Calculation of gravity due to a vertical cylinder using a spherical harmonic series and numerical integration

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1 CSIRO PUBISHING Exloration Geohysics htt://dx.doi.org/.7/eg43 Calculation of gravity due to a vertical cylinder using a sherical harmonic series and numerical integration Sung-Ho Na,3 Hyoungrea Rim,3,4 Young-Hong Shin Mutaek im Yeong-Sue Park,3 Sace Geodetic Observation Center, NGII, Sejong 339-8, Korea. Korea Institute of Geoscience and Mineral Resources, Daejeon 35-35, Korea. 3 Geohysical Exloration, Korea University of Science and Technology, Daejeon 35-35, Korea. 4 Corresonding author. rhr@kigam.re.kr Abstract. We develoed a sherical harmonic series that reresents the gravitational otential and its gravity field due to a buried right vertical cylinder. This series can be used at far- and intermediate-regions, and is fast and accurate, using only a few terms. We comared the values of the fields acquired by this new sherical harmonic series, with ones comuted by direct numerical integrations, using a fine-mesh structure for a vertical cylinder. Results of the calculations are shown and erformances of the two different methods are comared. Faithfulness of the sherical harmonic series is tested with an inversion examle. Key words: gravity, gravity otential, numerical integration, sherical harmonics, vertical cylinder. Received December 4, acceted December 4, ublished online 3 January 5 Originally submitted to KSEG 8 March 4, acceted 8 November 4 Introduction Gravity and magnetic fields due to various subsurface anomalous objects such as a shere, cylinder, or sheet, have been studied since the early days of geohysical exloration. Nettleton (94) gave several aroximate formulae, including one for gravity attraction by a vertical cylinder. His master curves are still useful because of the wide range of arameter variation. Nabighian (96) derived a closed formula for the vertical comonent of gravity due to a right vertical cylinder. His formula consisted of ellitic integrals and enables us to comute vertical attractions on the exterior of a vertical cylinder. For the case of a right vertical cylinder, it is ossible to exress the gravity otential as a sum of sherical harmonics. Once the sherical harmonic series for gravity otential is known, evaluation of the gravity field and its derivatives can be readily erformed. Convergence of this series is normally fast so that, in the case of gravity otential, only three or four terms are enough for 99% accuracy in most of the convergence region. With the aid of a fast comuter, direct numerical calculation for both the otential and gravity due to a vertical cylinder, can be erformed accurately within a few seconds. However, this tye of numerical integration requires the summation of at least hundreds or thousands of terms, if one desires reasonable accuracy. It is a merit of the sherical harmonic series reresentations of these fields that the series can readily yield the related reresentations for their artial derivatives with resect to the cylinder shae, whilst such cannot be romtly handled by direct numerical integration. Vertical cylinder and its mesh structure for numerical integration Gravity due to a cylindrical body is illustrated in Figure. The gravitational attraction vector at any observation oint (P) can be decomosed into horiontal and vertical comonents as shown. Journal comilation ASEG 5 For the direct numerical calculation in this study, the cylinder is divided into 55 cells by first dividing the cylinder into arallel circular discs, and then dividing each disc into concentric rings of which the k-th ring is subdivided into k identical sectors. In the figure, the smallest 9 rings of the central art of the disc are shown. The gravity otential, and the gravity due to any arbitrary set of mass elements (Dm i ), can be simly reresented as follows: the otential Uð~rÞ ¼G P Dm i i j~r~r i j and the gravity~gð~rþ ¼G P ð~r~r i ÞDm i i. In fact, we could notice that j~r~r ij 3 the numerical integration with the 55 cell model (Figure ) is accurate enough to acquire more than six significant figures at any oint. We also tried coarser numerical models and comared the calculation results. Gravity otential: sherical harmonic series reresentation If one considers the gravity otential (du) at the symmetry -axis is due to a ring element of radius (r), then the following occurs: rdr du ¼ GdD ffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðþ þ r where G, d, and D are the gravitational constant, mass density, and thickness of the disc, resectively (Figure ). The gravity otential (DU) at due to the disc is then exressed as follows: DU ¼ GdD ð R r¼ rdr ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ GdDð þ R Þ ðþ þ r And, the gravity field at is acquired as the gradient of otential as follows: Dg ¼ GdD ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð3þ þ R The gravity otential along the -axis due to a right vertical cylinder (Figure ) is now acquired as follows:

. Schematic illustrations of the gravity due to a vertical cylinder and its cylinder mesh structure for numerical integration: a right circular cylinder and the gravitational attraction vector at the

2 B Exloration Geohysics S.-H. Na et al. Z (c) P θ r Gravitational attraction x R Each disc Each disc is divided into circular rings of equal width The cylinder is divided into discs i-th ring is subdivided into i sectors Fig.. Schematic illustrations of the gravity due to a vertical cylinder and its cylinder mesh structure for numerical integration: a right circular cylinder and the gravitational attraction vector at the field oint P; illustrations showing how the cylinder mass is divided into discs and further to 55 small mass elements; and (c) close overview of 55 mass oints of each disc. U ¼ Gd þ ð ð ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ R Þd ¼ Gd ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ R þ R ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ logð þ þ R Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð þ Þ ¼ ð þ Þ þ R ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ R þ ð þ Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ R log þ þ ð þ Þ þ þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi A ð4þ þ R For the integration we used an identity exression 3. from Dwight (96), which is equivalent to exression.6 of Gradshteyn and Ryhik (98). ikewise, the magnitude of gravity due to the cylinder along the -axis can be exressed as: gðþ ¼Gdð þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ R ð þ Þ þ R Þ ð5þ At far-field ( > ffiffiffi R and þ R þ < ), equation 5 can be exressed as a series of negative owers of as follows: gðþ¼gm 3 þ 3 4 R 4 þ 3R 3 5 þ ð6þ where M is the total mass of the cylinder: M = R d. Then, after integration in we have: UðÞ¼GM þ R þ 3R þ ð7þ which may be directly acquired from equation 4. For the gravity otential off the -axis of the cylinder, the equation can be exressed as: Uðr;Þ¼GM r 3R þ 8 r P ðcos Þþ r 3 P ðcos Þ ð8þ R r 4 P 3ðcos Þþ where P n (cos y) is the egendre olynomial of degree n. The conversion from equation 7 to equation 8 is justified

3 Sherical harmonic series and numerical integration Exloration Geohysics C.6 R Z Fig.. Gravity along the vertical axis due to a circular disc. The disc is in the x-y lane, and its centre coincides with the origin. The field oint is on the -axis, and R and r are the radii of the disc and inner circular ring. D is the infinitesimal thickness of the disc. ρ ΔZ g g x Height Height Gravity otential U Height Distance x 4 5 Fig. 4. Two comonents of gravitational attraction due to a vertical cylinder calculated by the direct numerical integration as described in Figure. For brevity GM is taken as unity, and the cylinder radius R and height are also taken as unity. Selected heights of are between. and Distance x 8 Fig. 3. Gravity otential (U) due to a vertical cylinder calculated by the direct numerical integration as described (Figure ). For brevity GM is taken as unity (G is the gravitational constant and M reresents the mass of the cylinder), and the cylinder radius R and height are also taken as unity. Selected heights of are between. and. because P of: (i) the uniqueness of a sherical harmonic series, n¼ ða nr n þ bn r ÞP nþ n ðcos Þ, and (ii) equation 8 becomes equation 7 on the -axis, where P n (cos y))p n () = and r ) (Morse and Feshbach, 953:. 67; Jackson, 999: ch. 3.3). In fact, equation 8 is a onal harmonic series because of the aimuthal symmetry of the roblem involved. The gravity field at an arbitrary osition can be readily acquired as the gradient of equation 8: g r ðr;þ¼ qu qr and g ðr;þ¼ qu r q More exlicitly, the magnitude of g r (r, y) and g y (r, y) are found as follows: g r ¼ GM 3R þ g ¼ GM r r 3 P ðcos Þþ 3 r 5 P 3ðcos Þþ dp ðcos Þ r 3 d 3R R r 4 P ðcos Þ R 3 4 dp 3 ðcos Þ r 5 þ d dp ðcos Þ r 4 d ð9þ ðþ And, the magnitudes of the horiontal and vertical comonent, as shown in Figure, are given: g x ¼ g r sin g cos and g ¼ g r cos þ g sin : For near-field ( < R, ), only the otential U(r, y) is given here, and further descrition is avoided: Uðr;Þ¼Gd ðd RÞrP ðcos Þþ D r P ðcos Þ 6 R D þ r 3 P D 3 3 ðcos Þ ðd þ Þ 8 D 5 r 4 P 4 ðcos Þþ ðþ

4 D Exloration Geohysics S.-H. Na et al. Gravity otential U u4 u u u u Height = u u u4 Height = u u u4 3 Distance X Fig. 5. Gravity otential due to a vertical cylinder calculated using the sherical harmonic series: u, u,, and u4 stand for series of a single leading term, two terms, three terms, and four terms, resectively. For brevity GM is taken as unity, and the cylinder radius R and height are also taken as unity. The numerical calculation is shown as a solid line. The dotted line corresonds to convergence limit in the left ( = ) and series converges at all distance x in the right ( = ). Calculation result At seven different the height, from. to, the values of the gravity otential along the horiontal axis, between x = and, calculated by the direct numerical integration scheme of this study (Figure ) are illustrated in Figure 3. For brevity, both the radius and length of the cylinder are taken as unity. ikewise, the gravitational constant and the mass of the cylinder are taken as unity. The values of the two gravity comonents calculated by numerical integration are also shown in Figure 4. All figures shown in this study were based on far-field calculations. We found that the near-field sherical harmonics series aroximation can be converged only for a very limited circumstance: r << R,. The gravity otential, calculated using a sherical harmonic series, is illustrated in Figure 5. Each grah in Figure 5 (u, u,, and u4) corresond to the gravity otential values calculated as: u - the first term only, u - sum of the two leading terms, - sum of three leading terms, and u4 - sum of the leading four terms of the series. Chosen heights of are and. An numerical calculation is also drawn together and used as a reference value. The two sets of gravity otential values acquired by both the sherical harmonic series, and the numerical integration for height ( = ) and distance (x =,,,... ), are given in Table. The dotted vertical line Table. Comarison of sherical harmonic series u to four terms and numerical evaluation of gravity otential due to a right vertical cylinder ( is secified as ). u u u4 Numerical x = (.) (.5) ( ) (.78333).685 x = (.777) (.5333) ( ) (.537).5393 x = x = x = x = x = x = x = x = x = denotes the convergence limit for =. For comarison, the convergence limit for = is drawn by the dotted line in Figure 5, and shows that the series converges in the entire range of x (equation 5). Two comonents of gravitational attraction due to the vertical cylinder calculated by the sherical harmonic series are illustrated in Figure 6. In each grah, values denoted as gxi and gi show the sum of the first i terms (i =,, 3, or 4). The dotted vertical lines reresent the convergence limit. The convergence limit for both otential and gravity when height = is found to be x >, whereas, when = or higher there is no nominal limit for series convergence; i.e. convergence is attained everywhere. In the convergence region, the otential series with four terms shows greater than 99% accuracy. Series convergence is slower in gravity than in otential. Still, the gravity series of four terms shows 97% or greater accuracy in the region of convergence. The accuracies increase in farther regions, such as r > 3. Numerical integration by the scheme of this study is found to be stable, but requires millions of additional stes. This is simly due to the very-fine structure of the cylinder numerical model of this study, which is comosed of 55 minute cells. The sherical harmonic series reresentation of equation 8 can be readily modified to reresent gravity comonents g x and g or their artial derivatives with resect to source configuration, such as: qg x qr ; qg x q ; qg qr and qg q. The following inversion rocedure was erformed using these artial derivatives. First, a dataset of vertical gravity anomaly of g was generated for a buried cylinder of normalised dimension. Then, from the initial guess model and calculated artial derivatives qg qg qr and q, true values of source arameters were sought. In fact, values of R =. and =.9 were taken as the true ones, while R =. and =. were used as starting values. The total mass was assumed to be exactly known. Selected field ositions were at x =.5,.,.5, 3., and 3.5 with fixed as.. Four iterations yielded estimates as: () R =.859, =.88945, () R =.983, =.89834, (3) R 3 =.3, 3 =.8997, and (4) R 4 =.49, 4 = Further verification of calculation accuracy In this section, the comutational accuracy of the gravity due to a vertical cylinder by using sherical harmonic series is comared with the numerical integration scheme. As noted before, numerical integration of gravity and its otential due to a right vertical cylinder, were erformed by summation of a set of 55 small mass elements ( discs => each disc is divided into rings + each ring is divided into same sie sectors) (Figure ). In fact, other numerical calculations with much coarser

5 Sherical harmonic series and numerical integration Exloration Geohysics E.4.3 Height = gx gx gx3 gx4..8 Height = g g g3 g4..6 g x g x (c) Height = gx gx gx3 gx4 g g (d) Height = g g g3 g Distance x Fig. 6. Twocomonentsof gravitational attraction due to a verticalcylinder calculated byusing the shericalharmonic series: u, u,, and u4stand for series of a single leading term, two terms, three terms, and four terms, resectively. For brevity GM is taken as unity, and the cylinder radius R and height are also taken as unity. The numerical calculation is shown as a solid line. The dotted line corresonds to convergence limit in the left ( = ) and series converges at all distance x in the right ( = ). Table. Values of the gravity otential due to a right circular cylinder aroximated by n discs (each disc is divided into n circular rings, which are again comosed of n sectors). The selected values for n are, 3, 5,, 3, 5, and. Horiontal distances are taken as x =,,,... n = n =3 n =5 n = n =3 n =5 n = x = x = x = x = x = x = x = x = x = x = x = aroximation, such as 55 mass elements ( discs => each disc is divided into rings => each ring is divided into same sie sectors), are fairly accurate. Comared with the one described in Figure 3, the relative errors of the numerically calculated values of gravity otential ( = ) for the equivalent right vertical cylinder, aroximated with 55 mass elements, are found in the order of 3 to 7 (for x varying from to ). The gravity otential using a coarser set of aroximate N mass elements (N ¼ n nðnþþ ) is summarised in Table. The selected values of n are: n =, 3, 5,, 3, 5, and. The corresonding numbers of mass elements are: N =, 8, 75, 55, 3 95, 63 75, and 55. Conclusion We develoed a sherical harmonic series reresentation to evaluate the gravity otential and comonents of gravity field due to a right vertical cylinder. Values of the sherical harmonic

6 F Exloration Geohysics S.-H. Na et al. series with only four terms are accurate enough with at least three to six significant figures deending on the distance. Partial derivatives of the sherical harmonic series can also be evaluated with resect to source arameters such as radius or cylinder height. Acknowledgements This study was suorted by the Korea Institute of Geoscience and Mineral Resources (KIGAM) and funded by the Ministry of Science, Information and Communications Technology, and Future Planning of Korea. The first author thanks all the co-authors and others in KIGAM for their warm suort and encouragement. The authors also thank the two anonymous reviewers, whose various comments enhanced the manuscrit. References Dwight, H. B., 96, Tables of integrals and other mathematical data: The Macmillan Comany. Gradshteyn, I. S., and Ryhik, I. M., 98, Table of integrals, series, and roducts: Academic Press. Jackson, J. D., 999, Classical electrodynamics (3rd edition): John Wiley and Sons Inc. Morse, P. M., and Feshbach, H., 953, Methods of theoretical hysics - art II: McGraw-Hill Book Comany, Inc. Nabighian, M. N., 96, The gravitational attraction of a right vertical circular cylinder at oints external to it: Geofisica Pura e Alicata, 53, doi:.7/bf78 Nettleton, I. I., 94, Gravity and magnetic calculations: Geohysics, 7, doi:.9/

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