Automatic detection of position and depth of potential UXO using continuous wavelet transforms Stephen D. Billingsa and Felix Herrmannb a UBC
Geophysical Inversion Facility, 6339 Stores Rd, Vancouver, BC, V6T1Z4, Canada Earth and Ocean Sciences, 6339 Stores Rd, Vancouver, BC, V6T1Z4, Canada
b UBC
ABSTRACT Inversion algorithms for UXO discrimination using magnetometery have recently been used to achieve very low False Alarm Rates, with 100% recovery of detected ordnance. When there are many UXO and/or when the UXO are at significantly different depths, manual estimation of the initial position and scale for each item, is a laborious and time-consuming process. In this paper, we utilize the multi-resolution properties of wavelets to automatically estimate both the position and scale of dipole peaks. The Automated Wavelet Detection (AWD) algorithm that we develop consists of four-stages: (i) maxima and minima in the data are followed across multiple scales as we zoom with a continuous wavelet transform; (ii) the decay of the amplitude of each peak with scale is used to estimate the depth to source; (iii) adjacent maxima and minima of comparable depth are joined together to form dipole anomalies; and (iv) the relative positions and amplitudes of the extrema, along with their depths, are used to estimate a dipole model. We demonstrate the application of the AWD algorithm to three datasets with different characteristics. In each case, the method rapidly located the majority of dipole anomalies and produced accurate estimates of dipole parameters. Keywords: Wavelet Transform, Unexploded Ordnance, Detection, Discrimination.
1. INTRODUCTION A discrimination strategy for interpretation of magnetic data over areas contaminated by UXO was successfully demonstrated by Billings et al.1, 2 The basis of the method is inversion for the dipole moment of an anomaly. The difference between the recovered moment and that from the closest item in an ordnance library provides an estimate of the minimum remanent magnetization that the item must possess if it were a UXO. The lower the value of the remanence the more likely the source is an intact UXO. At sites in the Helana Valley, Montana it was found that prioritizing the dig-sheet on the basis of this remanent magnetization could return 100% of the detected UXO while leaving many identified anomalies in the ground. False-alarm rates (FARs) as low as two were found when using this method (that is two non-UXOs dug per UXO). Before discrimination can be attempted we first need to go through a detection phase. For magnetics this can be a challenging problem as anomalies are typically dipolar with adjacent positive and negative peaks. Once detected, we then need to estimate the size of the anomaly so that we know how large a data window to invert in order to recover an accurate estimate of the dipole parameters. In addition a good initial estimate of the dipole moment and position are required to avoid possible local minima in the misfit function. This can be obtained by an analytic method developed by McFee and Das3 once the location of the dipole maximum and minimum have been found. When there are many UXO and/or when the UXO are at significantly different depths, manual estimation of the scale and location of the peaks for each potential UXO, is a laborious and time-consuming process. The main difficulty with estimating the position of dipoles is that (except for vertical or near-vertical moments) the dipole has a positive and a negative peak, with the source located somewhere between the two. Send correspondence to Stephen Billings: E-mail:
[email protected], Telephone: +1 604 822 1819
The usual approach to overcome this problem is to calculate the analytic signal which has the effect of collapsing the two dipole peaks into a single peak approximately centered over the source.4 The main disadvantage of the analytic signal is that it is calculated as the square-root of the sum of squares of the three orthogonal gradients of the total field data. This means that noise in the data are amplified so that it can be difficult to push the detection limit down to the noise floor without picking a lot of spurious anomalies. In this paper we propose an alternative technique that is based on a wavelet transform. As we are dealing with a potential field we can use a Poisson wavelet that is derived from the physics of the problem.5 This wavelet has been extensively studied and applied to interpretation of potential-fields (see Moreau et al.5 and references therein), particularly for locating and characterizing edges in the data.6 A very useful property of this class of wavelets is that the wavelet transform can be simply obtained by upward continuation followed by horizontal differentiation. This means that the wavelet coefficients at multiple scales can be easily and rapidly computed. We refer to the method we develop in this paper as the Automated Wavelet Detection (AWD) algorithm. Individual peaks in the magnetic data are followed across multiple scales, with the decay in peak amplitude related to the depth to the source. Nearby positive and negative peaks in the image are joined together if they have comparable depth estimates. In this way, we can avoid incorrectly joining the peaks from nearby dipoles that are at different depths. In the last stage of the algorithm, the amplitudes of the peaks and their relative position are used to provide an initial estimate of the dipole parameters. We demonstrate the application of the AWD algorithm to three different datasets with different characteristics. The first is highly cluttered with multiple overlapping anomalies; the second has dipoles of widely varying depths and amplitudes; while the third contains few dipoles but has a ridge of high magnetic intensity running through the data. In each case, the AWD algorithm produces excellent results without the need for extensive tuning, or editing of the detection lists.
2. THE AUTOMATIC WAVELET DETECTION ALGORITHM In most UXO clean-up scenarios, the detector is in the far-field of the source, so that the anomaly looks like a dipole, µo 3(m · x)x B= [−m + ], (1) 4πr3 r2 where m is the dipole moment located at the point xm = (xm , ym , zm ), x = (x − xm , y − ym , z − zm ), r is the distance between the source and an observation point and µo is the permeability of free-space. The most common type of detector is the cesium-vapor magnetometer which measures the scalar valued total-magnetic field. The Earth’s magnetic field is usually 10 to 1000 times larger than the field due to a UXO, so that to a close approximation the total-field anomaly B = αBx + βBy + γBz ,
(2)
where (α, β, γ) is a unit vector in the direction of the Earth’s magnetic field. The principal difficultly with any method for detecting dipoles is that the anomaly usually consist of a positive and negative peak, with the source located somewhere in between. The AWD algorithm we have developed for locating potential UXO through their dipole anomalies consists of the following four steps: 1. Follow local maxima and minima as the image is zoomed to different scales with a wavelet transform; 2. Estimate the depth to the source of each peak by analyzing how the amplitude of the peak changes with scale; 3. Join nearby positive and negative peaks of comparable depth; and 4. Use the amplitudes and distance between adjacent peaks to provide an initial estimate of dipole parameters.
2.1. Step 1: The Wavelet Transform We are interested in a particular class of wavelet transforms that can be written as a multiscale differential operator, dn+m (3) Wxn ym [B](x, y, s) = sn+m n m (B θs )(x, y), dx dy where s > 0 is the wavelet scale and θs is a smooth function (such as a Gaussian) with the property that, θs (x, y) =
1 x y θ( , ) s2 s s
(4)
and (B θs ) represents convolution.7 The wavelet transform involves convolving the function under consideration, B, with the smoothing function θs at a particular scale s, and then differentiating with respect to position. The change in the amplitudes of the wavelet coefficients with scale provides useful information on the local signal structure. For instance, multiscale edge analysis involves following the modulus maxima of the wavelet transform (points where the modulus of the wavelet transform is locally maximal) as the scale changes. The way the amplitude changes with scale is related to the type of singularity in the source distribution. Hornby et al.6 demonstrated how this edge analysis capability could be used to map geological structures from potential field data. For our problem, we are not interested in edges but in the way local maxima and minima change across different scales. Local minima and maxima occur at points where Wx [B](x, y, s) = s
d (B θs )(x, y) = 0. dx
(5)
Thus, our task is to follow the zero crossings of the wavelet transform. We are interested in extrema in 2-D (we don’t want to locate valleys or ridges) so that we also require the derivative in the orthogonal direction to be zero, d (6) Wy [B](x, y, s) = s (B θs )(x, y) = 0. dy In addition, to ensure that the point is a local minima or maxima we require that Wyy [B](x, y, s)Wxx [B](x, y, s) − Wxy [B](x, y, s)2 > 0,
(7)
where Wxx , Wyy and Wxy are 2-nd order multiscale differential operators. For potential-fields the smoothing function arises naturally from the Laplace equation Green’s function,5, 6 θs (x, y) =
s 1 ; s > 0. 2 2 2π [x + y + s2 ]3/2
(8)
Convolution with this smoothing function is equivalent to upward continuation of the potential field by an amount z = s, which can be achieved in the Fourier domain with the operator, Θs (k) = exp(−2πsk),
(9)
where k is the 2-D frequency vector (in cycles per unit length). Thus, calculation of wavelet coefficients at different scales can be achieved by upward continuation followed by horizontal differentiation. Conceptually (the first two steps of) our AWD algorithm for finding maxima and minima at different scales s is based on the method advocated in Hornby et al.6 : 1. Upward continue to the height z = s; 2. Take the 2-D horizontal derivative and multiply the result by s; 3. Locate points where Wx [B](x, y, s) = Wy [B](x, y, s) = 0 and ensure that the point is a local maxima or minima via Equation (7). Given these minima and maxima at different scales, our next task is to estimate the depth of the dipole source.
2.2. Step 2: Following Local Maxima and Minima Across Scales After applying the first-step of the AWD algorithm, we have a number of maxima and minima at various different scales. These extrema can be easily followed across multiple scales as their position changes only slowly with height (see Mallat7 for theorems governing this rate of change). We have found it more efficient to follow the extrema from the coarsest scale down to the finest and not the other way round as is usual in wavelet analysis. In addition, we usually don’t follow the minima all the way down to the finest scale (essentially the raw image). This allows us to avoid picking, as much as possible, peaks that occur due to noise in the data. A common method for estimating the depth to a source at (xm , ym , zm ) is via Euler’s equation,8 r · ∇B = −nB,
(10)
where r = (x − xm , y − ym , z − zm ) and n is the structural index of the source. The structural index controls the rate of decay of the amplitude of the anomaly with distance. There are two factors that contribute to the value of this exponent. The first is the decay of the Green’s function with distance away from the source. The second is the rate of increase of source material within the field of influence of the Green’s function which broadens with increasing distance from the source. For a dipole, the structural index is n = 3. At a maximum or minimum ∂B/∂x = ∂B/∂y = 0, so that (z − zm )
∂B = −nB, ∂z
(11)
B(z) =
A , (z − zm )n
(12)
which has as a solution
where A is related to the magnitude of the source. Consider upward continuation to the two levels z = s1 and z = s2 and assume that we measure zm < 0 as the depth of the source relative to the sensor. Define z1 = s1 −zm and z2 = s2 − zm in which case the fields are related by the equation n z1 B(s1 ). (13) B(s2 ) = z2 We note that this conventional analysis is very similar to that used for Poisson wavelets by Moreau et al.5 In wavelet notation the above equation is equivalent to n z1 (B θs1 ), (14) (B θs2 ) = z2 and note that this equation involves convolution with the smoothing function but has no differentiation with respect to horizontal position (more on this presently). The structural index of the source corresponds to the exponent governing the rate of decay of the wavelet coefficients in Equation (14). Some straightforward manipulation shows that (15) log(B θs ) − log B(0) = n[log(−zm ) − log(s − zm )]. Therefore, in a log-log plot of the amplitude of the wavelet coefficients against scale, the slope of the line will be equal to the negative of the structural index of the source, while the intercept will be related to the source depth. It is thus a simple matter to estimate both the source type and the depth for each anomaly. We note, that when multiple anomalies lie close together that the amplitudes of adjacent peak increasingly interfere with height. In that case, we estimate the structural index and depth to source from the first few levels of upward continuation. Note further, that we get very good results in most surveys by fixing the structural index at n = 3 for a dipole source, and only using Equation (15) for estimating the depth zm . Strictly speaking the wavelet transformation we use in Equations (14 and 15) occurs with a non-admissible wavelet. That is, a wavelet which does not have vanishing moments (specifically a zero average). We note that
we can convert the wavelet into an admissible form by recognizing that we are dealing with a field, B derived from a potential φ. Thus in 1-D we could write B = −∂φ/∂x and consequently, (B θs )(x) = −(
∂φ ∂θs θs )(x) = −(φ )(x), ∂x ∂x
(16)
and we note that [∂θs /∂x]dx = 0 Thus, if we view everything in terms of potentials, the wavelet becomes admissible. Furthermore, when we follow zero crossings of the wavelet transform of the magnetic field, we are effectively following modulus maxima in the potential field. Viewed in this way, our techniques are essentially equivalent to the standard wavelet methodologies used for multi-scale edge analysis.
2.3. Step 3: Joining Adjacent Maxima and Minima At this point the AWD algorithm will have identified maxima and minima in the data and followed them across multiple scales to obtain a depth estimate. The next task is to join adjacent minima and maxima that have comparable estimated depths. This is achieved after first ranking extrema in decreasing order of amplitude. For the first identified peak we then search for a peak of opposite sign that lies within a certain radius determined by the depth estimate. Typically, we need to search out to a distance of 1.5 times the depth estimate to ensure that we find the other dipole peak. If a peak of opposite sign is found within this radius the estimated depths of the two peaks are compared. If they agree to within a certain tolerance (usually 50%) the peaks are joined together and both peaks are taken out of the search list. If more than one peak lies within the search radius, then the peak with the closest depth estimate is chosen. The process is repeated for all positive and negative peaks. Typically, we find that there are a certain number of extrema that can’t be joined together. These can include (i) noise; (ii) adjacent peaks of opposite sign where the joining algorithm has failed typically due to noise or interference and (iii) vertically or near vertically oriented dipoles which have a large peak of one sign and a small or non-existent peak of the other sign. In addition there are usually a small number of peaks that have been inappropriately joined together. Therefore a post-processing step is required to clean-up the results of the automatic picking algorithm. We note that the level of manual intervention is usually very small compared to the effort required to do full manual target picking. In addition, manual interventions are minimized by appropriate selection of the parameters required by the algorithm. Moreau et al.5 in a 1-D analysis of potential fields exploits the way the peak positions change with scale to join adjacent extrema. The modulus maxima of the potential (which corresponds to the locations of the peaks) essentially point towards each other and intersect at the depth to the source. This procedure works quite well in uncluttered datasets, but once anomalies start to overlap the paths of adjacent peaks are no longer diagnostic and the algorithm becomes inefficient. As a post-processing step (alluded to in the previous paragraph), we can often use the relative peak movements to decide that certain peaks have been inappropriately joined together. At this point, the detection phase of the AWD characterization task is complete and we then move onto the discrimination phase.
2.4. Step 4: Estimating Dipole Parameters The last step of the AWD algorithm is to estimate the dipole parameters from the location and amplitudes of the dipole peaks. Six-parameters are required consisting of three for the location xm = (xm , ym , zm ) and three for the dipole moment m = (mx , my , mz ). For single peaks we assume the dipole is vertically oriented, so that mx = my = 0 and that the source is located directly below the peak at the depth returned by the wavelet analysis. It is then a simple matter to estimate the vertical component of the dipole moment by scaling the unit dipole response by the amplitude of the peak. For anomalies with both a positive and a negative peak, we assume that the source is located along the path between the two peaks, and that the dipole moment is oriented in this direction. This assumption is valid to a close approximation, although depending on the ambient field-direction, the peak positions can deviate slightly from this ideal. By rotating and translating the coordinate system so that the y-axis points from the positive peak to the negative peak, there are now only four-parameters in our dipole model, xm = (0, ym , zm )
7
1
6.5
0.5
6
0
Normalized profile of dipole
Distance between peaks (m)
Depth = 4.13 m Depth = 4.22 m
5.5
5
4.5
4
−0.5
−1
−1.5
−80
−60
−40
−20 0 Dip angle of dipole (deg)
20
40
60
−2 −2
−1
0
1
2 Distance (m)
3
4
5
6
Figure 1. Estimation of dipole parameters from peak separation and relative amplitude, for item 7 in dataset 2 (see Figure 4). On left, the variation in the peak separation for two depths are shown. On the right, the normalized profiles for the two solutions at each depth are compared to the observed ratio of peak amplitudes (marked by a *). The best fit is achieved with z = 4.22 m and θ = 1.7o .
and m = (0, my , mz ), one of which we already know, zm , from the wavelet analysis. The dipole magnitude can be obtained by a scaling argument once the other parameters have been determined, so we only need to find ym , the horizontal distance of source from the positive peak and θ = arctan(my /mz ), the dip angle of the dipole (θ = 0 for a horizontal dipole). We have two pieces of information that can be used to constrain these parameters; (i) the distance between the dipole peaks; and (ii) the ratio of the peak amplitudes. At a given depth, the distance between the positive and negative peaks changes as the dip of the dipole changes (Figure 1a). In general, there are two dip angles that can produce the observed peak separation, one above the horizontal and the other below. We chose the dip angle that gives the closest fit to the observed ratio of peak amplitudes (Figure 1b). In general, we find that neither of the dip angles can reproduce both the correct peak separation and the correct ratio of peak amplitudes. This is because the depth returned by the wavelet analysis is imperfect and we usually need to make small adjustments to this depth to achieve a perfect match (Figure 1b). To summarize the dipole estimation algorithm consists of the following: 1. Input the peak positions x+ and x− and their corresponding amplitudes B + and B − ; 2. Determine the angle ψ of x− relative to x+ and rotate and translate the coordinate system so that x+ → (0, 0) and x− → (0, d), where d is the distance between the two peaks; 3. Adjust zm to achieve the best match between the peak separation, d and the ratio of peak amplitudes B − /B + which involves finding the optimum dip angle θ at each depth (and trivially the horizontal distance to the source ym ); 4. Scale the dipole moment so that the model matches the observed peak amplitudes and calculate m = (0, my , mz ); and 5. Rotate and translate xm = (0, ym , zm ) and m back to geographic coordinates using the location of x+ and the azimuthal angle of the dipole ψ.
We mentioned at the outset that the main purposes of the AWD algorithm were firstly, automated detection of potential UXO; and secondly, to provide initial estimates of dipole parameters for an inversion based discrimination procedure. The initial dipole model provided by the wavelet based method allows potential local minima in the inversion to be avoided. In addition, the depth estimate and/or the peak separation provides a sensible estimate of the window of data that need to be inverted.
3. APPLICATION OF THE METHOD TO AUTOMATIC TARGET SELECTION To illustrate the performance of the algorithm we will consider magnetic surveys collected over three different areas. The first is a highly cluttered dataset with multiple overlapping anomalies; the second has dipoles of widely varying depths and amplitudes; while the third contains few dipoles but has a ridge of high magnetic intensity running through the data. All surveys covered an area of approximately 50 m × 50 m. The magnetic field of the Earth at the location of the surveys was Bo = (3646, 20874, −49405) nT. All the data were collected with a quad-sensor array of cesium vapor magnetometers. Digital data were acquired along a regular pattern of parallel transects using visual markers to assist in navigation. Positioned measurements were recorded at nominally 10 cm intervals along each transect and each sensor was separated by 37.5cm perpendicular to the transect direction. A base-station magnetometer was used throughout the survey and sampled the magnetic field at 5s intervals. This provided a baseline measure of temporal variations in the magnetic field which was used to correct the field data. In all survey areas, the magnetic data were positioned using a Trimble 5700 differential GPS system. The magnetic data were corrected for temporal variations using the data acquired at the magnetic basestation. The data from each sensor were then high-pass median filtered to remove sensor heading off-sets and disturbance from geological sources deeper than 5 m. The corrected data were then interpolated to a 10 cm regular grid.
3.1. Survey 1: Detection in a Cluttered Environment The first survey illustrates how the method performs in a cluttered environment where the anomalies from adjacent items overlap. In the first step of the algorithm we upward continue the images with increasing height and locate local minima and maxima at each stage (Figure 2). As the level of upward continuation is increased we note that: (i) the number and amplitude of the peaks decreases significantly; (ii) the location of the peaks change slightly; and (iii) anomalies from adjacent items increasing overlap. The fact that adjacent anomalies increasingly interact means that the rate of decay of the extrema amplitudes can deviate from n = 3 at large heights. In these situations we can avoid biasing the predicted depths by only using small values of the upward continuation height in Equation (15). Only high amplitude peaks persist to a height of z = 1.6 m or greater. By monitoring how long peaks persist with upward continuation we can be selective of the number of anomalies that we pick (Table 1 and Figure 3). With a minimum persistence of 1.6 m, only 70 targets are picked but the algorithm has a hard-time matching dipole peaks. As the persistence height is decreased to 0.2 m around 300 anomalies are selected and 74 dipole targets are formed. At this stage the algorithm is aggressively seeking out local extrema in the image and proposes a relatively large number of single-peak anomalies. One can go a long way towards reducing the number of these by enforcing a minimum peak amplitude that a single peak must possess to be considered a potential UXO (Table 1). For this survey, we found that enforcing a minimum persistence of 0.4 m and a minimum amplitude of 20 nT gave the best results, any lower and the number of unmatched and incorrectly joined peaks increased substantially.
3.2. Survey 2: Detection When Depths and Amplitudes Vary Widely The second survey has a wide range of different sized items from 500 pound aircraft bombs to 20 mm projectiles at a wide range of depths from very near surface (< 10 cm) down to almost 4 m (Figure 4a). The AWD algorithm is able to reliably join adjacent extrema across a large range of scales and amplitudes. There are two unmatched adjacent peaks that on closer inspection should have been joined (between items 7 and 10).
Height = 0.2 m
Height = 0.4 m
50
50
45
45 nT
40
nT
40 100
100
35
35
0
25
20
50 Northing (m)
Northing (m)
50 30
30 0
25
20
−50
15
−50
15 −100
−100
10
10
5
5
0
0
5
10
15
20
25 30 Easting (m)
35
40
45
0
50
0
5
10
15
20
Height = 0.8 m
25 30 Easting (m)
35
40
45
50
Height = 1.6 m
50
50
45
45 nT
40
nT
40 100
100
35
35
0
25
20
−50
15
50 Northing (m)
Northing (m)
50 30
30 0
25
20
−50
15 −100
−100
10
10
5
5
0
0
5
10
15
20
25 30 Easting (m)
35
40
45
50
0
0
5
10
15
20
25 30 Easting (m)
35
40
45
50
Figure 2: Upward continuation of a cluttered magnetic image with the locations of maxima marked with crosses.
In addition, there are three sets of extrema (items 17, 18 and 20 in Figure 4a), all with low amplitudes, that should not have been joined. Two of these sets of extrema had peak movements in very different directions so could easily be identified as inappropriately joined. For this survey, we predicted dipole parameters using the AWD method before doing a full dipole inversion using the Billings et al. method2 (Table 2). Inspection of the table reveals an excellent agreement between the parameters recovered by the full and AWD methods. Horizontal positions generally agree to within 10 cm (worst 33 cm), depths to within 5 cm (worst 7 cm), dipole magnitudes to within 20 % (worst 40%), and azimuths and dips to within 5o (worst 24o and 17o respectively). The results demonstrate that the AWD predictions are more that adequate as initial models for the inversion algorithm.
Table 1. Change in the number of targets picked with the minimum height that a peak persists. Also shown are the number of unmatched single peak anomalies as the minimum amplitude threshold is changed.
Height
Dipole targets
Unmatched
Unmatched
Unmatched
10 nT
20 nT
30 nT
1.6 m
8
62
60
59
0.8 m
30
117
107
102
0.4 m
59
194
161
133
0.2 m
74
227
183
145
Table 2: Comparison of dipole parameters recovered by the AWD and full inversion methods.
1
Position Diff. (m) 0.09
Full Depth (m) 0.83
AWD Depth (m) 0.89
Full Moment (Am2 ) 94.2
AWD Moment (Am2 ) 109.4
Full Azimuth (deg) 179.3
AWD Azimuth (deg) 176.4
Full Dip (deg) -2.9
AWD Dip (deg) -1.1
2
0.10
1.05
1.09
124.8
147.7
-178.4
-176.8
0.5
0.0
3
0.09
0.12
0.12
9.61
9.47
-174.6
-166.0
2.2
0.0
4
0.02
0.26
0.20
10.4
8.40
4.6
6.3
0.9
0.0
5
0.06
0.45
0.40
7.86
6.72
-1.5
-10.3
4.3
3.3
6
0.16
0.52
0.51
8.24
8.38
82.8
80.5
1.9
4.3
7
0.14
3.46
3.52
192.7
199.4
7.5
9.5
3.3
1.7
8
0.15
0.23
0.23
1.15
1.15
-82.2
-90.0
-30.1
-29.3
9
0.33
1.63
1.69
29.1
40.6
-4.5
-4.8
-8.1
-20.7
10
0.06
0.21
0.22
1.16
1.43
12.9
12.5
-13.5
-19.2
11
0.13
0.59
0.60
4.08
3.77
-1.0
-4.4
-3.9
0.0
12
0.08
0.67
0.69
3.47
4.00
178.1
175.9
3.5
0.0
13
0.12
1.14
1.12
5.06
5.02
-1.6
11.9
14.0
17.3
14
0.22
0.02
0.02
0.49
0.46
113.8
90.0
32.7
35.8
15
0.03
0.56
0.60
1.28
1.39
4.3
8.7
5.1
3.7
16
-
1.42
1.28
-
-
-5.4
0.0
-
-
19
0.26
0.44
0.51
0.19
0.20
160.7
157.4
24.6
41.0
Item
3.3. Survey 3: Detection in the Presence of Linear Anomalies The third and last data-set that we consider has a linear feature running through the survey area (Figure 4b). This linear feature represents a small magnetic high along a dirt track. Several previous detection algorithms we trialed confused the ridge with local minima and maxima and picked a large number of targets along its margin. The AWD algorithm identifies one dipole-like anomaly and four single peaks on the verges of the road. The dipole anomaly turned out to be a small piece of scrap, while the four single peaks were geologically related. Thus, the ridge does not cause the AWD algorithm to create many spurious anomalies. In addition, all magnetic anomalies were found and adjacent peaks successfully joined together.
Height = 0.2 m
Height = 0.4 m
50
50
45
45 nT
40
nT
40 100
100
35
35
0
25
20
50 Northing (m)
Northing (m)
50 30
30 0
25
20
−50
15
−50
15 −100
−100
10
10
5
5
0
0
5
10
15
20
25 30 Easting (m)
35
40
45
0
50
0
5
10
15
20
Height = 0.8 m
25 30 Easting (m)
35
40
45
50
Height = 1.6 m
50
50
45
45 nT
40
nT
40 100
100
35
35
0
25
20
−50
15
50 Northing (m)
Northing (m)
50 30
30 0
25
20
−50
15 −100
−100
10
10
5
5
0
0
5
10
15
20
25 30 Easting (m)
35
40
45
50
0
0
5
10
15
20
25 30 Easting (m)
35
40
45
50
Figure 3. Performance of the target selection algorithm when picking down to a specified height. In each case, joined diamonds represent the positive and negative peaks of a single dipole. Crosses mark the locations of single peaks.
4. DISCUSSION In this paper we have developed an algorithm for automatically detecting potential UXO and determining their approximate dipole parameters. The AWD method is based on following minima and maxima in the magnetic field across multiple scales as the image is zoomed using a Poisson wavelet. Zooming with this wavelet is equivalent to upward continuation at different heights followed by horizontal differentiation. Thus, the wavelet coefficients can be easily and rapidly calculated. The AWD algorithm is able to efficiently detect and follow extrema in the data, even in the presence of noise. The noise has different spatial characteristics than the signal and its energy tends to dissipate rapidly with continuation height. We note that the method is particularly well suited to detection in high-noise levels because continuation is a smoothing operation. The only sharpening operation is the differentiation required to locate the extrema at different levels. We contrast this to methods based on the analytic signal, where derivatives are explicitly required to calculate the transformation. Hence noise tends to become amplified.
50
50 14
17
3
45
45
7
20 nT
40
nT
40
30.00 16
35
13
200
2
35
22.38 14.76
100
30 1
10
0
25 12 20
Northing (m)
Northing (m)
8 30
7.14 −0.48
25
−8.10
−100
20
−200
15
−15.71
9 4
15
−23.33
6 10
10
15 5
11
5
0
−30.00
19
0
5
10
15
18 20
25 30 Easting (m)
5
35
40
45
50
0
0
5
10
15
20
25 30 Easting (m)
35
40
45
50
Figure 4. Automatic estimation of the position and depth of potential UXO for two other datasets. In each case, joined diamonds represent the positive and negative peaks of a single dipole. Crosses mark the locations of single peaks.
The algorithm has two primary purposes. The first is the automatic detection of potential UXO in noisy, cluttered datasets. We have seen that the AWD algorithm is both a fast and an effective detection tool. The second purpose of the algorithm is to provide an initial model and scale estimate for an inversion based discrimination algorithm. The results for dataset 2 demonstrated that the AWD algorithm produced initial models that were in close agreement to the full inversion results. The combination of efficient detection and accurate estimation of dipole parameters, substantially reduces the time required to process and interpret a magnetic dataset with advanced discrimination algorithms.
REFERENCES 1. S. D. Billings, L. R. Pasion, and D. W. Oldenburg, “Inversion of magnetics for UXO discrimination and identification,” in Proc. 2002 UXO Forum, Orlando September 3-6, 2002. 2. S. D. Billings, J. M. Stanley, and C. Youmans, “Magnetic discrimination that will satisfy regulators,” in Proc. 2002 UXO Forum, Orlando September 3-6, 2002. 3. J. E. McFee and Y. Das, “Fast non-recursive method for estimating location and dipole moment components of a static magnetic dipole,” IEEE Trans. Geosci. Rem. Sens. 24, pp. 663–673, 1986. 4. I. N. MacLeod, K. Jones, and T. F. Dai, “3-d analytic signal in the interpretation of total magnetic field data at low magnetic latitudes,” Austr. Soc. Expl. Geophys. 24, pp. 679–688, 1993. 5. F. Moreau, D. Gilbert, M. Holschneider, and G. Saracco, “Identification of sources of potential fields with the continuous wavelet transform: Basic theory,” J. Geophys. Res. 104(B3), pp. 5003–5013, 1999. 6. P. Hornby, F. Boschetti, and F. G. Horowitz, “Analysis of potential field data in the wavelet domain,” Geophys. J. Int. 137, pp. 175–196, 1999. 7. S. Mallat, A wavelet tour of signal processing, Academic Press, 2nd ed., 2001. 8. R. J. Blakely, Potential theory in gravity and magnetic applications, Cambridge University Press, 1996.
