49 1 006 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,,,,- 0109 %5 %.,,, 0001-5733(006)01-044 - 05 P631 004-11 - 3,005-10 - 1 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. - 0109 %5 %. Errors are Keywords Cosine-transform spectrum of gravity potential field, Derivative of anomaly, Cosine transform, Accuracy (XQ0070705).,,1969,,1995,,. E-mail : zhangfx @jlu. edu. cn 1994-010 China Academic Journal Electronic Publishing House. All rights reserved. http://www.cnki.net
1 : 45 1,., ;,,, [1, ]..,, [3 ] [4 ],. Fourier [57 ],,,,,. 1974 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 1-1994-010 China Academic Journal Electronic Publishing House. All rights reserved. http://www.cnki.net - = 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) (,, )
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 = 0. 3 311 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) 1994-010 China Academic Journal Electronic Publishing House. All rights reserved. http://www.cnki.net
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 0 018 116 14 31 410 418 516 614 71 810 818 916 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 ) ( %) ( %) - 1186-115 - 151-195 - 3151-4119 - 5101-5190 - 6153-5174 010 18113 46151-189 - 148-17 - 3111-3163 - 419-5109 - 5195-6157 - 5177-0104 18107 46147 601 13135 8137 514 3141 139 1160 0185 0161 015 - - 0133-0109 0178 0198 117 1167 16 3114 415 6176 10149 16175 610 33199 19138 0143 019 115 1167 17 3117 4156 6180 10154 16181 617 34106 19143 44187 611 1157 0 0144 0195 0188 0159 0148 0136 018 011 015 1994-010 China Academic Journal Electronic Publishing House. All rights reserved. http://www.cnki.net
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.,,,- 0109 %5 %. 5,.,, Fourier, Fourier.,. [ 1 ],.. :, 19911 139154 Luo X K, uo S Y. Applied eophysics Course ( in Chinese ). Beijing : eological Publishing House, 19911 139154 [ ],,.. :,0031 19690 Wang Q S, An Y L, Zhang C J, et al. ravitology (in Chinese). Beijing : Seismological Press, 0031 19690 [ 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 : 159174 [ 7 ] Lourenco J S, Morrison H F. Vector magnetic anomalies derived from measurements of single component of the field. eophysics, 1973, 38 : 395368 [ 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) : 073078 [11 ].. :,0031 363 375 Hu S. Digital Signal Processing (in Chinese). Beijing : Tsinghua University Press, 0031 363375 [1 ] Cvetkovic Z, Popovic M V. New fast algorithms for the computation disctete cosine and sine transform. IEEE Trans. Signal Process., 199, 40 (8) : 083086 [13 ] Vladim r Brita k. A unified discrete cosine and discrete sine transform computation. Signal Processing, 1995, 43 : 333339 [14 ] Jain A K. A fast Karhunen-Loeve transform for a class of stochastic processes. IEEE Trans. Commun., 1976, 4 : 103109 ( ) Oil 1994-010 China Academic Journal Electronic Publishing House. All rights reserved. http://www.cnki.net