sensors Article

Full Tensor Eigenvector Analysis on Air-Borne Magnetic Gradiometer Data for the Detection of Dipole-Like Magnetic Sources Boxin Zuo

ID

, Lizhe Wang * and Weitao Chen

School of Computer Science, China University of Geosciences, Wuhan 430074, China; [email protected] (B.Z.); [email protected] (W.C.) * Correspondence: [email protected]; Tel.: +86-027-6788-3714 Received: 22 July 2017; Accepted: 26 August 2017; Published: 29 August 2017

Abstract: The detection of dipole-like sources, such as unexploded ordnances (UXO) and other metallic objects, based on a magnetic gradiometer system, has been increasingly applied in recent years. In this paper, a novel dipole-like source detection algorithm, based on eigenvector analysis with magnetic gradient tensor data interpretation is presented. Firstly, the theoretical basis of the eigenvector decomposition of magnetic gradient tensor is analyzed. Then, a detection algorithm is proposed by using the properties of the tensor eigenvector decomposition to locate dipole-like magnetic sources. The algorithm can automatically detect magnetic dipole-like sources without estimating the magnetic moment direction. It performs well for locating weak, anomalous dipole-like sources in air-borne magnetic data through quantitative interpretation. The effectiveness of the proposed algorithm has been demonstrated in the designed synthetic experiment. Finally, an air-borne magnetic field data taken at high altitude with exact source position information is used to validate the practicality of the proposed algorithm. All of the experiments prove that the proposed algorithm is suitable for magnetic dipole-like source detecting and air-borne magnetic gradiometer data interpretation. Keywords: magnetic gradiometer; ordnance detection

eigenvector analysis;

magnetic dipole;

unexploded

1. Introduction An estimated million acres of land throughout the world are contaminated with various types of unexploded ordnances (UXO), landmines, and other steel objects [1]. To clean up these areas, magnetic surveys are one of the most common geophysical methods used to detect these metallic targets [2]. However, discriminating these magnetic sources with magnetic field data is a challenging task. To characterize and locate potential UXO targets precisely, many methods for automatic detection, localization, and classification (DLC), using total magnetic intensity (TMI) data, have been developed [2,3]. Kristofer et al. proposed using an extended Euler deconvolution algorithm for automatic UXO detection based on the concept of the structural index (SI) [4]. Zalevsky et al. defined a wavelet mother function with the physical parameters of a magnetic dipole for detecting marine field magnetic targets [5]. Phillips et al. suggested using the Helbig method to estimate magnetization direction before applying the Euler analysis to locate compact magnetic sources [6]. Li et al. formulated a downward continuation algorithm as an inverse problem with Tikhonov regularization for enhancing magnetic data acquired for UXO discrimination [7]. Recent advances in magnetic gradiometry have significantly increased the signal to noise ratio (SNR) of magnetic gradient tensor (MGT) data. Magnetic gradiometry has been increasingly used in wide area assessments to rapidly locate potential UXO contamination [8]. Schneider et al. utilized Sensors 2017, 17, 1976; doi:10.3390/s17091976

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the Wynn Frahm algorithm for inverse full-tensor data from dipole-like sources with a ground-based magnetic SQUID gradiometer system [9]. Sanchez et al. presented the numerical modelling of UXO and related metallic objects with a nonlinear integral equation [10]. Their results show that the presence of significantly higher order moments of asymmetric objects are obviously above the noise level of ground observed magnetic gradient data. Beiki et al. proposed the normalized source strength (NSS) method to deal with the remanent magnetization effect for quantitative interpretation of magnetic gradiometer data [11]. Yin et al. proposed modified tilt angles and using the NSS method to detect multiple dipole sources by using magnetic gradient tensor data [8]. Munschy et al. proposed a non-linear dipole inversion algorithm for ground magnetometer-measured profiles data in order to locate and characterize dipoles generated by UXO [12]. Hiergeist et al. built a ground level test facility to analyze signal resolution and the relative spatial resolution of MGT measured with unexploded bomb detection applications [13]. Bracken and Brown used a prototype tensor magnetic gradiometer system and designed a data-reduction procedure to detect UXO [14]. These advanced discrimination methods have been successfully deployed in UXO survey activities with ground-based magnetometry and magnetic gradiometry systems. In order to expand the magnetic gradiometry survey over large areas and over areas that are unsuitable for deploying ground-based systems, airborne magnetic gradiometry systems have been developed. Airborne magnetometer surveys can rapidly screen both surface and subsurface steel objects without being affected by the geological background noise. Billings and Wright deployed a low-level helicopter magnetometer survey and proved its ability to perform wide-area UXO assessment [15]. Doll et al. showed that the source of noise for low-altitude air-borne systems generally comes from the helicopter rotor [16]. Xia et al. suggested a moving hum filter to suppress the rotor noise in helicopter-based airborne systems for the detection of UXO [17]. Burazer et al. suggested using a FIR filter to extract weak anomalies from data that were measured at high-altitude air-borne magnetometer system [18]. As Billings and Wright analyzed, a large sensor and target stand-off distances of an airborne magnetometer system will result in a reduction in the amplitude anomalies [15]. Therefore, to process data obtained at high altitudes, it is necessary to extract the weak anomalous information caused by small buried steel objects. Miller and Singh suggested that when the observation distance is 2.5 times larger than the largest size of the source, the magnetic source can be considered as a magnetic dipole [19]. Therefore, UXO and other metallic targets in high-altitude air-borne magnetometer surveys can be considered to be dipole-like sources. In recent years, eigenvalue analysis has become one of the most popular methods applied to geophysical data processing. Fakiris et al. presented an automated target detection approach with Principal Component Analysis (PCA) for marine magnetic data interpretation [20]. Zou and Nehorai derived the analytical expressions of magnetic gradient tensor and presented an efficient ship wake detection method for airborne magnetic data [21]. Zuo and Hu proposed an eigenvalue analysis method to detect geological structures in air-borne gravity gradiometer data processing [22]. Beiki and Pedersen developed an eigenvector analysis method for air-borne gravity gradient tensor data to interpret geologic structures [23]. In this paper, we propose a high-order tensor eigenvector method to detect magnetic dipole-like sources in air-borne magnetic gradient tensor data. The reliability and efficiency of the proposed method is analyzed with a designed synthetic data experiment. Finally, the method is applied to the air-borne magnetic field data of the Fort Peck reservation, Montana, USA. 2. Methods The magnetic gradients present the spatial rates of change of the magnetic field B (Vs/m2 , SI Unit) in Equation (1) [24]. Z 1 B = −Cm ∇ ϕ = −Cm ∇ ∇(ri ) · m(ri )dv (1) | r − ri | V

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where m(ri ) is the magnetization vector of the sources, r and ri denote observation and source positions, respectively. Cm = 10−7 (Henry/m, SI units). Then, the magnetic gradient tensor Γ can be written as shown in Equation (2).    Γ = −Cm  

∂2 ϕ ∂x2 ∂2 ϕ ∂y∂x ∂2 ϕ ∂z∂x

∂2 ϕ ∂x∂y ∂2 ϕ ∂y2 ∂2 ϕ ∂z∂y

∂2 ϕ ∂x∂z ∂2 ϕ ∂y∂z ∂2 ϕ ∂z2



 Φ xx     =  Φyx  Φzx

Φ xy Φyy Φzy

 Φ xz  Φyz  Φzz

(2)

where Φij (i = x, y, z; j = x, y, z; ) denotes the elements of the magnetic gradient tensor. 0 Assuming the magnetic gradient tensor matrix Γ can be diagonalized as Λ = V ΓV, and eigenvector V(V = [V1 , V2 , V3 ]) forms a new coordinate system. Physically, this diagonalized procedure rotates the coordinate system to align with the equivalent point source direction [25]. Γ can be projected to a new form coordinate system with the diagonalized procedure. The rotation matrix can be expressed as Equation (3): 

cos ψ cos θ cos φ − sin ψ sin φ  V =  − sin ψ cos θ cos φ − cos ψ sin φ − sin θ cos φ

cos ψ cos θ cos φ − sin ψ sin φ − sin ψ cos θ sin φ − cos ψ cos φ − sin θ cos φ

 cos ψ sin θ  − sin ψ cos φ  cos θ

(3)

where ψ, θ, φ are the rotation angles about x-, y-, and z-axis, respectively. After decomposition, the non-diagonal elements are much smaller than the diagonal elements and the magnetic dipole direction information is extracted to the rotation matrix V. Hasan et al. suggested a method to calculate the matrix V through the eigenvector decomposition for 3-D tensor data as Equation (4) shows:      Φ xy Φyz − Φ xz (Φyy − λi ) Φ xz Φyz − Φ xy (Φzz − λi )      vi =  Φ xz Φyz − Φ xy (Φzz − λi ) Φ xz Φ xy − Φyz (Φ xx − λi )     Φ xy Φyz − Φ xz (Φyy − λi ) Φ xz Φ xy − Φyz (Φ xx − λi )

(4)

where the largest eigenvalue λi can be estimated with the tensor components Φij [26]. Li suggested that this diagonalized decomposition rotates the local coordinate system into a user-defined system, and the Euler angle of the rotated coordinate is θ 0 = 0, ψ0 = 0 [25]. In this paper, we propose a magnetic dipole-like location method using eigenvector analysis. According to Equation (3), the eigenvector rotates angle φ along the z-axis after eigenvector decomposition. If the angle φ is not equal to zero, it means that the observation point has a vertical direction bias when compared to the position of the magnetic dipole source, while the magnetic dipole moment will be represented by the corresponding eigenvalue. From this view, we think that the purpose of the eigenvector analysis is to decompose the magnetic gradient data into two data sets: the eigenvector data set contains the source position information, and the magnetic moment is represented by the eigenvalues data set. Through this procedure, noise and interference error can be suppressed because they do not have an anomalous response in all five independent tensors (Φ xx , Φ xy , Φ xz , Φyy , Φyz ). According to this deduction, it is possible to locate magnetic dipole-like sources by using tensor eigenvector decomposition. To make the value of the eigenvector inversely proportional to the position of the magnetic dipole, the direction of eigenvector v1 = [ x 0 , y0 , z0 ] is defined through the angle φ, as shown in Figure 1.

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position of the magnetic dipole, the direction of eigenvector v1 =[x', y',z' ] is defined through the Sensors 2017, 17, 1976

angle

φ , as shown in Figure 1.

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Figure 1. 1. Coordination Coordinationrotation rotationbybyeigenvector eigenvectorvV1 decomposition; decomposition;Angle Angle φ is coordination the coordination Figure φ is the 1 rotation angle. rotation angle.

Theoretically, the eigenvector v is a unit vector. The amplitude will not commonly equal 1 with Theoretically, the eigenvector 1v1 is a unit vector. The amplitude will not commonly equal 1 the noise interference. But the error of amplitude will not influence the direction of eigenvector v1 . with the noise interference. But the Equation error of amplitude will not influence the direction of eigenvector The angle φ can be calculated with (5). v1 . The angle φ can be calculated with Equation (5).   Φ xy Φyz − Φ xz (Φyy − λ1 ) Φ xz Φ xy − Φyz (Φ xx − λ1 )   (5) ctanφ = (Φ λ1xz) )( ( Φ xz−Φλxy ) ) Φyz (Φ xx − λ1 )) Φ xz Φyz − Φ xy (Φzz (−Φλxy1Φ) yz (−ΦΦxyxzΦ (ΦΦyyxz Φ − xyλ1−))Φ+ yz ( Φxx 1 − yzyy−−Φ ctanφ = (5) Φ xz Φ yz − Φ xy ( Φ zz − λ1 ) )( ( Φ xy Φ yz − Φ xz ( Φ yy − λ1 )) + ( Φ xz Φ xy − Φ yz ( Φ xx − λ1 )) ) ( A small angle φ rotation will result in a relative large ctanφ value to illustrate that the position of the is close the observation Conversely, the is nearly equal to 0, ctan φ value will resultpoint. in a relative large if value of to ctanφ illustrate that the position Adipole small angle φ torotation it indicates that there is no source nearby to this observation position. Through this transformation, of the dipole is close to the observation point. Conversely, if the value of ctanφ is nearly equal to 0, the position of the source is projected to a source position map, and it is then able to be visually it indicates that there is no source nearby to this observation position. Through this transformation, analyzed. Here, we define the Magnetic Dipole Tangent Angle (MDTA) according to the rotation the position of the source is projected to a source position map, and it is then able to be visually angle φ, as expressed in Equation (6). analyzed. Here, we define the Magnetic Dipole Tangent Angle (MDTA) according to the rotation angle φ , as expressed in Equation (6). MDTA = dctanφeα (6)

MDTA = data ctanφwill (6)  α not present a local peak value in MDTA, Although, theoretically, the noise in full tensor a high-level noise and low quality data will create false alarms in the MDTA map in practice. Therefore, Although, theoretically, the noise in full tensor data will not present a local peak value in MDTA, a threshold filter d·eα was set to remove these small false alarms, which are much smaller than the a high-level noise and low quality data will create false alarms in the MDTA map in practice. peak value that is caused by a source. Therefore, a threshold filter ⋅α was set to remove these small false alarms, which are much smaller 3. Method Validation than the peak value that is caused by a source. In this section, three synthetic experiments were designed to validate the proposed algorithm. 3. Method Validation The performance of the proposed algorithm with different dipole moment directions and multiple-source scenarios are discussed in detail.were designed to validate the proposed algorithm. In this section, three synthetic experiments Firstly, a synthetic wasalgorithm designed with with different nine magnetic that are buried inmultipledifferent The performance of the model proposed dipoledipoles moment directions and depths, as shown in Figure 2. source scenarios are discussed in detail. Firstly, a synthetic model was designed with nine magnetic dipoles that are buried in different depths, as shown in Figure 2.

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Figure2.2. 2.Plain Plain sectional viewofof ofthe thesynthetic syntheticmodel. model. Figure Figure Plainsectional sectionalview view the synthetic model.

Themoment momentofofthe thedipoles dipoleswas was20 20Am Am2,2,and andthe thedeclination declinationand andinclination inclinationofofthe thedipoles dipoles The The moment of the dipoles was 20 Am2 , and the declination and inclination of the dipoles  magnetization vectorwere were setas as I = 90, ,DD===000◦ .. .The The simulatedmagnetic magnetic fieldwas was sampledatat magnetization magnetizationvector vector wereset set asII ==90 90◦ , D The simulated simulated magnetic field field was sampled sampled at an interval of 1 m in the horizontal plane. The full magnetic gradient components of the magnetic an aninterval intervalof of11m min in the the horizontal horizontal plane. plane. The Thefull fullmagnetic magneticgradient gradientcomponents componentsof ofthe themagnetic magnetic fieldwere werecalculated calculatedaccording accordingtotothe theequations equationsthat thatwere wereproposed proposedby byPedersen Pedersenand andRasmussen Rasmussen field field were calculated according to the equations that were proposed by Pedersen and Rasmussen [27]. [27].These Thesemagnetic magneticgradient gradienttensors tensorsare aredisplayed displayedininFigure Figure 3. [27]. These magnetic gradient tensors are displayed in Figure 3. 3.

Figure3.3. 3.The The simulated magneticgradient gradienttensor tensordata. data. Figure Figure Thesimulated simulatedmagnetic magnetic gradient tensor data.

Forthe thereason reasonofofthe thesymmetry symmetryofoftensor tensorcomponents, components,we weonly onlydraw drawfive fiveindependent independenttensors tensors For For the reason of the symmetry of tensor components, we only draw five independent tensors (ΦΦxxxx,, ,Φ , ΦΦxz, xzΦ and ΦΦzzzz according according tothe theexpression expression Equation (2)inin3. Figure As ΦΦ ((Φ , ,yyΦΦ ) )and ofofEquation (2) Figure 3.3.As xyxy ,,Φ , yyΦyy,yz, Φ )Φ and to theto expression of Equation (2) in Figure As Figure 3 yzyz Φ zz , according xx xy xz ,, shows, the anomalous caused by the magnetic dipoles decay quickly with the buried depth increasing. Figure 3 shows, the anomalous caused by the magnetic dipoles decay quickly with the buried depth Figure 3 shows, the anomalous caused by the magnetic dipoles decay quickly with the buried depth The proposed method was applied toapplied this synthetic model, and theand sources detected are shown increasing. Theproposed proposed method was applied thissynthetic synthetic model, andthe thesources sources detected are increasing. The method was totothis model, detected are in Figure 4. The numerical range of the MDTA map is 0–10,853. For visibility, we limited the maximum shown in Figure 4. The numerical range of the MDTA map is 0–10,853. For visibility, we limited the shown in Figure 4. The numerical range of the MDTA map is 0–10,853. For visibility, we limited the value to 500, andtoto the threshold αparameter was set toαα 0. was maximum value 500, andthe theparameter thresholdparameter wasset settoto0.0. maximum value 500, and threshold

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Figure 4. Magnetic Dipole Tangent Angle (MDTA) map of magnetic gradient tensor data. Black text Figure 4. Magnetic Dipole Tangent Angle (MDTA) map of magnetic gradient tensor data. Black text represents the coordination of detected magnetic dipole Figure 4. Magnetic Dipole Tangent Angle (MDTA) map location. of magnetic gradient tensor data. Black text represents the coordination of detected magnetic dipole location. represents the coordination of detected magnetic dipole location.

All of the locations of the magnetic dipole sources were correctly found, as shown in Figure 4. All ofestimation thethe locations ofof dipole sources wereby correctly found, shown Figure The 1All m error inthe east direction, is likely caused the datafound, sample procedure. Analytic of locations themagnetic magnetic dipole sources were correctly asas shown in in Figure 4. 4. The 1 1mm estimation error east direction, is likely bycompare the data sample procedure. signal (AS) is a useful method in magnetic data interpretation [28]. to it, we drew an AS The estimation error inin thethe east direction, is likely causedcaused by theTo data sample procedure. Analytic Analytic signal (AS) is a method useful method in magnetic data interpretation [28]. To compare to it, we drew map (Figure signal (AS) is5). a useful in magnetic data interpretation [28]. To compare to it, we drew an AS an map AS map (Figure (Figure 5). 5).

Figure 5. Total field anomalies of the synthetic model. Figure5.5.Total Totalfield fieldanomalies anomalies of of the Figure the synthetic syntheticmodel. model.

As Figure 5 shows, the position of the shallow magnetic dipole sources (1–5) can be located, but the deep magnetic dipolethe source wasof not detected withdipole the AS map for thiscan application, asbut an As Figure 5 shows, position theclearly shallow magnetic sources (1–5) be located, As Figure 5value shows, the position ofthe the shallow magnetic dipole sources (1–5) can be located, obvious did not appear fornot deep source regions. the deeppeak magnetic dipole source was clearly detected with the AS map for this application, as an butobvious the deep magneticdid dipole sourcefor was not clearly with the AS map for this application,  peak not declination appear the sourcedetected regions. Then, wevalue changed the anddeep inclination of this model (Figure 2) to I = 40 ,D = 10 . as an obvious peak value did not appear for the deep source regions.   The total anddeclination the MDTAand mapinclination are drawnofinthis Figure 6. (Figure 2) to I = 40 ,D = 10 . Then,field we anomaly changed the model Then, we changed the declination and inclination of this model (Figure 2) to I = 40◦ , D = 10◦ . Although Figure 6band shows a muchmap wider range The total field anomaly the MDTA aresource drawnlocation in Figure 6. than Figure 4 for every dipole Thelocation, total field anomaly and thepeak MDTA map drawn in Figure 6. than there exists values atare thesource corresponding dipole locations, weevery markdipole these Although Figureobvious 6b shows a much wider location range Figureand 4 for Although Figure shows a much wider location range than Figure 4 for every dipole positions in Figure with black text. The numerical range of the MDTA map (Figure 6b)mark is limited location, there exists6b6b obvious peak values at thesource corresponding dipole locations, and we these location, there exists obvious peak values at the corresponding dipole locations, and we mark these  positions in Figure 6b for with black text. numerical range ofthe thevertical MDTA inclination map (Figure 6b) is from [0, 355] to [0, 20] visibility. AsThe Figure 6a shows, with angle ( Ilimited = 40  positions in355] Figure 6banomaly with black The numerical range numerical ofthe thevertical MDTA map (Figure 6b)(does )from bias,[0, the magnetic fieldtext. displays an asymmetrical distribution, this not to [0, 20] for visibility. As Figure 6a shows, with inclinationbut angle Iis=limited 40 ◦ from [0, 355] to [0, 20] for visibility. As Figure 6a shows, with the vertical inclination angle (I = 40 the precision the proposed algorithm. Although the Reduce to Polebut (RTP) )severely bias, theaffect magnetic anomalyoffield displays an asymmetrical numerical distribution, thisalgorithm does not ) bias, magnetic anomaly displays analgorithm. asymmetrical numerical distribution, butfurther this does canthe reduce thisthe bias, RTP isfield beyond the scope of this study, and wethe doReduce not discuss it any here.not severely affect precision of the proposed Although to Pole (RTP) algorithm severely affect thebias, precision the proposed algorithm. Although to Pole (RTP) algorithm We reduce also compared the proposed algorithm with other UXO detection algorithms this model can this RTP isof beyond the scope of this study, and wethe do Reduce not discuss itwith any further here.(  this bias,  RTP is beyond the scope of this study, and we do not discuss it any further here. canWe reduce compared proposed algorithm I = also 40 , D = 10 ), the as shown in Figure 7. with other UXO detection algorithms with this model (  WeIalso algorithm = 40compared ,D = 10 the ), asproposed shown in Figure 7. with other UXO detection algorithms with this model (I = 40◦ , D = 10◦ ), as shown in Figure 7.

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(a) (a) (a)



(b) (b) (b)



◦ and Figure 6. Total anomaly (a)=( 40 I ◦= 40 ) and MDTA (b).text Black text in (b) Figure 6. Total fieldfield anomaly (a) (I 10D map (b).map Black in (b) represents Figure 6. Total field anomaly (a) ( I =, D 40=, , D )= = 10 10MDTA ) and MDTA map (b). Black text in (b)   α 00 .. (b). Black text in (b) represents the coordination of detected theFigure coordination offield detected magnetic = 0. MDTA 6. Total anomaly (a) ( dipole I magnetic = 40locations, ,Ddipole = 10αlocations, ) and α = = map represents the coordination of detected magnetic dipole locations, represents the coordination of detected magnetic dipole locations, α = 0 .

(a) (b) (a) (b) (a) signal (ASS); and, (b) Normalized Source Strength (b) Figure 7. (a) Analytic (NSS) [11]. Figure 7. (a) Analytic signal (ASS); and, (b) Normalized Source Strength (NSS) [11]. Figure 7. (a) Analytic signal (ASS); and, (b) Normalized Source Strength (NSS) [11]. Figure 7. (a) Analytic signal (ASS); and, (b) Normalized Source Strength (NSS) [11]. As Figure 7a,b shows, analytic signal and normalized source strength do not perform very well As Figure 7a,b shows, analytic signal and normalized source strength do not perform very well forAs detecting the deep sources in this experiment. Figure shows, analytic signal and sourcestrength strengthdo donot notperform performvery verywell well for detecting the deep sources in this experiment. As Figure7a,b 7a,b shows, analytic signal and normalized normalized source To prove the robustness of the proposed algorithm, we designed a multiple-source scenario with forfor detecting the deep sources ininthe this experiment. To prove the robustness proposed algorithm, we designed a multiple-source scenario with detecting the deep sourcesof this experiment. three magnetic dipoles overlapping and closealgorithm, to each other, as shown ainmultiple-source Figure 8. ToTo prove the robustness of the proposed we designed designed scenario with three magnetic dipoles overlapping and close to each other, as shownain Figure 8. prove the robustness of the proposed algorithm, we multiple-source scenario with three magnetic dipoles overlapping and close to each other, as shown in Figure 8. three magnetic dipoles overlapping and close to each in Figure 8.

(a) (a)

(b) (b) 



(a) view of synthetic model where source A: I =(b) Figure 8. (a) Plain sectional 70 ,D = 30  , and source Figure 8. (a) Plain sectional view of synthetic model where source A: I = 70 ,D = 30 , and source     2; and (b)   are 20 Am B: I = 60 ,D = 40  , and source C: I = 80 ,D = 20  ; dipole moments 2; and ◦, ,and Figure (a)Plain Plain view ofofsynthetic source A: A: I moments =I 70 ,◦D 3030 source C: I =model ; dipole 20 Am (b) B: I 8.=8.(a) 60 , D sectional =sectional 40 , and 80 ,where Dwhere = 20 Figure view synthetic model source = 70 ,are D== andsource source Magnetic anomaly field.     ◦ , Danomaly ◦ , and ◦ ◦ 2 ; and (b) 2 Magnetic field. B: B: I =I 60 = 40 source C: I = 80 , D = 20 ; dipole moments are 20 Am Magnetic = 60 ,D = 40 , and source C: I = 80 ,D = 20 ; dipole moments are 20 Am ; and (b) anomaly field. Magnetic anomaly field. Then, the proposed algorithm was applied to the data (Figure 8b), and the result is displayed in

Then, the proposed algorithm was applied to the data (Figure 8b), and the result is displayed in Figure 9. Figure 9. the proposed algorithm was applied to the data (Figure 8b), and the result is displayed in Then, Then, the proposed algorithm was applied to the data (Figure 8b), and the result is displayed Figure 9. in Figure 9.

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Figure 9. MDTA map. Black text denotes detected magnetic magnetic dipole dipole locations. locations. Figure MDTA map. coordination of detected Figure 9. 9. MDTA map. Black text denotes the the coordination of detected magnetic dipole locations. (α = 0). ((α α == 00).). Figure there 9. MDTA text denotes theshown coordination detected magnetic locations. Although are map. someBlack non-zero values in theofregion region that arefar fardipole awayfrom from the source Although that are away the source Although there there are are some non-zero values shown in the region that are far away from the source ( α = 0values ). position, these are relatively small and distributed smoothly. It is easy to distinguish them from position, smoothly. It It is is easy easy to to distinguish distinguish them them position, these these values values are relatively small and distributed smoothly. the large peak values that arethat related the actual Overall,Overall, the proposed algorithm indicates from the large peak are to related sources. Overall, the proposed algorithm from the large peak to shown thesources. actual proposed Although therevalues are some non-zero values in thesources. region that are farthe away from thealgorithm source the position of all of sources with minimal bias, as shown in Figure 9. We also compared the proposed indicates the position of are all of sourcessmall withand shown in in Figure 9. We We also also compared compared indicates thethese position minimal bias, as shown Figure 9. position, values relatively distributed smoothly. It is easy to distinguish them algorithm with AS and NSS, as shown in Figure 10. the proposed algorithm with AS and NSS, 10. thefrom proposed algorithm as shown in Figure 10. the large peak values that are related to the actual sources. Overall, the proposed algorithm

indicates the position of all of sources with minimal bias, as shown in Figure 9. We also compared the proposed algorithm with AS and NSS, as shown in Figure 10.

(a)

(b) (b)

Figure 10. (a) AS map of Figure 8 data, and (b) NSS 88 data. Figure NSS map map of of Figure Figure data. (b) Figure 10. (a) AS map of Figure 8 data, and (b) NSS map of Figure 8 data. 10. (a) AS map of Figure 8 data, and (b) NSS map of Figure 8 data. As Figure Figure 10 10Figure shows, As shows, the analytic signal method successfully successfully detects detects all all of of the the sources sources As Figure 10 the analytic signal method detects all of thesource sources (Figure 10a). (Figure 10a). The Theshows, NSS method locationsuccessfully precision for source A and BB detection. (Figure 10a). NSS shows more for source A and source detection. As Figure 10 shows, the analytic signal method successfully detects all of the sources The NSS method shows more location precision for source A and source B detection. However, both However, both of these two methods could not indicate the exact position of the magnetic dipoles. However, position of the magnetic dipoles. of (Figure both 10a). of Thethese NSS method shows more location precisionexact for source A and source B detection. these two methods could not indicate the exact position of the magnetic dipoles. The numerical TheHowever, numerical distribution smooth, and to the The numerical distribution inmethods Figure 10a,b smooth, and itit is isofhard hard to mark markdipoles. the exact exact both of these two could is notrelatively indicate the exact position the magnetic coordinates of the sources. of Figures 9 and 10 are calculated and displayed distribution in Figure 10a,b is relatively smooth, and it is hard to mark the exact coordinates the coordinates of the sources. As proof, the gradient and 10 are calculated and displayed in The numerical distribution in Figure 10a,b is relatively smooth, and it is hard to mark the exactofin Figure 11. sources. As proof, the gradient of Figures 9 and 10 are calculated and displayed in Figure 11. Figure 11. coordinates of the sources. As proof, the gradient of Figures 9 and 10 are calculated and displayed in

Figure 11.

(a) (a)

(a)

(b) (b) (b)

(c) (c) (c)

Figure 11.11. The gradient ofofthe the detection results Figure 9, (b) Figure 10a, and Figure 10b. Figure The gradient thedetection detectionresults results for for (a) (a) and (c)(c) Figure 10b. Figure 11. The gradient of for (a) Figure Figure9, 9,(b) (b)Figure Figure10a, 10a, and (c) Figure 10b. Figure 11. The gradient of the detection results for (a) Figure 9, (b) Figure 10a, and (c) Figure 10b.

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Analytic signal and NSS methods perform well in many UXO detection applications. Especially when the observation is close to theperform sources,well the in range of UXO the anomalous sources will Especially be relatively Analytic signalpoint and NSS methods many detection applications. when thethe observation point close to the sources, thethe range of the anomalous sources be small, and sources easy to is locate. However, when observation point is far from will the UXO, relatively small, and the sources easy surveys to locate.or However, whenburied the observation is far from the such as during high-altitude air-borne with deeply UXO, it ispoint necessary to develop UXO, such as method during high-altitude surveysprecisely. or with deeply buried it isthat necessary a corresponding to locate the air-borne source position Moreover, weUXO, believe locatingtothe develop a corresponding method to locate the source position precisely. Moreover, we believe that in source’s horizontal position with a high precision may also increase the computational accuracy locating the source’s horizontal position with a high precision may also increase the computational a source depth estimate. accuracy in a source depth estimate.

4. Study Region and Field Data

4. Study Region and Field Data

We applied the proposed algorithm to the magnetic field data that was acquired in eastern We applied the proposed algorithm to the magnetic field data that was acquired in eastern Montana. This helicopter magnetic survey was conducted by the U.S. Geological Survey (USGS), which Montana. This helicopter magnetic survey was conducted by the U.S. Geological Survey (USGS), includes the East Poplar oil field on the Fort Peck Indian Reservation to assess the ground-water quality. which includes the East Poplar oil field on the Fort Peck Indian Reservation to assess the groundThe magnetic data were acquired using the Geometrics G-822 magnetometer, which was mounted water quality. The magnetic data were acquired using the Geometrics G-822 magnetometer, which on the SkyTEM 301 airborne electromagnetic system. The raw magnetic data were corrected for the was mounted on the SkyTEM 301 airborne electromagnetic system. The raw magnetic data were diurnal drift,for removing thedrift, International Reference Field Reference (IGRF). The reduced-to-pole corrected the diurnal removing Geomagnetic the International Geomagnetic Field (IGRF). The (RTP) magnetic field was calculated according to the inclination and the declination parameters that reduced-to-pole (RTP) magnetic field was calculated according to the inclination and the declination were providedthat in the raw file (Figure 12). parameters were provided in the raw file (Figure 12). TheThe flight altitude was 850 distancewas wasmuch muchlarger largerthan than size of the flight altitude was 850m, m,and andthe theobservation observation distance thethe size of the metallic oil oil and water/brine can be beconsidered consideredasasthe themagnetic magnetic dipoles metallic and water/brinewells, wells,so sothat that the the wells can dipoles in in thisthis application. The anomalouscaused causedby bythese these dipole-like dipole-like sources locations of the application. The anomalous sourcesare areweak. weak.While Whilethe the locations of the targets have been well markedwith withexact exactground ground prior-information, prior-information, as 13.13. wellwell targets have been well marked asshown shownininFigure Figure

Figure Magneticfield fielddata data of of part part of Figure 12.12.Magnetic of the the East EastPoplar Poplaroil oilfield. field.

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Figure 13. The location ofof the wells in the research region marked bybyred rectangle. Figure Figure 13. 13. The The location location of the the wells wells in in the the research research region regionmarked marked byred redrectangle. rectangle.

The survey area lies in the Williston Basin, which is a large structural basin with an area of 800 The survey survey area lies structural basin with an area of 800 The lies in in the theWilliston WillistonBasin, Basin,which whichisisa large a large structural basin with an area of x 500 kilometers. The sedimentation of the Williston Basin formed mostly in the Paleozoic and early x 500 The sedimentation theWilliston WillistonBasin Basinformed formedmostly mostlyininthe thePaleozoic Paleozoicand and early early 800 × kilometers. 500 km. The sedimentation ofof the Mesozoic eras, and ceased by the Cretaceous era. The thickness of the entire sedimentary section is Mesozoic eras, eras, and ceased by the Cretaceous Mesozoic Cretaceous era. era. The Thethickness thicknessofofthe theentire entiresedimentary sedimentarysection sectionis about three kilometers. Although there is a subsidiary domal structure in the study area, named the area, named named the the isabout aboutthree threekilometers. kilometers.Although Althoughthere thereisisaasubsidiary subsidiarydomal domal structure structure in the study area, Poplar Dome, the geological framework of the study area in the East Poplar oil field is relatively clear PoplarDome, Dome,the thegeological geologicalframework frameworkof ofthe thestudy studyarea areain inthe the East East Poplar Poplar oil oil field field is is relatively relatively clear clear Poplar and simple [29]. andsimple simple[29]. [29]. and 5.5.Results 5. Results Results The corresponding gradient tensor maps of this data were calculated bybythe formulae suggested The Thecorresponding correspondinggradient gradienttensor tensormaps mapsof ofthis thisdata data were werecalculated calculated by the the formulae formulae suggested suggested bybyPedersen and Rasmussen [27]. The MDTA map of the field data and the corresponding horizontal byPedersen Pedersenand andRasmussen Rasmussen[27]. [27]. The The MDTA MDTAmap mapof of the the field field data data and and the the corresponding corresponding horizontal horizontal gradient map are displayed ininFigure 14. gradient gradientmap mapare aredisplayed displayed inFigure Figure14. 14.

(a) (a)

(b) (b)

Figure 14. (a) MDTA map of the field data ( α = 20 ); (b) Horizontal gradient map of (a). α ==20); 20 (b) Figure ); (b) Horizontal gradient map Figure14. 14.(a) (a)MDTA MDTAmap mapof ofthe thefield fielddata data((α Horizontal gradient map of of (a).(a).

This magnetic field data was also analyzed with other UXO detection algorithms, as shown in This magnetic field data was also analyzed with other UXO detection algorithms, as shown in FigureThis 15. magnetic field data was also analyzed with other UXO detection algorithms, as shown Figure 15. in Figure 15.

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(a)

(b)

(c)

(d)

Figure 15. (a) NNS of the field data; (b) horizontal gradient of (a); (c) analytic signal of the field data; Figure 15. (a) NNS of the field data; (b) horizontal gradient of (a); (c) analytic signal of the field data; and, (d) horizontal gradient of (c). and, (d) horizontal gradient of (c).

Both the NSS and analytic signal methods successfully located the primary oil wells in the range, Both in theFigure NSS and signal methods successfully located themethod primaryperforms oil wells in the range, as shown 15. analytic However, as displayed in Figure 15a, the NSS poorly with as shown in Figure 15. However, as displayed in Figure 15a, the NSS method performs poorly with weak sources detection in this application, showing some confusion when identifying and describing weak sources detectionsource. in this Figure application, showing some when identifying and describing the weak anomalous 15b shows that the confusion weak anomalous sources do not have an the weak anomalous source. Figure 15b shows that the weak anomalous sources do not have anlocation intense intense response on the gradient map. Similarly, the analytic signal shows a relatively precise response on the gradient map. Similarly, the analytic signal shows a relatively precise location for the for the primary oil wells, as shown in Figure 15c,d, but there is severe interference when the sources primary wells, as shown Figure 15c,d, there is MDTA severe interference when the are close are close oil to each other. In thisinapplication, thebut proposed was not influenced bysources these problems, to each other. In this application, the proposed MDTA was not influenced by these problems, while it while it marked almost all of the weak anomalous sources and separated the sources located in close marked almost all of the weak anomalous sources and separated the sources located in close proximity proximity to one another. To clearly show this, we marked these weak anomalous sources that were to one another. To clearly show detected by MDTA in Figure 16.this, we marked these weak anomalous sources that were detected by MDTA in Figure 16. This data contains high-level noise, and it was further amplified in the calculation of tensor This data contains this high-level noise,shows and itthat was further in the calculation of tensor components. However, experiment MDTA is amplified not obviously influenced by the noise components. However, this experiment shows that MDTA is not obviously influenced by the noise and error. Actually, we believe that real field data acquired by a full tensor magnetic gradiometry and error. Actually, we believe that real field data acquired by a full tensor magnetic gradiometry system will provide better results with the proposed algorithm. system will provide better results with the proposed algorithm.

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Figure 16. marked with thethe detected weak anomalous sources, identified with awith white Figure 16. MDTA MDTAmap map marked with detected weak anomalous sources, identified a rectangle. white rectangle.

6. Discussion Discussion 6. A common common problem problem with with air-borne air-borne magnetic magnetic surveys surveys is is that that the the anomalies anomalies are are weak weak when when the the A distance between between targets targets and and observation observation point point isis large. large. The by aa magnetic magnetic distance The anomalies anomalies produced produced by dipole-like source decay with distance cubed inversely. Like the field data in this paper, the dipole-like source decay with distance cubed inversely. Like the field data in this paper, the amplitude amplitude of a weak anomaly may be confused with observation noise and cannot be directly of a weak anomaly may be confused with observation noise and cannot be directly extracted with extracted with traditional detection methods.in AsFigure analyzed Figure 1,dipole a magnetic dipole can display traditional detection methods. As analyzed 1, ainmagnetic can display an obvious an obvious direction information in the eigenvector. While the noise contained in the data will not direction information in the eigenvector. While the noise contained in the data will not show similar show similar physical properties in eigenvector decomposition. So, the proposed algorithm extracted physical properties in eigenvector decomposition. So, the proposed algorithm extracted more of the weak more of thesources weak anomalous efficiently in the fieldThe data experiment. The core of theis proposed anomalous efficiently sources in the field data experiment. core of the proposed method rotating method is rotating the coordination of the observation plane, and we used the new coordination the coordination of the observation plane, and we used the new coordination system to locate the position system to locate the Basically, position of proposed cannot be classified of magnetic dipoles. themagnetic proposeddipoles. methodBasically, cannot bethe classified as amethod mathematic transformation, as a mathematic transformation, such as Independent Component Analysis (ICA), or Principal such as Independent Component Analysis (ICA), or Principal Component Analysis (PCA). Component Analysis (PCA). There were some reasons for selecting this magnetic survey data in the field experiment section. There were some reasons for objects, selecting thisasmagnetic survey in the field experiment Firstly, the location of the metallic such the oil wells anddata water/brine oils, were wellsection. known Firstly, the location of the metallic objects, such as the oil wells and water/brine oils, were known prior to this study. This information was critical for validating the proposed algorithmwell in the field priorexperiment. to this study. This information critical for validating proposed algorithm in contains the field data Secondly, the altitudewas of this magnetic survey isthe high, at 850 m, so the data data experiment. Secondly, the altitude this magnetic survey is high, atof 850 m,proposed so the data contains some weak anomalous sources that canofbe used to prove the efficiency the algorithm. some weak anomalous sources that can be used to prove the efficiency of the proposed algorithm. Although the detected sources are not totally buried under the surface, the covered layer do not have Although the influence detected sources are not totally buried under theof covered layer do not have an important on the air-borne magnetic survey forthe thesurface, detection the metallic objects. an important influence on the air-borne magnetic survey for the detection of the metallic objects. Many methods have been developed to detect metallic targets, most of which consider these Many methodsdipoles. have been developed to detect dipole metallichas targets, of of which consider these objects as magnetic Commonly, a magnetic three most degrees freedom related to objects as magnetic dipoles. Commonly, a magnetic dipole has three degrees of freedom related to the source location, while another three degrees are present in the direction and magnitude of the the source location, while another three degrees are present in the direction and magnitude of the magnetic dipole. In practice, the uncertainty of the direction of the magnetic moment significantly magnetic dipole. practice, the uncertainty of themagnetization, direction of theitmagnetic moment significantly complicates UXOIndetection. Unlike the remnant is impossible to estimate the complicates UXO detection. Unlike the remnant magnetization, it is impossible to estimate the direction of every dipole. So, many methods have been proposed to extract the amplitude of the direction of every dipole. So, many methods have been proposed to extract the amplitude of the magnetic gradient data, such as the analytic signal and NNS, which are used independently of the magnetic gradient data, such as the these analytic signal are anddesigned NNS, which are used independently of the magnetization direction. However, methods to limit the influence of remanent magnetization direction. However, these methods are designed to limit the influence of remanent magnetization in an inversion procedure. They build an envelope of magnetic data for all moment magnetization in an the inversion procedure. They builddirection an envelope ofremoved magneticwith datathese for all moment directions. Although uncertainty of magnetization can be transforms, directions. Although the of uncertainty magnetization direction can be removed with the positioning information the sourcesof also will be reduced. In UXO detection applications, thethese goal transforms, the positioning information of the sources also will be reduced. In UXO detection of the algorithm is to extract the source location parameters, and the direction of the magnetic dipoles is applications, the goal This of the algorithm is tothe extract theprocedure source location parameters, and the direction not our main concern. study analyzed physical of tensor eigenvector decomposition of the magnetic dipoles is not our main concern. This study analyzed the physical procedure of tensor eigenvector decomposition and directly calculated the horizontal position of a magnetic dipole

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and directly calculated the horizontal position of a magnetic dipole source. Compared to the analytic signal and NNS methods, the proposed method performs with higher precision in locating the source, and is not severely influenced by the direction of magnetic moment. 7. Conclusions In this paper, an attempt to locate metallic objects using eigenvector analysis for full tensor magnetic gradient data is presented. The physical meaning of the magnetic gradient tensor data decomposition has been expressed in detail. We used the characteristics of tensor data decomposition to locate the dipole-like magnetic source. The synthetic experiments show that the proposed method is not severely interfered with by the direction of the magnetic moment, and it can be used as an efficient tool to locate weak anomalous sources. We also validated that the method works well when the sources are close or overlapping each other in the synthetic experiment. In practice, the proposed algorithm can be used for both the total-field-magnetic data and magnetic gradient tensor data, and can also help provide information on the horizontal position of the source in modelling or inversion. Future improvements to this work should consider estimating the depth parameter of sources for UXO data interpretation. Acknowledgments: This research is funded by the National Natural Science Foundation of China (No. 41630317, No. 41674110). Survey of Basic Geology in Northeast China No. 12120115063201, No. 17-H863-01-ZT-002-015-01. Author Contributions: Lizhe Wang and Boxin Zuo conceived and designed the experiments; Boxin Zuo performed the experiments and analyzed the results; Weitao Chen analyzed the data; Boxin Zuo wrote the paper. Conflicts of Interest: The authors declare no conflicts of interest.

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Full Tensor Eigenvector Analysis on Air-Borne Magnetic Gradiometer Data for the Detection of Dipole-Like Magnetic Sources.

The detection of dipole-like sources, such as unexploded ordnances (UXO) and other metallic objects, based on a magnetic gradiometer system, has been ...
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