Laplacian fields can be continued either by the use of Taylor series, or by spectral analysis and synthesis. While the first method is applicable to non-Laplacian fields as well, the second, as it is presently developed, is not. The convergence properties of downward continuation by these methods have been investigated for some source geometries that frequently occur in nature. The cases treated include gravity, magnetic, electrical, and electromagnetic fields. When the spectral method is used, the boundary between regions of convergence and divergence is always a plane that is parallel to the plane of observation. With the Taylor series method, on the other hand, the boundary is a hyperboloid or a combination of hyperboloids. The position of these boundaries--plane or hyperboloid--depends upon the shape of the anomaly-causing body. For bodies with corners and edges, the plane passes through the shallowest corner or edge, or there is a hyperboloid apexed at each corner or edge so that the outermost envelope constitutes the boundary of convergence. For smooth bodies, like spheres and cylinders, these boundaries are positioned at some internal point, which, in the case of artificial-field methods, may correspond to an image. For instance, for the gravity effect of a sphere or a two-dimensional horizontal circular cylinder, or their magnetic and electrical effects under the influence of a uniform primary field, the plane or the hyperboloid passes through or is apexed at the center or the axis, as the case may be.The results of this paper provide the theoretical justification for using the method of continuation in interpretation of mining geophysical data, as reported empirically earlier.

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