Computing the gravitational and magnetic anomalies due to a polygon: Algorithms and Fortran subroutines

Abstract

We present two algorithms for computing the gravitational and magnetic anomalies due to an n-sided polygon in a two-dimensiona l space. Both algorithms have been implemented as subroutines coded in Fortran-77, and listings are provided. Because references to trigonometric functions have been almost completely eliminated, these codes run substantially faster than mosts codes now in existence. Furthermore, anomalies can be computed at any point outside, on, or inside the polygon. Unlike other codes, these algorithms can be used to model subsurface observations. below the polygon. Most existing computer programs are not as general as those given here. Obtaining the correct answer within the polygon is essential when modeling gravity and magnetic anomaly measurements obtained in tunnels, boreholes, or submarines.Although there' are many codes based on the method of Talwani et al. (1959), we feel that those presented here are worthy of attention because they run an order of magnitude faster and generate correct answers at every point in the 2-D space. THE GRAVITY ANOMALY DUE TO A POLYGONHubbert (1948) showed that the gravitational attraction due to a 2-D body can be expressed in terms of a line integral around its periphery. Talwani et al. (1959) considered the case of an n-sided polygon and broke the line integral up into n contributions, each associated with a side of the polygon. We follow Talwani et al. (1959) by placing the point at which the gravity anomaly is to be computed (i.e., the station) at the origin of the coordinate system ( Figure 1) and expressing the vertical and horizontal components of the gravity anomaly as where Z, and Xi are line integrals along the ith side of the polygon, G is the gravitational constant, and p is the density of the polygon. Talwani et al. (1959) derived expressions for Z, and Xi that make extensive references to trigonometric functions. Grant and West (1965) reformulated the expression for Z i by making more references to the vertex coordinates {Xi' Z;}i=l,n and fewer references to angular quantities, and thus reduced the number of trigonometric expressions involved in the computation. We follow Grant and West's approach, and produce a formula for Xi as well as Zi. We com- INTRODUCTIO N

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