A ne w method for spectral analysis of the potential field and conversion of derivative of gravity-anomalies : cosine transform

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1 CHINESE JOURNAL OF EOPHYSICS Vol. 49, No. 1 Jan., 006,,..,006,49 (1) :4448 Zhang F X, Meng L S, Zhang F Q, et al. A new method for spectral analysis of the potential field and conversion of derivative of gravityanomalies : cosine transform. Chinese J. eophys. (in Chinese),006,49 (1) :4448 1, 1, 1,, 1, 1, 1 1, 13006, 13003,,.,,.,Fourier,,,, %5 %.,,, (006) P , A ne w method for spectral analysis of the potential field and conversion of derivative of gravity-anomalies : cosine transform ZHAN Feng- Xu 1, MENLing- Shun 1,ZHAN Feng-Qin 1,,LIU Cai 1, WU Yan- ang 1, DU Xiao-Juan 1 1 eo- Exploration Science and Technology Institute, Jilin University, Changchun 13006, China College of Physics, Jilin University, Changchun 13003, China Abstract In order to improve the accuracy of derivative conversion of gravity anomalies and reflect anomaly characteristic of geologic bodies effectively, we propose a new method of calculating anomaly derivative using cosine transform. Two theorems are put forward and proved and the common expression of the cosine transform spectrum of the gravity potential field and the formula to calculate derivatives of gravity anomalies are deduced. So the theory of cosine- transform- spectrum of the potential-field is established. In model experiments, we find that the deviation of the first derivative calculated by Fourier transform is very large to comparing with the theoretical derivative, but the fitting effect of the derivative of anomalies calculated by cosine transform is very good. The calculation accuracy of data are all very high except that errors of several data on the boundary are large because of the residual ibbus effect induced by finite truncation of gravity anomalies %5 %. Errors are Keywords Cosine-transform spectrum of gravity potential field, Derivative of anomaly, Cosine transform, Accuracy (XQ ).,,1969,,1995,,. edu. cn China Academic Journal Electronic Publishing House. All rights reserved.

2 1 : 45 1,., ;,,, [1, ]..,, [3 ] [4 ],. Fourier [57 ],,,,, Ahmed et al. [8 ] (DCT),DCT [9,10 ]. 30,,,. Fourier., K- L ( Karhunen-Lo ve transform),, [11 ], DCT [1,13 ], Fourier.,. 11,,,,. 1 f ( t) g ( t),, f ( t) g ( t), C[ f ( t) 3 g ( t) ] = f C ( ) g C ( ), (1) C,= f. :F () = C [ f ( t) 3 g ( t) ], f ( t) g ( t),,, F( ) = f ( ) g ( t - ) d cos( t) d t. - - t 1 = t -,t = t 1 +,d t = d t 1, F( ) = f ( ) g ( t 1 ) d cos( t China Academic Journal Electronic Publishing House. All rights reserved. - = f ( ) d t 1 ) g ( t 1 ) d(cos t 1 cos - sin t 1 sin ) d t 1 = g ( t 1 ) cos - t 1 d t 1 f ( - ) cos d - g ( t 1 ) sin t 1 d t 1 f ( ) sind. - - f ( t) g ( t),,, F( ) = C[ f ( t) 3 g ( t) ] = f C ( ) g C ( ), 1. f ( t), Fourier,f ( t) C[ f ( n) ( t) ] = n CΠ [ f ( t) ] n = 1,3,5, ( - 1) n n C[ f ( t) ] n =,4,6, CΠ [ f ( t) ] = f ( t) cos - t d t /.. (),. 1 oxyz, z,,,,,, P ( x, y, z) V ( x, y, z) = (,, ) ddd [ (- x) + (- y) + (- z) ] 1Π. (3) (,, )

3 46 (Chinese J. eophys. ) 49, = 0, (3), R ( x, y, z) = ( x + y + z ) 1Π, V ( x, y, z) = ( x, y, z) 3 R ( x, y, z). (4), R ( - x, - y, - z) = R ( x, y, z), R, (4) 1,(1) V C ( u, v, w) = C ( u, v, w) R C ( u, v, w), (5), u, v, w x, y, z. (,, ), V = 9 V 9 x + 9 V 9 y + 9 V 9 z = - 4( x, y, z), (6),(5) C( V) = - 4 ( u + v + w ) V C ( u, v, w) = - 4 C ( u, v, w), (7) C ( u, v, w) V C ( u, v, w) = ( u + v + w ). (8) (8),,,z.,w, (1) V ( u, v, z) = R ( u, v, z) 3( u, v, z), R ( u, v, z) = = ( u + v + w ) cos(wz) d w - u + v e V ( u, v, z) = - z z u + v, ( u, v, ) e u u + v. + v ( z- ) d, (9) (9) z n, n 9V n ( u, v, z) 9 z n = u + v 13 n V ( u, v, z). (10),,,, (10), (), z ( u, v, z) = u + v n g ( u, v, z), (11) x ( u, v, z) = y ( u, v, z) = (u) n C / [ g ( x, y, z) ] n = 1,3,5,, ( - 1) n (u) n C[ g ( x, y, z) ] n =,4,6,, (1) (v) n C / [ g ( x, y, z) ] n = 1,3,5,, ( - 1) n (v) n C[ g ( x, y, z) ] n =,4,6,, CΠ [ g ( x, y, z) ] = - - g ( x, y, z) cos - ( ux + uy) d xd y. (13) (11) (1) v = Ahmed et al. [8 ]. { x ( n) : n = 0,1,, }, X C ( k) = x ( n) = N c ( k) x ( n) cos n = 0 N k = 0 c ( k) X C cos k, n = 0,1,,,, ( n + 1) k, (14a) N ( n + 1) k, (14b) N c ( k) = 1Π k = 0 1 k 0, (14c), x ( i, j) : i = 0,1,, ; j = 0,1,, M - 1, X C ( m, n) = x ( i, j) = NM c ( m, n) cos NM M- 1 i = 0 j = 0 ( i + 1) m cos N M- 1 m = 0 n = 0 x ( i, j) ( j + 1) n, M c ( m, n) X C ( m, n) (15a) China Academic Journal Electronic Publishing House. All rights reserved.

4 1 : 47 cos ( i + 1) m cos N ( j + 1) n, (15b) M m, i = 0,1,, ; n, j = 0,1,, M - 1 ; c ( m, n) = 1Π m = 0, n = 0 1. (15c) 31, (14a),(11), (14b) ;, (15a b).,,, n,nπ, (14a) (15a).,, (14a) (15a) ;,, X C / ( k) = X C / ( m, n) = N c ( k) x ( n) sin n = 0 NM c ( m, n) sin i = 0 M- 1 j = 0 x ( i, j) sin ( n + 1) k, N (16) ( i + 1) m N ( j + 1) n, (17) M c ( k) c ( m, n) (14c) (15c). (1) (13),, (14b) (15b) n., (16) (17), [14 ],. 4., Fourier ( g z g x ),. Fourier., ;,,,. Fourier,,. 1, : R = 110km ; ( x 0, d), x 0 = 10 km, d = 10 km;= 015gΠcm 3. 1 : g z ( x,0) = R [ d - ( x - x 0 ) ] [ ( x - x 0 ) + d ], (18) g x ( x,0) = 4R ( x - x 0 ) [ ( x - x 0 ) + d ]. (19) 1,, (1 c) a,fourier b,. 1 g z, g x xπkm Fourier,,1 g z g x Table 1 Accuracy of g z, g x of infinite cylinder calculated by cosine transform g z (10-9 Πs ) g x (10-9 Πs ) ( %) ( %) China Academic Journal Electronic Publishing House. All rights reserved.

5 48 (Chinese J. eophys. ) 49 (References) 1 (a) ; (b). a ; b Fourier ; c. Fig. 1 Contrast analysis of vertical first-order derivatives of an infinite cylinder calculated by different methods.,,, %5 %. 5,.,, Fourier, Fourier.,. [ 1 ],.. :, Luo X K, uo S Y. Applied eophysics Course ( in Chinese ). Beijing : eological Publishing House, [ ],,.. :, Wang Q S, An Y L, Zhang C J, et al. ravitology (in Chinese). Beijing : Seismological Press, [ 3 ],,..,1995,5() :3338 Liu B H, Zhang W, Meng En. A simple method for calculating the first vertical derivative. Chinese), 1995,5() :3338 Journal of Ocean University of Qingdao (in [ 4 ]..,1996,31(3) :4154 Wang B Z. Computing the vertical second derivative and upward continuation of gravity anomaly by spline function method. eophysical Prospecting (in Chinese), 1996,31 (3) : 4154 [ 5 ],,.. :,1987 Wu X Z, Liu H, Xue Q, et al. Analysis of Method and Application on Fourier Transform and Potential Field Spectrum ( in Chinese). Beijing :Surveying and Mapping Press, 1987 [ 6 ] Kevin L Michus, Juan Homero Hinojosa. The complete gravity gradient tensor derived from the vertical component of gravity : A Fourier transform technique. Journal of Applied eophysics, 001, 46 : [ 7 ] Lourenco J S, Morrison H F. Vector magnetic anomalies derived from measurements of single component of the field. eophysics, 1973, 38 : [ 8 ] Ahmed N T, Natarajan T, Rao K R. Discrete cosine transform. IEEE Trans. Comput., 1974, 3 (1) : 9093 [ 9 ] Rao K R, Yip P. Discrete Cosine Transfrom : Algorithms, Advantages and Applications. New York : Academic Press, 1990 [10 ] Dinstein I, Rose K, Heiman A. Variable block- size transform image coder. IEEE Tans. Comm., 1990, 38 (11) : [11 ].. :, Hu S. Digital Signal Processing (in Chinese). Beijing : Tsinghua University Press, [1 ] Cvetkovic Z, Popovic M V. New fast algorithms for the computation disctete cosine and sine transform. IEEE Trans. Signal Process., 199, 40 (8) : [13 ] Vladim r Brita k. A unified discrete cosine and discrete sine transform computation. Signal Processing, 1995, 43 : [14 ] Jain A K. A fast Karhunen-Loeve transform for a class of stochastic processes. IEEE Trans. Commun., 1976, 4 : ( ) Oil China Academic Journal Electronic Publishing House. All rights reserved.