Polygonal models which are of uniform density or magnetization contrast and which contain no cavities are sufficient to ensure the uniqueness of solutions in gravity and magnetic problems. Nonuniqueness is then attributed to a number of other factors which are discussed in detail. We study the problem of nonuniqueness using mainly a parameter hyperspace in which ambiguity takes the form of a scatter of local minima or a continuous domain bounded by a contour whose value is determined by the amplitude of observational errors. The possible solutions to the problem being examined are contained in a region which would have contained the unique solution under exact conditions and which decreases in extent as the factors causing ambiguity decrease. Thus, under favorable conditions, a large degree of uniqueness may occur. By starting the search from a good initial point based on all available information, we may obtain a satisfactory solution without specifying any of the parameters defining the model. Different solutions produced by different initial points vary in their emphases but are, generally, of the correct order of magnitude. When such favorable conditions are lacking, the extent of ambiguity may be reduced by specifying one or more parameters. With the aid of the parameter hyperspace, we may thus conceive why and how ambiguity is present and determine what means can be found to limit it.

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