Depth Estimation of 2-D Magnetic Anomalous Sources by Using Euler Deconvolution Method

Transcription

1 Aerican Journal of Applied Sciences 1 (3): , 2004 ISSN Science Publications, 2004 Depth Estiation of 2-D Magnetic Anoalous Sources by Using Euler Deconvolution Method 1,3 M.G. El Dawi, 1 Liu ianyou, 1 Shi Hui and 2 Luo Dapeng 1 Faculty of Geophysics and Space Inforation, China University of Geosciences, Wuhan, , China 2 Faculty of Electronic Engineering, China University of Geosciences, Wuhan, , China 3 School of Earth Sciences, Faculty of Science and echnology, Al Neelain University, Khartou, Sudan Abstract: Euler deconvolution ethod is the ethod of depth estiation which is best suited for anoalies caused by isolating and ultiple anoalous sources. In the present study, algorithic progras based on generalizing linear inverse theory, uses a least square sense to solve for Euler s equation is being written. he Euler technique that can estiate the location of a siple body fro easureents of the agnetic field could be applied to a long profile of easureents, by dividing the profile into the windows of consecutive easureents, each window providing a single estiate of depth and source location. Acceptable solutions for features of interest ay involve soe trial and error by adjusting the structural index and the window size. When all such easureents are plotted they tend to cluster around agnetization of geologic interest. Soe indication of the source type can be gained by varying the structural index for any particular feature. Shallow features can be deconvolved well by using sall window to reduce source interference. By using n = 1 and n = 1. 5, and deconvolved with sall window sizes, our progra yields good tight clustering and acceptable depths of the anoalous igneous body located along aeroagnetic profile in the south of Aenzi area, Inner Mongolia, North China. he depths obtained by using the structural index of n = 1. 5 varying between he ethod yielded useful solutions with an acceptable depth estiate. Key words: Euler Deconvolution, Structural Index, Window Size, Aenzi Area INRODUCION systes that the single-source algorith can only deal with approxiately. he advantages of this technique over ore conventional depth interpretation ethods (i.e. Characteristic curves, inverse curve atching), are that no particular geological odel is assued, and that the deconvolution can be directly applied and interpreted even when particular odel, such as a pris or dyke cannot properly represent the geology. he Euler deconvolution ethod is used for rapid interpretation of potential field (agnetic and gravity) data. It is particularly good at delineating contacts and rapid depth estiation. his technique belongs to autoatic depth estiate ethods and, is designed to provide coputer-assisted analysis of large volues of agnetic and gravity data. Euler s equation has been used by a nuber of authors for analyzing both agnetic anoalies [1, 2, 3] and gravity anoalies [4]. Euler s hoogeneity relationship offers a quasi-autoated ethod to derive the plan location and depth estiation of buried Ferro-etallic objects fro agnetic data. Euler s hoogeneity equation relates the agnetic field and its gradient coponents to the location of the source with the degree of hoogeneity expressed as a structured index, [1]. [1] (x x ) z = n (x) 0 0 Developed the technique and applied it to x z profile data. [3] Developed the technique ost widely used version for grid-based data. Also recent iproveents on the technique had occurred which included the estiation of the structural index [5]. [6] Developed a ultiple-source generalization of Euler x + z = x + n (x) 0 0 deconvolution, which is capable of handling coplex x z x 209 heory of euler equation: However, for applying the Euler s expression to profile or line-oriented data (2-D sources). In this case, the x-coordinate is a easure of the distance along the profile, and the y-coordinate can be set to zero along the entire profile: Rearrangeent of the above expression yields: (1) (2)

2 where (x 0, z 0 ) is the position of a 2-D agnetic source whose total field is detected on (x, z). he total field has a regional value of B, and n is a easure of the falloff rate of the agnetic field. N is directly related to the source shape and is referred to as the structural index [1]. By evaluating the total field and its derivatives (calculated or easured, he gradient can be calculated using standard potential theory in the space or wave nuber doain or the vertical gradient ay be calculated and can be used directly in the above equation) at all points on a agnetic data set, a syste of siultaneous linear equations is obtained with the unknowns being the x 0, z 0 represent the location and depth of the agnetic source, n, the structural index of the source and B the regional agnetic field. Using the derivatives of potential-field anoalies enhance the field associated with shallow features and de-ephasize the field fro deeper sources. Used appropriately, the ethod is suitable for characterizing sources of all potential-field data and/or their derivatives, as long as the data can be regarded atheatically as continuous [7]. he application of equation (2) directly to the observed data akes the exact solution unreliable and erratic. he proble of reoving the bias fro the observed data is solved in the following way by [1]. Assue the anoalous field is perturbed by a constant aount B in the window in which equation (2) is being evaluated. he observed quantity is: where, is the vector of unknown paraeters: X 0 = Z 0 G G And: G d,, x x x z =,, x z z z,x +,z + n, x x x z x =,x +,z + n, z x z z z (7) (8) (9) Steps of processing: he Following Flow Chart Shows the Input and Steps of Processing: (x) = (x) + B (3) where, B is a constant in the coordinate x over the portion of the profile where the analysis is being ade. Solving equation (3) for, substituting into equation (2) and arranging ters yield: x + z + NB = x + n 0 0 x z x (4) Equation (4) with three unknowns x 0, z 0, and B can be solved by using a least squares procedure. he proposed ethod involves setting an appropriate value of the structural index of the suspected source bodies and then solving the syste by least-squares inversion for an optiu x 0, z 0 and B. he inversion process also yields an uncertainty (standard deviation) for each of the fitted paraeters. hese uncertainties ay be used as criteria for acceptance or rejection of solutions. o solve for the source location x 0, z 0 the process norally solving the following least square noral equation associated with equation (4): G G = G d (5) 1 = G d(gg ) (6) least square technique. 210 Interpretation: Using MALAB software version 6.5 for prograing, a progra uses the generalized linear inverse ethod in a least square sense to solve the Euler s equation has been written. It will be shown that the general inverse can be constructed so as to get a better fit to the data (in a least square sense) becoe stabilized, so that it is less critical to choose a large nuber of odel paraeters than with the ordinary

3 o discuss our odel construction, fro the linear equation; G = d (10) M N N 1 M 1 Where: M = he nuber of observations N = he nuber of odel paraeters he least square solution to linear probles, objective function: he above is the theory of SVD of the physicist Lanczos [9], so any atrix G R M N can be factored as the equation (16), that is G can be factored as G = Ur V r Let : r G = V U (18) 1 L r r r = V U U U (19) 2 r r r r r r E (d G )(d G ) = (11) = G (U U ) (20) 2 1 r r r = dd Gd d G + G G (12) Let E = 0, then we have: d G + G G = 0 (13) 1 = (G G) G d (14) G L 1 = G (GG ) (21) Be the inverse of the atrix G, G L is called natural inverse by [9]. [10] was naed it as Lanczos inverse, and also as generalized inverse and is used based on the above equation for solving linear inversion probles. So the solution of the G = d is: = G d = V U d (22) 1 L r r r his is what is called the noral equation: = G d (15) L If M N = r, where r = rank of atrix G, an overdeterined syste, then soe constrains ust be added to effect ore inforation (M-N) fro the observation. his is what is called L. S solution. So that fro equation (6): 1 = G (GG ) d he above equation is called the generalized inversion. If G is M N atrix which is singular atrix. he atrix G can be factored as. By applying the atrix decoposition theory of [8] : G = U V (16) r r r he above equation is called Singular Value Decoposition (SVD) of singular atrix G, r is a diagonal atrix with r singular values on its ain diagonal and zero elsewhere: = diag( λ, λ,... λ ). r 1 2 r λ = λ 0 1 r λ r (17) 211 he generalized inversion based on the Lanczos natural inverse, the definition of natural inverse, G L d = r = N M, G = d is an over-deterined equation, in this case, the solution of the generalized inversion is: 1 G d (G G) G d L = = (23) his is the least square solution which provides the iniu length solution of linear systes. he provided algorith is based upon the above theory of the generalized inversion. he application of our algorith is based on the above steps of processing and was tested on soe typical agnetic structures, and on real field data exaples using an aeroagnetic data profile over Aenzi area, Inner Mongolia, North China, which targets the shallow igneous body source. Single anoalous sources: Figure 1-3 represent the total-field agnetic anoalies over single 2-D anoalous source bodies of known depths. By using the structural index of n=1, and varying the window size along the profile, the pattern of the clustering around agnetization of geologic interest derives source position and depth. Varying the window size along the profile will show roughly different degree of clustering and also different depths. For a sall window, the points give a good tight clustering on the top of the vertical and dipping dike bodies Fig. (1, 3) and for horizontal source body the two edges of the source body is indicated by tight clustering points (Fig. 2).

4 Fig. 1: (a) Shows the otal-field Anoaly (d) Over Siple Single Anoalous Vertical Dike; (b) he Depths Obtained by Decovolved with Different Window Size and by Using Structural Index n = 1 Model 1: Width = 4; Height = 40; op Depth: 10; Botto Depth: 50; Center Point Coordinate: (x0 = 50; z0 = 12) Fig. 3: (a) Shows the total-field Anoaly (d) Over Single Dipping Anoalous Source Body; (b): he Obtained Depths by Deconvolved with Different Window size and Using Structural Index n = 1 For a large window, the pattern is represented by only one point on the top of the vertical and dipping bodies and in the case of the horizontal body the pattern is represented by only one point in the center of the source body. In case of horizontal body the geological odel ay represent the situation where two sedientary units with different agnetic susceptibilities are juxtaposed which is referred to agnetized contact. Fig. 2: (a): Shows the otal-field Anoaly (d) Over Single Anoalous Horizontal Body; (b). he Depths Obtained by Deconvolved with Different Window Size and by Using n = 1 Model 2: Width = 40; Height = 4; op Depth: 10; Botto Depth: 14; Center Point Coordinate: (x0 = 50; z0 = 12) 212 Multiple anoalous sources: In 2-D ultiple anoalous sources' case of known depths, the deconvolution was run using different structural indices n = 0, 0.25, 0.5, and 1 and a nuber of window size along the profile. In case of n = 0, the progra gives poor biased depths, showing that n = 0 is a wrong index (Fig. 4a). Using n = indices, the progra gives poor clustering around anoalous bodies and biased depths (Fig. 4b). At n = 0. 5, the progra yields a good solution at the horizontal and dipping bodies, but at the vertical dike gives poor clustering around the geologic body (Fig. 4c). On n = 1, and deconvolution with different window sizes, our progra yields good clustering and acceptable depths of anoalous bodies varying between (Fig. 4d). A real field exaple fro aenzi area: Our algorith was applied to a real field exaple of aeroagnetic profile. he aeroagnetic profile has a length of 27 k located in the southern part of Aenzi area, Inner Mongolia, North China., with a north-south direction. he local anoaly in the south of the profile is interpreted as a shallow-buried igneous body of a depth less than 1000, which is corresponding to the positive local agnetic anoaly located in the south part of Aenzi area.

5 Fig. 5: Shows the otal Magnetic Field (d) and the Euler Response Over Field Exaple (Aenzi Area), Using Structural Index n=1 and n=1. 5 he anoalous igneous body was tested by Euler ethod, using two different structural indices, n = 1, and n = he result was shown in Fig. 5, which shows that the index n = 1.5 was yielding useful solutions and acceptable depth estiates. he depths vary between CONCLUSION A nuber of ethods are available for a fast analysis of large data sets. hey are used to deterine the average location of the ain anoalous bodies under a nuber of quite strict assuptions. Exaples include Euler deconvolution ethods [1], Werner deconvolution ethods [11], the analytic signal approach [12,13] statistical approach and siilar ethods [14,15]. Euler deconvolution has been widely used in autoatic aeroagnetic interpretations because it Fig. 4: Shows the otal-field Anoalies (d) of Multiodel Anoalous Source Bodies and their requires no prior knowledge of the source agnetization direction and assues no particular interpretation odel. Depths Deconvolved by Using Different he Euler deconvolution ethod is best suited for Structural Indices, a) n = 0, b) n = 0. 25, c) n = anoalies caused by isolating and ultiple anoalous 0. 5, d) By Using Correct Index n = 1 and sources. he ethod is applied to a long profile of Deconvolved by Using Different Moving aeroagnetic easureents, located over Aenzi area, Window Sizes Inner Mongolia, North China. Soe indication of the 213

6 source type can be gained by varying the structural index for any particular feature. It is concluded fro the above the deconvolution ust be run with different structural indices even with the interediate structural indices, which ight iprove clustering solutions, but they are not related to a known odel structure. he anoalous features of the field exaple are siple feature and located at shallow depths, so it is better to be deconvolved with a sall window size rather than a large window. REFERENCES 1. hopson, D.., EULDPH-new technique for aking coputer-assisted depth estiates fro agnetic data. Geophysics, 47: Barongo, J. O., Euler s differential equation and the identification of the agnetic point-pole and point-dipole sources. Geophysics, 49: Reid, A.B., J.M. Allsop, H.Granser, A.J. Millett and I.W. Soerton, Magnetic interpretation in three diensions using Euler deconvolution. Geophysics, 55: Marson, I. and E.E. Klingele, Advantages of using the vertical gradient of gravity for 3-D interpretation. Geophysics, 58: Barbosa, V.C.F., B. C. Joao and W.E. Medeiros, Stability analysis and iproveent of structural index estiation in Euler deconvolution. Geophysics, 64: Hansen, R.O. and L. Suciu, Multiple source Euler deconvolution. Geophysics, 67: Ravat, D., K. Kirkha and.g. Hildenbrand, A source-depth separation filter. Using the Euler ethod on the derivatives of total intensity agnetic anoaly data. he Leading Edge, 21: Penrose, R., A generalized inverse for atrices. Proc. Cabridge Phil. Soc., 51: Lanczos, C., Linear differential operators. D. Van Nostrand, Princeton, New Jersey. 10. Jackson, D.D., Interpretation of inaccurate, insufficient and inconsistent data. Geophys. J. R. Astr. Soc., 28: Ku, C. C. and J.A. Sharp, Werner deconvolution for autoated agnetic interpretation and its refineent using Marquardt s inverse odeling. Geophysics, 48: Roest, W.R., J. Verhoef and M. Pilkington, Magnetic interpretation using the 3-D analytic signal. Geophysics, 57: Spector, A. and F. S. Grant, Statistical odels for interpreting aeroagnetic data. Geophysics, 35: Blakely, R.J. and R.W. Sipson, Approxiating edges of such bodies fro agnetic or gravity anoalies. Geophysics, 51: Miller, H.G. and V. Singh, Seiquantitative techniques for the identification and reoval of directional trends in potential field data. J. Appl. Geophys., 32: