Computation of the Magnetic Anomalies Caused by Two-Dimensional Bodies

Abstract

The magnetic anomaly of a two-dimensional body of uniform magnetization and arbitrary shape can be expressed compactly in terms of the vertical and horizontal derivatives of the corresponding gravity anomaly and an angle p which incorporates both the direction of magnetization and the direction in which the observed magnetic anomaly component is measured. The formula is useful for computing families of curves. By an extended use of the formulation, computation of the conjugate harmonic function to the observed magnetic anomaly makes it possible to compute the anomaly for different values of p without knowledge of the shape of the body. The simplicity of the formulation cannot be extended to bodies of three-dimensional shape but computational short cuts are nevertheless available. The basic formulaIspir & KolCak (1969) have described a useful short cut in computing families of magnetic anomaly curves for two-dimensional bodies. A compact formula of similar type which relates the magnetic anomaly to the derivatives of the gravity anomaly is given here.The formula presented is most obviously useful for calculating families of magnetic anomaly curves over two-dimensional bodies with minimum computational effort, by computer or otherwise. It is a somewhat more general and more simple formulation than that given by Ispir and Kokak. It gives a much simplified approach to the evaluation of analytical expressions for magnetic anomalies caused by geometrical two-dimensional bodies, based only on differentiation of the corresponding gravity anomaly formulae. It also provides a useful starting point in the development of direct methods of magnetic interpretation, and is particularly well suited to the application of complex variable to interpretation theory.Let us consider a magnetic anomaly component produced by a two-dimensional body of uniform magnetization and arbitrary cross-section (Fig. I). The horizontal x-axis is chosen to be perpendicular to the strike of the body and lies entirely above the body. The y-axis points vertically downwards and the z-axis is parallel to the strike. Let the magnetization be J(J,, J,, JJ, its dip I,, its azimuth o r , measured from the x-axis in either sense, and the unit vector in the direction of the component of magnetization in the xy-plane m(cos p, sin p). Let the magnetic anomaly A(x, y ) be measured in the direction F(F,, F,,, FZ). 251

Identification of short polarity events by transforming marine magnetic profiles to the pole

“…However, Schouten's transformation constitutes a filter in the wave number domain, which greatly attenuates anomalies with wavelengths shorter than 5 km. Bott [1969] discusses a similar result using a different mathematical development.…”

“…constructed by least-squares fitting to observed profiles using plate models of specified thickness and depth, divided into cells of uniform width [Bott, 1967;Emilia and Bodvarsson, 1969;Bott and Hutton, 1970]. Although this method provides a computational procedure for stacking many profiles, it has the disadvantage that computational instabilities arise if the cell width is small.…”

Matrix Methods for Joint Interpretation of Two-dimensional Gravity and Magnetic Anomalies with Application to the Iceland-Faeroe Ridge

“…This requirement will be referred to as the Poisson condition. For two-dimensional bodies satisfying the Poisson condition, we have (Bott 1969 where the z-axis points vertically downwards, the x-axis is horizontal and perpendicular to the strike, g is the gravity anomaly, a is component of the magnetic anomaly, and F and p are related to the dip I,,, and azimuth a,,, of the direction of magnetization, and I, and a, of the direction of the magnetic anomaly component (azimuth being measured from the positive x-axis) by F = + (sin2 I,,, + COS'Z,,, cos' am)* (sin2 I, + cos2 I, cos' a,)*, B = P+O, P = arctan (tanZm/cos a,,,), u = arctan (tanI,/cos ae).…”