The minimum curvature surface stores the least strain energy of all twice-differentiable surfaces that pass through all the data-points. If we imagine bending an elastic plate to interpolate the data, then extra work must be done on the plate to create any interpolant other than the minimum curvature solution. Such surfaces occur naturally when the plate is under tension. In that case the plate can still pass through the data-points but in between it will take up a shape different from the minimum curvature solution. When the tension is infinite the plate will be rectilinear between data points. In Geosoft, the tension parameter, t, is normalised to lie between zero (for the minimum curvature solution) and one (infinite tension). Figure 62 illustrates the curves interpolating a set of six datapoints for cases of t=0 (minimum curvature), t=1 (membrane) and t=0.4 (intermediate tension). The large oscillations and extraneous inflection points of the minimum curvature solution have been observed in many geological applications (eg. Smith and Wessel, 1990). The use of non-zero tension is intended to eliminate these problems. 29

30 TABLES AND FIGURES Figure 1: Along-line, across-line and grid Nyquist cut-offs assuming a survey with 200m line spacing, 10m between along-line samples and a 50m grid cell. Notice the massive over-sampling along-lines (Sample Nyquist) and the under-sampling across-lines (Line Nyquist). Figure 2: Aeromagnetic image in the Middalya area (on left) and its corresponding power spectrum (on right). Figure 3: Colour table used for the display of power spectra, with blue indicating the lowest power and red the highest. 30

31 Figure 4: Fourier transforms of imaged data are symmetrical so that only one half (actually one-half plus one column) of the power spectrum is required. Within Geosoft only the portion in the right half of the frequency plane is saved to disk as shown here. Figure 5: Downward continuation of the Menzies survey by 40m (only the SE corner of the survey is shown). On the left the result for no padding is shown, while on right the image has been padded using the Matlab gap-filling algorithm. Notice the ripples propagating into the unpadded image. 31

32 Figure 6: Kaiser-Bessel Window function. 32

33 Figure 7: Menzies survey (50m line spacing) interpolated to a 25m grid cell. From top-left to right and then down, the images are (i) original, (ii) maximum entropy extension, (iii) inverse distance squared, (iv) Intrepid s PLUG routine, (v) triangular window and (vi) Kaiser-Bessel window with α =

34 Figure 8: Power spectra for the Menzies survey (50m line spacing) interpolated using BIGRID. From top left to right and then down the power spectra are (i) original image, (ii) maximum entropy extension, (iii) inverse distance squared extension, (iv) Intrepid s PLUG routine, (v) our Matlab thin-plate spline routine and (vi) Kaiser-Bessel window with a = 3.5. Figure 9: Radially averaged power spectral for the Menzies survey, including original image, Maximum Entropy extension, Intrepid Plug routine, Matlab thinplate spline and the Kaiser-Bessel window function. The Matlab thin-plate spline and Kaiser-Bessel spectra decay much more rapidly at higher frequencies than any of the other power spectra. 34

35 Figure 10: Power spectra for the Menzies survey interpolated to a 12.5m grid using either Akima splines (on left) or Cubic splines (on right). The same colour stretch is used for both images. Figure 11: Radially averaged power spectra for the Menzies survey interpolated to a 12.5m grid cell. The power spectrum for the Cubic spline decays much faster than the Akima spline. 35

36 Figure 12: Colour density images for BIGRID with cell sizes of 10 and 25m. 36

37 Figure 13: BIGRID cell size, 1/5, ¼, 1/3, ½ Figure 14: Effect of grid cell size on downward continuation of the Menzies survey by 30m. From left to right the images are for cell sizes of (1/3), 12.5 (1/4) and 10m (1/5). At 1/3 the line spacing the image looks reasonable, while at ¼ or 1/5 of the line spacing the amplification of high frequencies degrades the spatial detail. This will have important implications for any high pass filter operations on this grid. 37

38 Figure 15: Colour-density shaded images for the Menzies survey using RANGRID with tensions of 0, 0.2, 0.4, 0.6, 0.8 and 1.0. Figure 16: Power spectra for RANGRID for tensions of 0, 0.2, 0.4, 0.6, 0.8 and

39 Figure 17: Radial power spectrum for the Menzies survey and tensions of 0, 0.2, 0.4, 0.6, 0.8 and 1.0. Figure 18: Colour density images for RANGRID with cell sizes of 10 and 25m. 39

40 Figure 19: RANGRID cell size, 1/5, ¼, 1/3, ½. Figure 20: Fourier transform of the Menzies 100m survey using (on left) an FFT on the gridded image and (on right) the exact method for Fourier transformation of spline surfaces. Notice that the transforms are quite similar in the low frequency parts of the spectrum, but the grid based method has much more power at high frequencies. 40

41 Figure 21: IPS gridding of the Mortlock survey to a 50m pixel demonstrating the effect of smoothing iterations and radius. From top left to right and then down the images are (i) 0 iterations; (ii), 1 iteration, radius 1; (iii) 3 iterations, radius 1; (iv) 6 iterations, radius 1; (iv) 6 iterations, radius 2; and (vi) 6 iterations, radius 3. Figure 22: Power spectra for IPS gridding of the Mortlock survey to a 50m pixel, demonstrating the effect of smoothing iterations and radius. From top left to right and then down the images are (i) 0 iterations; (ii), 1 iteration, radius 1; (iii) 3 iterations, radius 1; (iv) 6 iterations, radius 1; (iv) 6 iterations, radius 2; and (vi) 6 iterations, radius 3. 41

42 Figure 23: Delaunay triangulation of an arbitrary distributed collection of reference points (green dots). 5 6 A Figure 24: Voronoi cells are formed by bisecting each of the lines connecting points in the Delaunay tessellation of Figure 23. The natural neighbours of an arbitrary point, A, are given by all the points that have Voronoi cells adjoining the Voronoi cell containing A. In this example the natural neighbours of A are points 1 to 6. 42

43 B 1 C A E D Figure 25: To interpolate the field at an arbitrary point (the red dot), a new Delaunay tessellation and set of Voronoi cells are calculated with the new point included. The amount of overlap of the new Voronoi cell with each original Voronoi cell determines the weight that is applied to each point in calculating the new field value. For example the area of the polygon ABCDEA over the total area of the new Voronoi cell (area coloured in yellow) gives the weight for point 1. Figure 26: Part of the Delaunay tessellation for a 50m line spacing survey, with 7m spacing between along-line samples. The much higher sampling rate along lines causes the Delaunay triangle to be very long and thin. 43

44 Figure 27: Colour-density images for BIGRID applied to the Menzies survey with line spacings of 50, 100, 150, 200, 250 and 300m. 44

45 Figure 28: Power spectra for BIGRID applied to the Menzies survey with line spacings of 50, 100, 150, 200, 250 and 300m. Figure 29: Colour-density images for RANGRID applied to the Menzies survey with line spacings of 50, 100, 150, 200, 250 and 300m. 45

46 Figure 30: Power spectra for RANGRID applied to the Menzies survey with line spacings of 50, 100, 150, 200, 250 and 300m. The black box in each image is the Nyquist frequency for the given line spacing. Figure 31: Colour density images for exact thin-plate spline gridding on the Menzies survey with line spacings of 50, 100, 150, 200, 250 and 300m. 46

47 Figure 32: Power spectra for exact thin-plate spline gridding on the Menzies survey with line spacings of 50, 100, 150, 200, 250 and 300m. Figure 33: Colour density images for IPS gridding on the Menzies survey with line spacings of 50, 100, 150, 200, 250 and 300m. 47

48 Figure 34: Power spectra for IPS gridding on the Menzies survey with line spacings of 50, 100, 150, 200, 250 and 300m. Figure 35: Colour density images for Natural Neighbour gridding on the Menzies survey with line spacings of 50, 100, 150, 200, 250 and 300m. 48

49 Figure 36: Power spectra for Natural Neighbour gridding on the Menzies survey with line spacings of 50, 100, 150, 200, 250 and 300m. 49

50 Figure 37: Radially averaged power spectra for the 100, 200 and 300m line spacing Menzies surveys using BIGRID, RANGRID, the exact thin-plate spline (TPS), Image Processing System (IPS) and natural neighbours (NN). The dashed line in the last two images would be the Grid Nyquist if the data were interpolated to a pixel size of one-quarter the line spacing. 50

51 Figure 38: Grey-scale images of a section of the Mortlock survey interpolated using (from top left to right), BIGRID, RANGRID and IPS. 51

52 Figure 39: Power spectra for the Mortlock survey interpolated (from left-to-right), BIGRID, RANGRID, exact TPS and IPS. 52

53 Figure 40: Colour-shaded images for BIGRID with every point, every second, fourth and eighth, sixteenth and thirty-second point included. Figure 41: Power spectra for BIGRID with every point, every second, fourth and eighth, sixteenth and thirty-second point included. 53

54 Figure 42: Colour-shaded images for RANGRID with every point, every second, fourth and eighth, sixteenth and thirty-second point included. Figure 43: Power spectra for RANGRID with every point, every second, fourth and eighth, sixteenth and thirty-second point included. 54

55 Figure 44: Radially averaged power spectra for BIGRID (left) and RANGRID (right) with every point, every second, every fourth, every eighth, every sixteenth and every thirty-second point included. Figure 45: Part of the Gunanya survey area, demonstrating the performance of the Matlab gap-filling algorithm. Note the colour tables for the two images differ as the gap-filled image has also had a linear trend removed from the field. Notice that the gaps are smoothly filled by the Matlab method but lack the finer scale variation in other parts of the survey. 55

56 Figure 46: Part of the Block F survey which was flown at 300m line spacing and interpolated to a 75m grid using BIGRID. The survey was not levelled correctly which cause the N-S banding and the concentration of power along the horizontal axis in the power spectrum. The solid box is the Nyquist frequency corresponding to the 300m line spacing. 56

57 Figure 47: Part of the Block C survey which was flown at a heading of N45 0 Eat 300m line spacing and interpolated to a 75m grid using BIGRID. The survey was not levelled correctly, and this causes the diagonal banding and the region of high power along the diagonal in the power spectrum. The solid box is the Nyquist frequency corresponding to the 300m line spacing. Figure 48: Part of the Argyle survey with (on left) a grid supplied by the contractor and (on right) a RANGRID output. 57

58 Figure 49: Power spectra for the Argyle survey with (on left) a grid supplied by the contractor and (on right) a RANGRID output. The contractor s grid has been smoothed considerably which is evident by the more rapid fall off in power with increasing frequency (also see Figure 48). Both power spectra have the same logarithmic colour scale. Figure 50: Radially averaged power spectra for the Argyle survey for BIGRID, RANGRID, IPS and the original grid supplied by the contractor. The contractor s grid has obviously undergone some smoothing. 58

59 Figure 51: Middalya survey with contractor s image on left and BIGRID output on right. Figure 52: Power spectra for the Middalya survey, on left contractor s grid and on right BIGRID image. The same scale has been used for both power spectra. 59

60 Figure 53: Radially averaged power spectra for the Middalya survey for the contractor s grid and the BIGRID image. The contractor s grid has been significantly smoothed. Figure 54: Part of the Duketon survey interpolated to a 12.5m grid cell using BIGRID and its associated power spectrum. 60

61 Figure 55: Warriedar survey, which was broken into three separate pieces for the analysis. Figure 56: Power spectra for the three segments of the Warriedar survey given in Figure 55. From left-to-right the spectra are for areas 1, 2 and then 3. 61

62 Figure 57: Gunanya image which was split into four pieces for the analysis. Area 1 is the NW corner, area 2 is NE corner, area 3 is the SW corner and area 4 is the SE corner. 62

63 Figure 58: Power spectra for the four areas of the Gunanya survey identified in Figure 57. From top left to right and then down the images are for Area 1, 2, 3 then 4. The same logarithmic stretch has been applied to each image. Figure 59: Radially averaged power spectra for the four areas of the Gunanya survey identified in Figure

64 Figure 60: The Matlab thin-plate spline method attempts to create an image that is periodic, with a seamless join between opposite image edges. Here an image with initial dimensions of 256 x 412 pixels is extended to an image with 336 x 504 pixels (the objective is to interpolate values to all pixels in the two boxes, ABDEA and HEFGH). The parts of the image outside the red box are the periodic extension of the original image. Figure 61: An example of a power spectrum displaying St George s Cross contamination due to edge mismatch (on left), and the power spectrum of the same image when the edges are smoothly joined by the Matlab thin-plate spline routine (on right). The Matlab image still has some edge artefacts as evidenced by the vertical strips in the power spectrum. 64