Fractals and the accuracy of geographical measures

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The problems of estimating line length, area, and point characteristics in the earth sciences have generated substantial but independent literatures. All three problems are of increasing concern given the current interest in digital capture, processing, and the storage of geographically referenced data. In the case of qualitative maps, all three are shown to be related to Mandelbrot's fractional dimension D (Mandelbrot, 1977) which allows the dependence of each on sampling density to be predicted. The general results are confirmed by simulation on surfaces of constant D. They also imply that certain improvements can be made in a number of previously proposed methods.