Identification of sources of potential fields with the continuous
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1. 1 Zi i Xoitl 2 2 222 alh sin I Wyx 6r 2 0 20 a z1 zo 2a ei 21 35 Car a 5 22 0 5 7 34 Calling zo the mean depth we obtain the following Taylor expansion for z9 a gt gt h 2 zo a gt gt l 2 large dilations or large average depth and hy 1 tan 0 Zl Wyajsr 2 0 20 a 4 225 alh 21 435 1 h 2 sec 8 cos 20 1 2 e 2 lair 2 Gora 35 19 466 SAILHAC ET AL WAVELET TRANSFORMS OF MAGNETIC PROFILES w 4 Thin Dipping Prism Almost Isometnc Vertical Pnsm Almost Isometric Dipping Prism Wide Prism With Vertical Edges 1 os f f so 0 05 4a 15 124 122 120 118 116 44 42 40 38 36 36 38 40 42 44 1416 118 120 122 124 x x x x wf Thin Dipping Prism Almost Isomeine Vertical Pnsm Almost Isometric Dipping Prism Waide Prism With Vertical Edges 15 k i Q Or 4 1T o i EOR j E 15 24 122 420 118 116 44 42 40 38 3 3 3 40 42 44 116 118 120 122 124 x x a a 120 80 40 0 40 80 120 x Figure 9 Vertical and dipping 0 40 prisms of magnetization at depth zo 1 with apparent inclination I 29 16 bottom Total magnetic field and corresponding modulus of top real or middle complex wavelet coefficients Isovalues are shown shaded with regular intervals Modulus maxima lines are shown in black their continuation down to the source is shown f2. Note that the search for the height from 22 or 28 assumes that one has first determined the dip angle m 2 0 This could be done via the direction of the modulus maxima lines see equation 25 When the angle remains unknown an estimation from equation 22 instead of 28 would be useful anyway as it gives the correct height within a factor of 2 for any 0 value except for values of about 45 or 90 Other possible errors are due to the uncertainty in z location and modulus maximum determination which are involved in the Taylor expansion along the modulus maximum line Quantification of this remark is given by manipu lations showing that the next term in the expansion of 20 is 2 zo zo a so that H a may give an average value of h 4 x xo instead of the height h itself noise in positioning or continuation also leads to errors in the wavelet coefficients 3 5 Magnetization Strip Let us now consider the total field anomaly gener ated by an inclined strip with dip angle 7 2 6 with a limit between the two points x1 z1 and x2 z2 and middle point at o 41 amp 2 2 and zo 21 22 2 This strip is formally the horizontal derivative of a right step source So as shown by Moreau et al 1999 its wavelet coefficients can be obtained by a simple hor izontal derivative of 23 The wavelet coefficients of order y for the strip are those of order y 1 for the right step divided by
3. Otherwise a zo zo cot 26 2 cs cal ED o v 26 This quantifies a classical property of potential field anomalies over a dipping homogeneous source At low altitude the main contribution is due to the upper part of the source while at increasing altitude the relative contribution of deeper parts progressively increases and the location of the maximum effect moves to the vertical passing through the center of gravity of the source SAILHAC ET AL WAVELET TRANSFORMS OF MAGNETIC PROFILES Writing wavelet coefficients along modulus maxima similarly to 19 for a vertical step then Taylor expand ing for zo a gt gt h 2 large dilations or large average depth gives Wy2jsr 2 0 2 2 25 ah lick 1 tan 0 27 Thus scaling parameters at large dilations are those of the vertical step equation 20 except for the rela tive amplitudes between terms of the Taylor expansion which depend on the dip angle 7 2 6 Fitting a straight line as in 21 provides the slope a 1 and the log factor k In Kh These can then be used along the extrema lines to plot a function H a similar to that defined by 22 which converges for 0 7 4 to a limit which is the height h of the step see Figure 8 Hi oe 1 tan 6 2 E E 28 l IWyrpsry 2 0 70 4 in a The phase is 7 2 for large dilations and could be used to determine an unknown inclination J
4. Using variables X z 21 X2 z f Z1 z z z a and Z2 z2 Z Zz a prefac tor K 2 sin J sin I and angle 2 0 one gets the wavelet coefficients for the inclined step to be compared with 17 Wy3 67 220 2 a 1 1 Kacos bets lz zx 23 The argument and modulus can be written general izing equations 18 respectively as follows By1 67 2 0 Z a 2I r tan a tan7 a Wura a Teer Solving for 0 Wy x a 0x 0 implies that modu lus maxima are defined by solutions of a second order polynomial in Z z zo and fourth order in X xo with v X h 2 4 0 and 6 0 Solutions are 24 a z tan 1 4 1 v 1 vot j 25 Equation 25 tells us that for different dip angles 7 2 0 the modulus maximum line is not a straight line and its slope depends on the dilation a and more precisely on a 20 h 2 v and 0 It is possible to make a Taylor expansion of this equation near to the vertical of the source for v lt lt 1 this gives two types of asymptotic behavior For large dilations a for a zo gt gt h 2 the maximum line is a branch of a hyperbola whose asymptote is z zg For small dilations the maximum line is a branch of a polynomial in x zo whose first order corresponds to a straight line with slope cot 26 If a zo gt gt h 2 2 h 2 tan 6 ac Ry To a zo O z x9
5. 22 Steps between z 70 6 and 2 1 4 solid vertical dashed dipping 60 20 1 a a Z 10 0 0 15 Steps between 2 0 2 and z 1 0 solid vertical dashed dipping 60 JWia 7 0 6 0 5 0 4 0 3 0 2 0 1 100 g a 1 10 at2 0 1 0 4 1 0 3 oS 0 2 A 208 gt 0 1 I d 0 7 t 0 6 1 10 0 10 20 atz at2 Steps between z 0 7 and Zj 1 3 solid vertical dashed dipping 60 0 35 0 3 0 25 0 2 0 15 0 1 0 05 Wa Ria 10 atzo 0 1 0 63 0 08 0 62 0 06 0 61 g x 0 04 0 6 0 02 0 59 o 0 58 0 10 20 at2 Figure 8 Scaling relations for vertical and dipping 9 40 steps of magnetization with apparent inclination J 29 16 left to right Modulus of complex wavelet coefficients along z o first order scaling equation 25 residual due to the second order term equation 31 and function H a estimating the height of the step equation 32 Correspond to top anomaly in Figure 7 zo 1 k 0 8 middle to a lower depth zo 0 6 h 0 8 and bottom to a lower height zo 1 h 0 6 19 464 Let us now consider an inclined step with dip angle nx 2 6 with a limit between the two points z1 21 and x2 22 and middle point at ro 21 z2 2 and zo z 22 2 Similar algebraic manipulations as those used for the vertical step allow us to calculate the total field anomaly and its derivatives
6. Typical SSW NNE profile of the survey Total magnetic field anomaly and its wavelet transform coefficients maxima are in dark and minima in clear shading Position in m 4 a a a n a a A E gt a n J E he AIO moO OOo NO NWWL HOT MANOUOVIOVCIO ONOVIGCOOCOeo Depth in m NM 100000 120000 Position in m Figure 13 Imaging sources from the use of 1 D real wavelets Each profile anomaly provides one source location For the set of profiles in the area this maps horizontal positions and depths top as seen from above superimposed on the anomalies in grey scale and bottom its vertical cross section projection For the central profile one gets the following depths zo from the SSW to the NNE 370 m SSW dike 250 m middle dike and 930 m the fault SAILHAC ET AL WAVELET TRANSFORMS OF MAGNETIC PROFILES 19 471 Complex Coef along Modulus Maxima Lines Modulus Maxima Lines 150 1400 Modulus Maxime Lines Complex Wavelet order 1 100 1200 s 50 d 1000 x 40 50 z E 30 g 800 5 5 20 alts 0 geo 3 FA 10 a ao i a al 2 i 200 nN z f i fice 100 k 5 i 4000 100105 relative coortiinates akm gt kaa dilation a in m 150 relative coordinates X in km mn 9 degrees Modulus vs Scale Phase vs Scale LogLog plot tor the scalling ot Wavelet Coefticients Modulus Complex order 1 10 5 3 Wavelet Coetncients W a i
7. lated for the total magnetic anomaly field at any normal apparent inclination Then we have applied the technique to data profiles from French Guiana A set of 27 profiles each of 80 km long with flightline direction N30 E has been trans formed by wavelets for the purpose of this study we have analyzed three isolated features in the Cayenne Kourou area two dikes and one fault 2 Method 2 1 Wavelet Transform of Potential Fields First we briefly recall the basic theory Moreau et al 1997 1999 which we apply to the case of a two dimensional physical space This involves parameters that are listed in the notation section We define the continuous wavelet transform of a function 9 z R as a convolution product b z Waal E w dole Da o b 1 where y x R is the analyzing wavelet a R is a dilation parameter and the dilation operator D is defined by Day s 2 2 The source is modeled as a homogeneous function o z z This means for instance when the source is homogeneous at x 0 and z 0 with homogeneity degree a that for any positive number it follows that o ArAw Az A o a z By applying a Fourier multiplier homogeneous of de gree y equivalent to a derivative of order y and a di lation to the Poisson semigroup kernel Moreau et al 0 2 0 1 10 5 0 5 10 x SAILHAC ET AL WAVELET TRANSFORMS OF MAGNETIC PROFILES 1997 1999 we obtain a cl
8. 9 and the struc tural index is N 0 1 for the middle dike The SSW dike and the fault have an homogeneity degree which is in the range of that of a prism or a strip Moreau 1995 near to that of a horizontal pipe a 2 0 This corre sponds with magnetization anomalies as expected sec tion 4 2 a gabbro dioritic dike of high magnetization and a fault whose magnetic signature is the Paramaca volcano sedimentary series fault this can also be the signature of an edge of a sill like source created by the Paramaca volcano sedimentary abatting against the coastal sedimentary rocks The middle dike is seen with a surprising homogene ity degree which is near to that of a step a 1 0 rather than a prism A first interpretation of this value is that there is also associated with this dike a con trast in the magnetization of the background Indeed a contact with the Cayenne Series composed of amphi 19 472 bolites leptynites and migmatites has been recognized see Figure 11 they constitute a surface of migrated ferromagnetic elements This remark agrees with the shape of the observed anomaly see Figure 12 as com pared to the synthetic one appearing on Figure 7 A second interpretation is that this is a strip like source as expected but with an infinite height extent such that the first term in equation 34 tends to zero and the second term implies the observed exponent 2 In this case the signifi
9. an apparent normal inclination J normal field direction Thus the total magnetic field anomaly T at a z is given by two con jugated symmetrical and antisymmetrical functions T and dT Let us introduce the mean apparent inclina tion S T I 2 such that in the special case of normal apparent inclination one gets S I J this now reads Woz su 2 2 4 a7 sin J sin sin J T ae ain Te 6T cos23 46T gt sin23 A5 where T and T are defined by second order deriva tives of the Green s function V 67 x z 2V z z Ta x z 328V 2 2 Complex wavelet coefficients of T are 25V z 2 A6 Wyzjsr z 8 E sin sin In Pe y 1 128 Sgc e 2 0 A7 sin J sin J Appendix B Modulus Maxima Lines of Real Wavelet Coefficients We look for the geometry of modulus maxima lines from real wavelet coefficients and its relation to the mean apparent inclination S equation A5 Let us consider the following characteristic ratio function O1 1 T z a Oxvt1 8 1 T5 z a RY z a Oz1t1 B1 Now if we look for the modulus maxima of Wy157 2 maxima of a profile while dilation is kept constant the question is reduced to searching for the zero set 19 473 of Wyr41 67 2 Algebraic manipulations lead to the equation R7 z a tan 2S respectively RY z a cot 2S for the definition of the modulus maxima lines x
10. and Kourou 1 Introduction The high resolution of recent magnetic and grav ity surveys Gunn 1997 Bregert and Millegan 1998 Grauch and Millegan 1998 stimulates the development of specific interpretation techniques which emphasize the information of interest to the geologist and to the geophysicist Modern magnetometers measure the mag netic field to 0 01 nT the Earth s dipole magnetic field is 30 000 60 000 nT worldwide with an ampli tude of the anomaly due to the upper crust of hundreds of nanoteslas Global Positioning Systems GPS can give the position from a few meters in aeromagnetic surveys to a few centimeters in ground surveys We have explored the use of wavelet transforms as initially introduced in the analysis of potential fields by Moreau 1995 She has described the technique for lo cal and extended sources with the emphasis on gravity applications followed by preliminary results for mag netic cases The principle of this method is to inter pret potential fields data via the properties of the up ward continued derivative field Moreau et al 1997 Copyright 2000 by the American Geophysical Union Paper number 2000JB900090 0148 0227 00 2000JB900090 09 00 have demonstrated the general n dimensional theory for local homogeneous sources Moreau et al 1999 have analyzed the effects of noise and extent of sources on the properties of the wavelet coefficients We now present specific propert
11. gt 1 t 2 05 1 o j ae a AS eae Wo a 0 6 2 i i i baa M ti 4 95 15 T i i p 1 ost lt 2 E Os nd 7 See i___i_l45 ok t 44 42 40 38 36 36 38 40 42 44 a s x x 05t i MASE l baias l r aA 80 60 40 20 0 20 40 60 80 x Vertical Step Step dipping 60 a u 1 05 0 2 d sft EA T i 1t i n 05 ji Ji Sa ed at ee ea 44 42 40 38 36 k ehanas x 05 3 80 60 40 20 0 20 40 60 80 Figure 7 Vertical and dipping steps of magnetization at mean depth zp 1 with apparent inclination I 29 16 bottom left Total magnetic field and top left corresponding modulus of real or complex wavelet coefficients zoomed at right Isovalues are shown shaded with regular intervals Modulus maxima lines are shown in black their continuation to the source is shown in the lower half space analytically above the top of the source modulus maximum line which is straight and vertical a Asymptotic behavior is obtained with a Taylor expan well known property for the maximum of the analytic sion for z a gt gt h 2 large dilations or large average signal The phase along this maximum line equals t depth 27 ah e2 n W zo soca 2 81 es Wy3 67 2 0 20 0 ptT 0 0 a 355 z1 a z2 a sin I i 20 r h 2 19 2 84 ahe are lat zo a E 20 SAILHAC ET AL WAVELET TRANSFORMS OF MAGNETIC PROFILES The first ord
12. is the only access to information on depths of structures as we have shown for dikes and a fault in Guiana Further possibilities with other wavelets of this kind could be explored for reduction to the pole in the first instance This wavelet technique combines different di rections of derivative of the field Also note that this could be used for interpretation of surveys where differ ent components of the field have been simultaneously recorded The wavelet analysis of the magnetic poten tial U is equivalent to the analysis of the horizontal and vertical components H and Z that are upward contin ued to different levels This article is the second of a series forthcoming ones will relate to statistics of the sources automation of the technique considerations of large scale asymptotic be havior and application of 2 D wavelets to aeromagnetic maps Appendix A Wavelet Coefficients Versus Green Function Here we show useful relations between wavelet coeffi cients of a potential field due to a 2 D body of constant source density and the Green s function V which corre sponds to its gravity potential Factors corresponding to the nature of the field mass or magnetization in tensity have been omitted for the relations to be gen eral these are given by Telford et al 1990 or Blakely 1996 From equations 5 and 7 complex wavelet coefficients of V are given by Wy2v 2 4 A 7 aV e z a OV x z a PEN
13. of complex wavelets 2 in order for this to be done in a simple way Using the Hilbert transform H which changes WY into Y7 H P7 we define the complex wavelets Y Y iH 7 as combinations of the hori zontal and vertical wavelets y and 7 Moreau 1995 PI y ivy These are actually defined not only for y N but for y R4 with the help of frac tional derivatives They are progressive wavelets pro portional to the Cauchy wavelets Holschneider 1995 and their general expression is et Gy 41 m i v z 6 where I is the Gamma function Thus the complex wavelet coefficients of the potential field z are Wazig z 2 0 Wyr16 2 2 a Wy 21 6 2 2 ye 7 where T is the Gamma function 19 458 F 0 0 15 pe upg 5 o 5 x SAILHAC ET AL WAVELET TRANSFORMS OF MAGNETIC PROFILES F x 0 2I p 2 5p 6 T x L J 2 5 0 5 x Figure 3 Magnetic potential U and total field T due to a dipole with inclination J 7 6 at zo 0 and zp 1 analyzed with complex wavelet 1 Phase isovalues shown in white every m 6 converge to the source dotted lines the phase isovalue z 0 for x zo 0 gives the inclination J For a local dipole of normal inclination J analyzed with a complex wavelet of order 1 the argument of the wavelet coefficients is constant along the vertical to the source equal to J for the magnetic potential a
14. the source a 8 a y for the total geomagnetic field anomaly Once the depth zp of the anomaly field has been obtained the exponent is simply obtained with the wavelet coefficients W along modulus maxima lines as the slope of log Wa a7 ver sus log a zo Figure 2 The kernel of the Poisson semigroup which is used to build these special wavelets defines the well known upward continuation filter Pa x which transforms the harmonic field z from measured level z to the level z a Le Mou l 1970 Bhattacharyya 1972 Galdeano 0 05 0 05 0 1 0 15 0 2 0 25 0 3 10 5 0 5 10 x Figure 1 Typical wavelets belonging to the Poisson semigroup class x 1 m 22 2 1 and W x 1 m x 1 1 SAILHAC ET AL WAVELET TRANSFORMS OF MAGNETIC PROFILES 19 457 3 tog IWia a Oo 02 04 06 08 1 log a z Figure 2 bottom left Magnetic total field T due to a vertical dipole at ro 0 and z 1 analyzed with vertical wavelet y top left Modulus maxima lines converge at the location of the source at x zo and a zp with downward continuation in dotted lines right the modulus along the line at z zo follows a scaling relation with exponent 3 associated with the homogeneity degree of the local line source a 2 1974 Baranov 1975 Gibert and Galdeano 1985 x being a 1 D variable abscissa along the profile
15. this is 1 a P x z apr 4 Two typical real wavelets can then be considered the horizontal Ys and the vertical made by one horizontal or one vertical upward derivative of P respectively they are said to be of order 1 Fol lowing with y 1 derivatives over x gives wavelets of order y 2 z O7P z or in Fourier domain DI u 2mu e 2 14 Y7 2 993a Palz la 1 oF in Fourier domain 47 u 27u 1 2n ul e 27 1 Then dilating these wavelets with the dilation a trans forms y x into 7 z a a which is also the yth derivative of Pa multiplied by the scaling factor a7 Thus the convolution of the harmonic field z mea sured at level z with these dilated real wavelets gives the wavelet coefficients at scale a which are also the derivatives of the upward continued field at level z a whose dimension is that of the initial field in nT for the geomagnetic total field anomaly a 877 2 v z a Ory 1 a 8771 z z a Or y 1 Waze 2 2 5 where z and z are the horizontal and up Wo2 9 2 2 ward vertical derivatives respectively of the harmonic potential field z at level z 2 2 Complex Wavelets Any of the two real wavelets y and 7 can be used to determine the depth and the homogeneity degree a of a local dipole source Figure 2 but the determination of the inclination of this dipole needs the introduction
16. to the anomaly field reduced to the inclination J 45 Analytic expressions for the total field anomaly due to a given body are equivalent to those of south north profiles by replacing inclinations 7 with apparent inclinations J for a collection of analytic expressions see Telford et al 1990 In this paper calculations of the wavelet coefficients will be done by taking derivatives from equations 5 and 7 It can be shown that the modulus of the real and imaginary parts exhibit extrema controlled by the derivative order y and the mean apparent inclina tion I I 2 while the modulus of the complex coefficients exhibit extrema whose geometry is indepen dent of the mean apparent inclinations Appendix B A special formulation for extended sources and the corre sponding complex wavelet coefficients is possible using the complex variables method it is introduced in Ap pendix C for its potential applications in the numerical approach to the direct and inverse problems We have calculated analytical expressions for some typical simple 2 D bodies below For local sources at depth zo exact power laws of a zo are shown a is the dilation or continuation altitude for extended sources Taylor ex pansions are calculated to analyze the perturbation due to a finite extent sin I 3 2 Local Elementary Magnetization As a first synthetic case let us consider the mag netic total field anomaly produced at level
17. z positive downward by a local elementary magnetization vector located at xo zo For this elementary magnetization Vertical Plane of the Profile z Geomagnetic North F Apparent inchnation downward z Figure 4 Apparent inclination J versus inclination J The unit vector f respective fa defining the direction of the magnetization M respective inclination J respective 74 when projected onto of the magnetic field F gives the apparent the vertical plane of the profile 19 460 SAILHAC ET AL WAVELET TRANSFORMS OF MAGNETIC PROFILES W Ww 5 5 4 4 3 3 2 2 1 1 o i 3 sf fo 4 2f X 3 12 1 o o 1 1 15 10 5 0 5 10 15 10 5 o 5 10 x x Figure 5 Local elementary magnetizations at depth zo 1 and a z 1 with apparent inclinations J 90 at z 10 and J 29 16 at z 5 bottom Total magnetic field and left modulus of real wavelet coefficients or right complex wavelet coefficients Isovalues are shown in white with regular intervals Modulus maxima lines are shown in black with their downward continuation to the source which is concentrated on an infinite horizontal line the Green s function is V x z ln z zo zo z so 10 holds with z z0 zo z 2 20 zo z z 20 o 2 2 20 zo 2 67 a z 2 6To z 4 11
18. z3 and z 4 2 3 The dip angle is 7 2 8 the height is h z2 z1 and the horizontal length is l z 24 As 21 21 2 z2 define the right edge and z3 z3 x4 z4 define the left edge Figure 9 analytic expres sions for the wavelet coefficients are given by the differ ence between wavelet coefficients of two step sources Using variables Xo x 2 41 2 Xoo t 2241 2 Z 241 2 21 4 and Z3 z3 z z2 a prefactor K sin sin I and angle 2I 6 one gets the wavelet coefficients for the inclined prism to be compared with equation 23 Wy2jsr 2 0 2 a Kacos be agra BARAA B 33 For small length for l lt lt h this corresponds to the horizontal derivative of the inclined step so that we recover the strip Thus one expects the same kind of results as those obtained for the strip As shown on Figure 9 properties of the wavelet coefficients such as the homogeneity degree are the same as for the strip These are properties corresponding to the asymptotic expansions for large dilations a Nevertheless for small dilations for z9 a 1 2 modulus extrema lines and isoarguments exhibit branching they point toward two top edges step sources instead of one single edge ob tained from the strip source For clarity exact analytical expressions of complex wavelet coefficients on modulus maximum lines are given only for the vertical prism at z zo
19. 1997 A total dis tance of 135 000 km has been flown at a low speed 250 km h and a low flight level 120 m clearance The interline distance was 0 25 0 5 or 1 km 500 m in the area of interest to us Using differential GPS the res olution of data after processing is better than 10 m sampling interval is 7 m along profiles constant time sampling interval of 0 1 s In French Guiana the main magnetic field had inclination J 21 and declination SAILHAC ET AL WAVELET TRANSFORMS OF MAGNETIC PROFILES SIMPLIFIED GEOLOGY OF FRENCH GUIANA 3 ze ed Mesocenozoic Gabbro and dolerite dikes Acid Plutonisum Caraibe anodorte T Monzogranta and gantad Upper Detrital Formation Orapu Acid Plutonisum Guyanais Volcano Sedimentary Armina 19 469 N NE Aumai mesaroc sedments Faai tertiary Porphyroid gr Sandstone and quartzite Mono and polygenic congiomerste Grantosses and Megmavies Bisca sctust and mete graywacke Figure 11 Simplified geological map The boxed area between Cayenne and Kourou defines the area of interest to us shown in more detail on the right side D 16 5 and the lines were flown in a N30 E di rection which means an apparent inclination of 30 when striking anomaly structures of induced magnetiza tion perpendicularly The reference station for diurnal corrections was located in the Cayenne area The con fidence in the final magnetic data is bette
20. ETRE TES et ay s Oz i Oz A1 Now let us introduce a complex gravity field anomaly ge gr i g whose real part is given by the horizon tal gravity field anomaly dg OV 0z and whose imag inary part is given by the vertical gravity field anomaly g OV 0z where the z axis is downward oriented With this definition dg is actually a complex function of the complex variable x ia which is defined from gy with the Hilbert transform ge 1 H dgz Hence complex wavelet coefficients of V are y 1 Woriv a 4 a dgc z z a A2 SAILHAC ET AL WAVELET TRANSFORMS OF MAGNETIC PROFILES Similary the complex wavelet coefficients of g are ese aN Were lea a e E bele a A3 Then we consider the magnetic potential anomaly produced by a set of elementary magnetization vectors with declination D and inclination J within a normal field of declination D and inclination J Along pro files perpendicular to the sources and making an an gle p with geographic north Figure 4 a simplifica tion follows by introducing an apparent inclination J corresponding to a geomagnetic south north profile de Gery and Naudy 1957 Thus the magnetic potential anomaly U is given by an oblique derivative of V in direction I sin w aN e dgc x z a A4 Eventually we consider the magnetic total field anomaly Similar to the apparent inclination magneti zation direction one can introduce
21. JOURNAL OF GEOPHYSICAL RESEARCH VOL 105 NO B8 PAGES 19 455 19 475 AUGUST 10 2000 Identification of sources of potential fields with the continuous wavelet transform Complex wavelets and application to aeromagnetic profiles in French Guiana Pascal Sailhac and Armand Galdeano Laboratoire de G omagn tisme Institut de Physique du Globe de Paris Paris France Dominique Gibert and Fr d rique Moreau G osciences Rennes Universit de Rennes 1 Rennes France Claude Delor Bureau des Recherches G ologiques et Mini res Orl ans France Abstract A continuous wavelet technique has been recently introduced to analyze potential fields data First we summarize the theory which primarily consists of interpreting potential fields via the properties of the upward continued derivative field Using complex wavelets to analyze magnetic data gives an inverse scheme to find the depth and homogeneity degree of local homogeneous sources and the inclination of their magnetization vector This is analytically applied on several local and extended synthetic magnetic sources The application to other potential fields is also discussed Then profiles crossing dikes and faults are extracted from the recent high resolution aeromagnetic survey of French Guiana and analyzed using complex one dimensional wavelets Maps of estimated depth to sources and their magnetization inclination and homogeneity degree are proposed for a region between Cayenne
22. Thus using variables X z zo and Z z z zo a and prefactor K 4 sinJ sin 7 one gets the following wavelet coefficients Woyaler 2 0 4 Ka e X iZ 12 We have computed these wavelet coefficients for two orientations of magnetization in a south north profile at the pole J 90 where the anomaly is symmetri cal and in the SSW NNE profiles of the Guiana survey I 29 16 Figure 5 shows these wavelet coefficients Extrema lines of the real wavelet coefficients intersect at the point zo a zo inside the lower half plane There is only one modulus maxima line of the complex wavelet coefficients with equation z zo Wavelet coefficients for any derivative order y N read for z 20 Wy2 67 2 0 20 2 atel 2l 142 5 zo a 1 2 2 7 1 4 13 When the depth zo is known modulus W and phase along a modulus maxima line simply give the values of the homogeneity degree a 2 and the apparent inclination I 2 y 2 4 modulo z When the depth zp is unknown adjusting a straight line to the plots of log W7 a7 versus log zo a for a set of a priori depths zp and looking for the best least squares fit provides both zp and a Figure 6 3 3 Local Multipole Magnetization As a second synthetic case let us consider a multi pole magnetization vector located at zo zo which is formally an oblique derivative of the local elementary magnet
23. a of Wyx 57 z respectively Wy 57 2 Thus their geometry is controlled by the derivative order y and the mean apparent inclination S F I 2 Nevertheless when considering the modulus of the complex wavelets coefficients it is obvious from equation A7 that the derivative order y but not 9 controls the geometry of modulus maxima lines Appendix C Complex Variables Method and Numerical Applications A general formulation for the direct calculation of the complex wavelet coefficient due to a 2 D body of any shape can be obtained with the help of complex variables w X 7Z where X z zo and Z zo a are the horizontal and vertical coordinates respectively from the source at coordinates zo and zo as used in section 3 This method has been previously applied to the calculation of derivatives of any order from the vertical gravity component Kwok 1989 This allows us to write the coefficients at x a of the complex wavelets y for the total magnetic field anomaly due to the 2 D body of contour zo zo S Wo sr ak a L i2I dro dzg 4 a4 TI a e If c CENENE s n ai For 7 and any derivative order y R this becomes a dito dzy _ x0 i zo a 7 ka ei2l I a As shown by Kwok 1989 because the integrand is analytic the double integral above can be simplified using the complex variable w xz zo i zo a The complex form of Green s theorem gives integrations ove
24. a dilation factor a Note that a SAILHAC ET AL WAVELET TRANSFORMS OF MAGNETIC PROFILES calculation based upon 10 as done in sections 3 1 3 4 would imply the same results Thus using variables X z z1 X2 z 22 Zi z z z a and Z z2 Z 244 prefactor K 4 sin I sin 7 angle 27 0 and the height h z2 z1 the wavelet coefficients read Wy 6T 2 0 2 0 Kacos e lace gir 29 The shape of modulus maximum lines of these wavelet coefficients not shown is similar to that of the step source Figure 7 With real wavelets they converge either to the top of the strip or to its mean depth zo While analytical formulation of modulus maxima of complex wavelet coefficients for the inclined step are so lutions of a second order polynomialin Z z zo equa tions 25 and 26 the strip case implies a fifth order polynomial which does not simplify easily Neverthe less graphical solutions for modulus maxima plotted as zeros of x derivatives of complex wavelet moduli are accessible showing asymptotic behavior similar to that obtained for the step source Figure 7 In the case of a vertical strip analytical formulations are possible Complex wavelet coefficients on modulus maxima lines for 29 obey Ww3 6T z 0 0 2 gn DTE ao z1 a z2 a Calling h the height of the strip and zo its mean depth we obtain the following Taylor expansion for zo a
25. agnetic data Geophysics 47 31 37 1982 Vanderhaeghe O P Ledru D Thi blemont E Egal A Cocherie M Tegyey and J P Mil si Contrasting mech anism of crustal growth Geodynamic evolution of Pa leoproterozoic granite greenstone belts of French Guiana Precambrian Res 92 165 193 1998 C Delor Bureau des Recherches G ologiques et Mini res La Source Orl ans France A Galdeano and P Sailhac Labora toire de G omagn tisme UMR 7577 du CNRS Institut de Physique du Globe de Paris 4 place Jussieu 75252 Paris France galdeano ipgp jussieu fr sailhac ipgp jussieu fr D Gibert and F Moreau G osciences Rennes UPR 4661 du CNRS Universit de Rennes 1 Bat 15 Beaulieu 35042 Rennes Cedex France gibert univ rennes1 fr moreauQuniv rennes fr Received March 17 1999 revised January 3 2000 accepted March 15 2000
26. ass of wavelets 7 Figure 1 for which the wavelet coefficients of a potential field due to local homogeneous sources exhibit simple prop erties For the potential field z z 0 measured at level z 0 which is due to a local homogeneous source located at x 0 and depth z zo the wavelet coeffi cients in the upper half plane of positions and dilations a z gt 0 obey a double scaling law with two expo nent parameters Worle 2 0 2 ayi fad tz a z lt zo W714 z 0 2 at zp a 3 In 3 z and a are the position and the dilation re spectively for the left hand side wavelet coefficient x a zo a zo and a are the position and the di lation respectively for the right hand side wavelet co efficient This defines a set of lines x a which satisfy z zo a const For various constants we obtain a family of lines that intersect at the point 0 zo inside the lower half plane therefore the wavelet transform ex hibits a cone like structure where the top of the cone is shifted to the location of the source Using the modulus maxima lines on which the signal to noise ratio is the best this constitutes a geometrical procedure to obtain the location of a homogeneous local source with no a pri ori idea of its homogeneity degree see Figure 2 The first exponent y is the order of the wavelet 7 which has been used the second exponent is associated with the homogeneity degree of
27. ation Therefore this vertical dike is probably of large extent with a top at 250 m depth and a magnetization with apparent inclination of 15 modulo 180 5 Conclusion and Perspectives The wavelet technique is aimed at analyzing both the geometry and locations of the sources of potential fields via relations of derivatives of the upward contin ued field This can be applied to any potential fields and components A geometrical interpretation of the scaling over the cone like structure of the real wavelet coefficients induced by the source can be used for mod eling An automatic depth to source determination is based upon adjustments of a scaling law for the mod ulus of complex wavelet coefficients of Wa a versus a zo This can be applied for some nonlocal sources such as prismatic bodies where residuals of a simple power scaling law form an estimate of the extent of the source this has been recently succesfully applied SAILHAC ET AL WAVELET TRANSFORMS OF MAGNETIC PROFILES to the interpretation of a gravity profile crossing the Himalayas Martelet et al 2000 The application to high resolution aeromagnetic pro files of French Guiana has shown the utility of the technique for the interpretation of magnetic structures There is no need to reduce to the pole and the phase of complex wavelet coefficients indicates the inclination of the magnetization This is helpful especially in ar eas where geophysical prospecting
28. cance of estimated depths is not the middle of the object but for the top of the dike as in the classical analytic signal interpretation method 4 5 Inclination From the Phase The plots of the phase of the complex coefficients ver sus scale exhibit a different behavior for the dikes and the fault see Figure 14 A constant phase of 120 is observed for the two vertical dikes A moving value from 150 to 120 is observed for the inclined fault These differences are due to the dependence of the phase on the dip angle of the source that is significant for small scales and disappears for large scales when the source has a finite extent In this case the limit for large scales can be used to obtain the inclination equation 43 The value of 120 gives an apparent inclination of about 15 modulo 180 for the vertical dike and the fault whose homogeneity degree is a 2 0 This im plies that values of inclination J and declination D of the source are similar to the inclination J and decli nation D of the normal field for which 30 in the case that the profile is perpendicular to the source For the vertical dike whose homogeneity degree is a 1 0 one obtains the same value for the apparent inclination by assuming this is a dike of infinite extent while one obtains a value of 30 modulo 180 for a finite step However it is unlikely that the Cayenne Se ries has a reversed remanent magnetiz
29. champ 4 partir des mesures de l intensit Ann G ophys 26 229 258 1970 Martelet G P Sailhac F Moreau and M Diament Char acterization of geological boundaries using 1 D wavelet transform on gravity data Theory and application to the Himalayas Geophysics in press 2000 Mil si J P E Egal P Ledru Y Vernhet D Thi blemont A Cocherie M Tegyey B Martel Jantin and P Lagny Northern French Guiana ore deposits in their geological setting in French with English extended abstract Chron Rech Min 518 5 58 1995 Moreau F M thodes de traitement de donn es g ophysiques par transform e en ondelettes th se de doctorat 177 pp Univ de Rennes I Rennes France 1995 Moreau F D Gibert M Holschneider and G Saracco Wavelet analysis of potential fields Inverse Probl 13 165 178 1997 Moreau F D Gibert M Holschneider and G Saracco Identification of sources of potential fields with the contin uous wavelet transform Basic theory J Geophys Res 104 5003 5013 1999 Nabighian M N The analytic signal of two dimensional magnetic bodies with polygonal cross section Its prop erties and use for automated interpretation Geophysics 37 507 517 1972 Nabighian M N Additional comments on the analytic sig nal of two dimensional magnetic bodies with polygonal cross section Geophysics 39 85 92 1974 Paul M K S Datta and B Banerjee Direct interpre tation of two dimensi
30. ed to obtain the height of the source equations 32 and 36 and Figure 10 This can also be used as a measure of the uncertainty in the estimated location of sources The phase along modulus maxima lines at large dila tions tends to a limiting value which is not dependent on a possible dip angle of these extended sources with finite height Syrer 2l ke3 modulo 27 38 where kg corresponds to the nature of the source and the order y of the wavelet kg y 2 for a line a strip or a prism and kg y 1 for a right step and kg y 1 fora left step More precisely kg is linked to the homogeneity degree In some cases this is equal to y a and this corresponds to a multipole mag netization whose sum of angles of successive derivatives is a multiple of 2r or O 0 modulo 2r in equation 15 For other extended sources having a very large ex tent h oo Taylor expanding for zo a gt gt h is not possible It is nevertheless possible to expand wavelet coefficients for z a gt gt z1 this shows that the depth as previously estimated is the top depth z instead of the average zo and both the homogeneity degree and limit phase do not obey the above laws Let us con sider the infinite strip which is a typical model used in the classical analytic signal method The expression for its wavelet coefficients is given by 29 whose first term with z2 oo equals 0 it is governed by the second ord
31. er term with z1 for any dilation a This implies that the homogeneity degree a is increased by 1 1 in stead of 2 for a finite strip and the phase is decreased by the dip angle 2 0 7 instead of 2 37 2 We have shown analytical results for the total mag netic field anomaly as due to a 2 D body of homoge neous magnetization with normal apparent inclination I These can also be used on other kinds of poten tial field data To put them into analytical results that would be obtained for a magnetic potential anomaly U a vertical gravity field anomaly gz or a gravity potential anomaly dV one can use duality equations between wavelet coefficients as shown in Appendix A 2 Wallet z 2 2 39 22L ei 21 Woatrjav 2 a 4 SAILHAC ET AL WAVELET TRANSFORMS OF MAGNETIC PROFILES so that y 2 order wavelet coefficients of 5V have the same properties as y order wavelet coefficients of 6T for inclination I 1 2 Similarly Wy2sr 2 4 1 T a0 Fear Wy 3 189 2 a 40 so that y 1 order wavelet coefficients of 6g have the same properties as y order wavelet coefficients of T for inclination I 71 4 Also Wy su 2 2 4 aT eT g a Wo2 59 z 2 0 so that y order wavelet coefficients of 6g have the same properties as y order wavelet coefficients of SU for in clination 7 1 2 4 Aeromagnetic Data 4 1 Survey of French Guiana and Geological Se
32. er term is that of a local source located at depth zo its homogeneity degree given by the scal ing exponent 2 is a 1 that of a step source whose structural index is known to be N a 1 0 The second order term is a pertur bation of 100 x h 2 zo a this is another local source located at depth zo but with homogeneity degree a 3 dipole of magnetization Thus 17 to 20 do not only provide the mean depth zo at the convergence of modulus extrema lines of real wavelet coefficients the apparent inclination J from the phase of the complex wavelet coefficients and the characteristic homogeneity degree a 1 using the slope in the plot of log Wa a versus log a zo along modulus extrema lines but they also provide a way to estimate the height h of the step from residuals in the determination of 19 463 Indeed moduli of 20 may be first approximated in a form valid for very large dilations Wya z in Dealeri 2 0 0 a k b ln zo a 21 Fitting a straight line to this approximation provides the slope 8 a 1 and the log factor k In Kh where K is the prefactor used in 17 which includes information on J J and the intensity of magnetization These then can be used to plot the following function H a which converges rapidly to a limit which is the height h of the step see Figure 8 H a 2 z a ie In Wy rier aanle k Pln zo a
33. ermany 1975 Barbosa V C F J B C Silva and W E Medeiros Stability analysis and improvement of structural index estimation in Euler deconvolution Geophysics 64 48 60 1999 Bhattacharyya B K Design of spatial filters and their application to high resolution aeromagnetic data Geo physics 37 68 91 1972 Biegert E K and P S Millegan Beyond recon The new world of gravity and magnetics Leading Edge 17 41 42 1998 Blakely R J Potential Theory in Gravity and Magnetic Ap plications 441 pp Cambridge Univ Press New York 1996 Deckart K Etude du magmatisme associ au rifting de PAtlantique Central et Sud G ochronologie Ar Ar et g ochimie des intrusions Jurassiques de Guin e et de Guyane fran aise Surinam et Cr tac au Br sil th se de doctorat 250 pp Univ de Nice Nice France 1995 Deckart K G F raud and H Bertrand Age of Juras sic continental tholeiites of French Guiana Surinam and Guinea Implications for the initial opening of the central Atlantic Ocean Earth Planet Sci Lett 150 205 220 1997 de Gery J C and H Naudy Sur l interpr tation des anomalies gravim triques et magn tiques Geophys Prospect 5 421 448 1957 Delor C J Perrin and C Truffert Campagne de g o physique a roport e en Guyane frangaise Magn tisme et radiom trie spectrale Rap 39625 102 pp Bur de Rech G ol Min Orl ans France 1997 Fedi M and T Quarta Wavel
34. et analysis for the regional residual and local separation of potential field anomalies Geophys Prospect 46 507 525 1998 Galdeano A Traitement des donn es a romagn tiques M thodes et applications th se de doctorat 153 pp Univ de Paris VI Paris 1974 Gibert D and A Galdeano A computer program to per form the upward continuation of potential field data Comput Geosci 11 533 588 1985 Grauch V J S and P S Millegan Mapping intrabasinal faults from high resolution aeromagnetic data Leading Edge 17 53 55 1998 Gunn P J Ed Airborne magnetic and radiometric sur veys J Aust Geol Geophys 17 216 pp 1997 Holschneider M Wavelets An Analysts Tool 423 pp Clarendon Oxford England 1995 Hornby P F Boschetti and F G Horovitz Analysis of potential field data in the wavelet domain Geophys J Int 137 175 196 1999 Hsu S K D Coppens and C T Shyu Depth to magnetic source using the generalized analytic signal Geophysics 63 1947 1957 1998 Huang D Enhancement of automatic interpretation tech niques for recognising potential field sources Ph D thesis 216 pp Univ of Leeds Leeds England 1996 Kwok Y K Conjugate complex variables method for the computation of gravity anomalies Geophysics 54 1629 1637 1989 SAILHAC ET AL WAVELET TRANSFORMS OF MAGNETIC PROFILES Le Mou l J L Le lev a romagn tique de la France Cal cul des composantes du
35. f order y whose first derivative is in z pl z complex wavelet of order y wz x wavelet of order y dilated by dila tion a Wy wavelet coefficient of o by convolu Wy7 6 2 Dy7 9 2 2 4 Sint 1 a sint y i 21 435 Woater 2 0 4 Tw 5 ry ell 21 3 n X pracena j 1 e7i 2 8 w541 tion of o x with Y z for x R wavelet coefficient of z by con volution of z z with Y7 z for z ER argument or phase of the wavelet coefficient Wy z 2 T x Gamma function of the real variable gz for n EN T n 1 n w conjugate of the complex variable w I D inclination and declination of the magnetization vector In Dn inclination and declination of the normal field I apparent inclination of the magneti zation vector I apparent inclination of the normal field S mean apparent inclination equal to I 1 2 oe SAILHAC ET AL WAVELET TRANSFORMS OF MAGNETIC PROFILES Acknowledgments This paper benefited from com ments made by Associate Editor Kathy Whaler and two reviewers Alan Reid and Dhananjay Ravat We also had nu merous discussions with our colleagues Matthias Holschnei der Guillaume Martelet and Ginette Saracco This is IPGP contribution number NS 1680 References Baranov W Potentzal Fields and Their Transformations in Applied Geophysics Geoexplor Monogr Ser vol 6 121 pp Gebruder Borntraeger Stuttgart G
36. gt gt h 2 large dilations or large average depth Wyi T z 0 0 s 2 i 21 4 35 4 BE ahi CHD ohy tE 31 sin J zo a gt This is similar to 20 except for a factor 2 and for the powers of z a The same estimation technique as that shown for the vertical step applies Equations 21 and 22 now apply with a y 3 Here the first order term is that of an elementary local magne tization of homogeneity a 2 which characterizes how the strip transforms into a horizontal line for h small Similarities between the vertical step and strip cases also apply for both inclined cases in Taylor expansions along modulus maxima lines so that a function H a similar to that defined in 22 and 28 which converges for almost all dipping angles to a limit which is the height h exists H a 2 zo a f 9 1 Wy rer 2 0 70 a In 32 k b ln zo a 19 465 where k In Kh and f 0 2 1 tan 6 Along modulus maximum lines phases for large dilation a con verge to 2 37 2 which is the value for an elemen tary local magnetization again this characterizes how the strip transforms into a horizontal line for h small 3 6 Magnetization Prism As a final synthetic case let us consider the total field anomaly generated by an inclined prism whose limits are the four points 21 21 Z2 22 23 2a and 4 24 clockwise with z1 z4 z2
37. h see Figure 10 Along modulus maxima line phases for large dilation a converge to 21 37 2 as for the strip Note that f reaches infinity for 6 0 with bodies having almost the same horizontal and vertical extents for which higher order Taylor expansion is necessary For d h I lt lt h44 1 zo a this implies the fourth power of h and I Wy3 sT 2 0 20 4 1X2 i 2I 35 1 4 a7 alhe i 2 lake en s Ber zo a 7 37 3 7 Summary and Discussion Analysis of magnetic potential fields anomalies using the continuous wavelet transform can be done without the need to reduce the data to the pole or to the equa tor As shown by the above results summarized below the interpretation is made via wavelet coefficients of the anomaly field itself Geometrical analysis of modulus maximum lines of either the real or imaginary parts of complex wavelet coefficients gives a convergence point at the location of a local source horizontal line When sources are ex tended step strip and prism and have uniform mag netization there is convergence at the center of the ob ject for modulus maxima lines at large dilations while the convergence is near the upper borders of the object at smaller dilations when the source is not too deep For extended sources this also depends on the choice of the derivative order Moreau et al 1999 The modulus maximum line of complex wavelet co efficients is a vert
38. ho mogeneous sources that have been estimated by pick ing the two dikes and the fault on all profiles these values have been corrected for flight altitude and are almost all positive a very few have very small values of lt 50 m corresponding to hills having strongly vari able ground level The fault right side on Figure 13 nas a deeper magnetic response 200 m down to 1 km than the two dikes lt 400 m in the central profile one gets the following depths zo from the SSW to the NNE 370 m SSW dike 250 m middle dike and 930 m the fault Note that horizontal positions follow what would have been obtained by classical methods based on the locations of the extrema of the analytic signal or the gradient of the field reduced to the pole This method gives realistic depth estimates with regard to the geological nature of the sources Other computa tions using the Euler deconvolution method not shown have given lower depth estimates with negative values when corrected for flight altitude Satlhac et al 1997 This traduces the problem of noise effect in Euler de convolution which is automatically solved by upward continuation in the wavelet method 19 470 SAILHAC ET AL WAVELET TRANSFORMS OF MAGNETIC PROFILES N nN g kad A a N Q 3 S oO o o to Dilation elevation of the upward continuation in m N o D o Inn 100 Signal 70000 60000 30000 26000 Position from sea coast in m Figure 12
39. ical through the location of a local source such as a horizontal line and the modulus along this maximum line allows one to find the depth and homogeneity degree by linear regression of bilog arithmic plots classical least squares can be used When sources are extended step strip and prism and have uniform magnetization the modulus maxima at large dilations form a vertical line pointing to the cen ter of the source at small dilations the modulus maxima do not form vertical lines except for a vertical step or strip and possibly at vertical borders of a prism but rather curves pointing near to the upper borders of the object with an angle linked to the dip angle For these extended sources with finite height h a 19 468 generic law extracted from a multipole expansion gives an estimate of the extent of the source The first order expansion is associated with linear fits in the bilogarith mic plot corresponding to an equivalent local source this depends on three parameters the depth zo the slope linked to the homogeneity degree and a third term k linked to the intensity The second order term implies a transformation of residuals from the previous linear fits in bilogarithmic plots and provides an alter native means of regression which is deterministic equa tion 28 This gives a function H a z0 8 k which should become a constant for large a for the best a pri ori model zo 8 k A correction factor must be appli
40. ies of the one dimensional 1 D complex wavelet coefficients of the total field mag netic anomaly Apparent inclination of magnetization in addition to depth vertical extent and dip angle of sources can be estimated This wavelet technique ap plied to magnetic studies is not only a filtering as used in recent advances in aeromagnetic processing Fedi and Quarta 1998 Ridsdill Smith and Dentith 1999 but ac tually gives the derivatives and analytic signal of the upward continued anomaly field This is more like the continuous wavelet analysis developed for the location of singular features of the source distribution Hornby et al 1999 which is improved when scaling relations of the wavelet coefficients are analyzed and when vertical derivatives are used in addition to the horizontal deriva tives This complementary development to existing up ward continuation techniques Paul et al 1966 and to recent advances in the interpretation of the gradi ents of potential fields Pedersen and Rasmussen 1990 Pilkington 1997 Hsu et al 1998 provides a theoreti cal framework to enlighten properties of the sources via their scaling character 19 455 19 456 First we have applied this technique to several lo cal and extended synthetic magnetic sources Local ele mentary magnetization dipole local multipole magne tization vertical and inclined steps strips and prisms of magnetization Wavelet coefficients have been calcu
41. ization vector previously analyzed Asymptotic expansions of the far field due to extended sources im ply the sums of multipoles For instance an elementary pole plus a dipole mass can be used as a model for an inclined gravimetric border Moreau 1995 Let us call 6 the direction of the oblique derivative corresponding to the dipole magnetization vector Then 62 is the di rection of the second oblique derivative corresponding to the quadrupole magnetization vector and 6 is the direction of the nth oblique derivative corresponding to the nth multipole magnetization vector We assume that the structure of this multipole source is still infinite in the horizontal direction perpendicular to the profile so that angles have to be considered in the vertical plane of the profile apparent angles 6 are equal to 0 Then for this nth multipole magnetization n additional oblique derivatives in directions 0 for 1 lt j lt n have to be performed in 10 12 and 13 SAILHAC ET AL WAVELET TRANSFORMS OF MAGNETIC PROFILES 20 slope B a y A o N wo A n c a 2 3 a pnori depth Zo 19 461 Best B 2 985 and 2 0 988 fog W a 1 log a z Figure 6 From the wavelet coefficients of Figure 5 right computation of the best slope 8 and depth zo is accomplished by least squares linear regressions of log W a7 versus log a zo left A set of a priori depths 0 1 5 has been tested this gives good estimates f
42. ler deconvolu tion the structural index N must be assumed a priori except when applied in conjunction with other proce dures Huang 1996 Ravat and Taylor 1998 Barbosa et al 1999 Using wavelets the homogeneity degree of the source a can be determined from the scaling ex ponent without a priori value Hereafter we give the exponent of the 1 D wavelet coefficients as a function of a We also give the homogeneity degree of the as sociated magnetic potential N in different situations depending on the type of anomaly field or potential which is analyzed V gravity potential or Green s function Py y a 2 y N 1 g VV gravity field or U VV M magnetic po tential due to the dipole M By Bu y ta 1 y N T VU magnetic field or 0 9 vertical derivative of the gravity field Pr y ta y N 1 Classically one defines N from the potential field but this gives a value which depends on the analyzed data SAILHAC ET AL WAVELET TRANSFORMS OF MAGNETIC PROFILES and this involves a shift of 1 between the gravity and magnetic cases Instead we consider its definition given by Huang 1996 this is more reliable as it gives a value which depends only on the geometry of the source N is the opposite to the homogeneity degree of the magnetic potential and is also that of the corresponding gravity field 3 Synthetic Examples 3 1 Total Magnetic Field Anomaly in Profiles We co
43. n ni m amp 10 dilation apparent depth a zo in m Figure 14 Characterization of the sources fr Plot for the scalling of Wavelet Coefficients Phase Complex order 1 Phasis Arg W in degrees S o 8 8 8 8 150 200 400 100 1200 600 800 10 1400 dilation a in m om the use of 1 D complex wavelets top The modulus maxima lines and the phase along these lines obey bottom scaling relations character izing the two dikes dotted and dashed and the fault plain bottom left homogeneity degree is given by the plot of log Wa a versus log a zo and bottom right inclination is given by the limit of Arg Wa for large a 4 4 Homogeneity Degree From the Modulus Using depth estimated from the geometrical proce dure on modulus maxima lines of the real wavelet coeffi cients section 4 3 we have estimated the homogeneity degrees from the modulus maxima lines of the complex coefficients calculated on the central profile of Figure 12 As shown on Figure 14 modulus maxima lines are both vertical for the two dikes and display an angle 6 50 for the fault this gives a rough estimate of its angle dipping NNE Using the depths zo es timated from real wavelets the linear regression of log W a a versus log a zo implies a homogeneity degree of a 1 9 the slope is 8 2 9 and the struc tural index is N 0 9 for the SSW dike and the fault and of a 0 9 the slope is 8 1
44. nd to 2I 7 2 for the magnetic total field anomaly Figure 3 Equation 3 shows that the isovalues of the phase of the complex wavelet coefficients also draw a cone like structure pointing to the source Figure 3 2 3 Comparison With Classical Techniques The complex wavelet coefficients Wyr 1 4_ 2 4 are associated with the upward continued analytic sig nal as early introduced to the interpretation of geophys ical potential fields Nabighian 1972 1974 2 w rreta 2 a Jart Etele Erdale za 8 In the classical use of the analytic signal one consid ers the modulus of the analytic signal but the inter pretation of its phase is missing Recent improvements are due to the interpretation of the phase Smith et al 1998 Within the theory of continuous wavelet trans forms Holschneider 1995 the use of both the modulus and phase and both the real and imaginary parts of the upward continued analytic signal at different levels complex wavelet coefficients for different dilations is natural and allows interesting properties regarding the geometry of the singularities to be taken into account Besides the exponent defined in 3 depends on the derivative order y and the homogeneity degree of the source Thus it relates to the structural index N used in Euler deconvolution which is also based upon an ho mogeneity property that of the field Thompson 1982 Reid et al 1990 Huang 1996 Using Eu
45. nsider the magnetic total field anomaly pro duced by aset of elementary magnetization vectors with declination D and inclination J within a normal field of declination Dn and inclination J To clarify the ana lytic expressions in the case of profiles we use two sim plifications First when profiles are striking geographic north with an angle y and perpendicular to the sources Figure 4 one can introduce an apparent inclination 7 and an apparent normal field inclination J correspond ing to a geomagnetic south north profile de Gery and Naudy 1957 tan I cos D tan I cos Dn 9 Second one can consider magnetization vectors with declination D and inclination J equal to those of the normal field D and In respectively as if there were only induced magnetization In this case apparent in clinations are equal F Hence the total magnetic field anomaly T at x z is given by two conjugated symmetrical and antisym metrical functions 67 and dT associated with second order derivatives of the Green s function V see detailed expressions in Appendix A 1 tan l tan J 9 Horizontal Plane Vertical Plane of the Magnetic Meridlan Geographic Geomagnetic North North Magnetic Meridian Magnetic Mendian x z 19 459 sin I 2 T 6T cos27 T sin27 10 T is equal to the anomaly field reduced to the pole J 90 and T is equal
46. onal structural faults from gravity data Geophysics 31 940 948 1966 Pedersen L B and T M Rasmussen The gradient tensor of potential fields anomalies Some implications on data collection and data processing of maps Geophysics 55 1558 1566 1990 Pilkington M 3 D magnetic imaging using conjugate gra dients Geophysics 62 1132 1142 1997 Ravat D and P T Taylor Determination of depths to centroids of three dimensional sources of potential field 19 475 anomalies with examples from environmental and geolog ical applications J Appl Geophys 39 191 208 1998 Reid A B J M Allsop H Grasner A J Millet and I W Somerton Magnetic interpretation in three dimensions us ing Euler deconvolution Geophysics 55 80 91 1990 Ridsdill Smith T A and M C Dentith The wavelet trans form in aeromagnetic processing Geophysics 64 1003 1013 1999 Sailhac P A Galdeano D Gibert and F Moreau Wavelet Based Method for the Interpretation of Aeromagnetic Sur vey in French Guyana Eos Trans AGU 78 46 Fall Meet Suppl F35 1997 Smith R S J B Thurston T F Dai and I N MacLeod iSPI The improved source parameter imaging method Geophys Prospect 46 141 151 1998 Telford W M L P Geldart and R E Sheriff Magnetic methods in Applied Geophysics pp 62 135 Cambridge Univ Press New York 1990 Thompson D T EULDPH A new technique for making computer assisted depth estimates from m
47. or both homogeneity degree a p y 1 985 and depth zo 0 988 Thus using variables X z zo and Z zo 2 zo a prefactor K 12 sin sin I and angle 2I 6 one gets the following wavelet coefficients for the dipole magnetization source in direction 6 Wy3 671 2 0 a Ka e f X4iZ 14 We have also calculated the wavelet coefficients for any derivative order y N and any multipole degree n N which reads for x zo Wy26Tn 2 0 Zo a i j avei 2l On 1 n 2 5 snI 2 y n 1 zo a vtn 2 sin I 15 where On X characterizes the combination of directions in this multipole magnetization Again there is only one modulus maximum line of the complex wavelet coefficients with equation z zo The modulus W and phase along this maximum line simply give the values of the homogeneity degree a n 2 and either the apparent inclination J O 2 y a r 4 modulo r or the sum of directions O 21 y a x 2 modulo 2r 3 4 Magnetization Step As a first example of a nonlocal source let us consider the total field anomaly generated by a vertical step lo cated at xo and depths z1 z2 with height A z2 z1 of elementary magnetization with normal apparent in clination J source on the north for z gt zo Equation 10 holds with 6T1 x z 2 an BOZ a a z To Xo x zo
48. or small dilations in the lower half space This is similar to 31 except for prefactor l and addi H a similar to that defined in 32 which converges tional term in l The first is linked to the change in the for almost all dip angles 1 2 0 to a limit which is source density per volume instead of per surface which the height h of the prism is hidden for the calculations the second is due to the change in source geometry It is difficult to interpret the H a 2 zo a f 8 i prefactors except in cases of well known a priori source Wy r7 2 lt 0 20 4 I density where conventional techniques apply Never Ss ae theless it is worth using a multiscale technique based upon the multipole expansion to obtain estimates for where k log 2Klh 8 3 and f l h 2 1 depth and source extensions There exists a function tan 6 I h gt is a factor which depends on the Nie k B in zo a j 36 SAILHAC ET AL Vh 0 inclined strip WAVELET TRANSFORMS OF MAGNETIC PROFILES 19 467 8 Vh 2 1 tane I h gt inclined prism 4 3 2 1 0 0 2 1 tan e o wi L 12 x AL E3 4 0 05 0 vertical prism 1 15 2 Vh Figure 10 Multiplicative correction factor f 0 l h for strip and prism to be applied to the height estimator defined in equation 26 undefined in the dashed curve neighborhood ratio between l and h and equals 1 V2 for 0 0 and l lt lt
49. r the closed contour C in the complex plane x9 izo Iro dzo _ _ _ _ S J ro i zo a Yt Im f C1 Wy2er z 0 8 AV y 2 S44 dzo dz w dw wIt2 drodza _ w dw 2 Im J eie 20 i zo a 7 Re wt C3 Hence equation C2 reads Wy2zsr z 0 z a sin I 2 1 21 34 w dw r y 2 a7 a e write C4 Also the real and imaginary parts give the real wavelet coefficients Wyz x a and Wy7 z a respectively For numerical applications on closed n sided polyg onal 2 D bodies one may consider the case of integer 19 474 derivative order y N To the coordinates x z of each vertex j 1 n one associates the series of complex numbers w x xj i z a and Wn41 w1 This is used to transform equation C4 into Jwj 7 Jw C5 where w Arg w is the argument of w Notation f x function of a real variable z f u Fourier transform of f z equal to ee dz f a e7 74 H Hilbert transform HINE JP de Ey D dilation operator Def x 1 a f 2 a P x Poisson kernel upward continuation filter at level 1 equal to P x P x dilated Poisson kernel upward con tinuation filter at level a equal to DaP 2 i square root of 1 Oz 2 partial differential operator with re spect to x w2 x real wavelet of order y whose first derivative is in z wy x real wavelet o
50. r than 3 8 nT standard deviation of misfit between crosslines and lines over the whole survey before leveling 4 2 Profiles and Anomalies of Interest A set of 27 profiles each of 80 km length has been transformed using wavelets see central profile in Figure 12 We have analyzed the extrema lines for three sin gular features two dikes from the Permo Triassic and a sinistral strike slip fault of the Upper Detrital Forma tion The geology is revealed magnetically because of the high magnetization of gabbro and diorite of dikes contrasting with the low magnetization of Caraibe and Guyanais acid plunonic rocks The origin of the magnetic anomaly of the fault is in the high magnetiza tion of the Paramaca volcano sedimentary series Fig ure 11 4 3 Depth From the Real Part We first computed the real wavelet coefficients of each profiles using 2 Figure 12 shows typical features of the region obtained on the central profile Above the dikes and the fault sources the wavelet coefficients plotted as a function of altitude exhibit tracks forming cone like structures In wavelet theory these are typical cones for local singularities located at their top The dikes and the fault are local singularities characterized in the wavelet domain by these cone like structures Interpretation by geometrical continuation of the ex trema lines down to their intersections allows one to model the sources Figure 13 shows the depth of
51. tting The region of interest for the application of the wavelet transform technique is located between Cayenne and Kourou in French Guiana Figure 11 To introduce the geological setting let us first recall that this forms part of the Guiana Shield which is made up of an Archean complex and a widely developed Paleoprotero zoic succession including metamorphosed sedimentary and volcanic formations and granitic and medium to high grade metamorphic terrains and Mezoproterozoic formations at its northern and southern edges Mil si et al 1995 Vanderhaeghe et al 1998 The opening of the North Guiana Trough began with the formation in a sinistral strike slip setting of pull apart basins in which the Upper Detrital Formation was deposited af ter 2120 Ma Associated with more recent activity due to the extensive tectonics of the early opening of the Atlantic Ocean 200 Ma Permo Triassic magmatism has emplaced clusters of dolerite dikes striking NNW and NNE Deckart 1995 Deckart et al 1997 This ge ological setting is similar to that found in West Africa with large ore deposits As Guiana has been greatly un derexplored in comparison recent exploration interest led to a new survey An airborne survey including radiometric and aero magnetic data total field intensity has been carried out by CGG G oterrex Compagnie G n rale de G o physique for BRGM Bureau des Recherches G ologiques et Mini res in 1996 Delor et al
52. z 22 x zo z 21 Thus using variables X g zo Zi 2 1 2 z a and Z2 z2 z z2 a and prefactor K 2 sin I sin T one gets the following wavelet co efficients for the vertical step Ta x z In 16 pw 2s l aes ee Wo 1 67 2 0 2 Kae z4x Fy iX 17 Figure 7 shows these wavelet coefficients The shape of modulus maxima lines for real wavelets is not as sim ple as for a local source in section 3 2 For small dila tions a these are not straight but converge at about the top of the step source for dilations a which are large enough typically zo a gt gt h 2 these are straight lines and converge at about the mean depth of the step zo z2 21 2 as if the source was local The argument and modulus are Pyijsr 2 0 a 2 r tan oa tan a Kah ana C9 Wy2jsr z 0 2 Equations 18 show that there is only one modulus maximum line for which 0 Wy is zero defined by X 0 vertical to the step source at x o Thus when we consider modulus from neither the real nor imaginary part Wy nor Wy but rather from the complex coefficient itself Wy there is one single 19 462 SAILHAC ET AL WAVELET TRANSFORMS OF MAGNETIC PROFILES WJ a 8 j 7 i 4 6 j Vertical Step Step dipping 60 ee l 5 1 i f d 4 i oO f 1 3
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