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Length Scale as a Fourth Spatial Dimension in Geology and Geophysics

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Abstract

Three-dimensional coordinate systems define the location of zero-dimensional points. Higher dimensional spatial objects such as lines and planes become apparent from sets of adjacent points. Any point defined in three-dimensional space within a sedimentary basin could be simultaneously located on the surface of a sand grain and a bedding plane in a kilometer-scale fold, and structure at any other scale. Therefore, like many other aspects of geology and geophysics, structural geology is multiscale which is why filtering, smoothing, scale bars and the like are prerequisites for any kind of geological mapping and outcrop imagery. It is argued here that the association of points with higher-dimensional objects that form the spatial building blocks of geology singles out length scale as a geometrical parameter that is fundamental to the definition of an object. Properties such as curvature, porosity, color and so on can only be attributed after location and scale are defined. It is proposed that length scale be treated as a fourth spatial dimension in geological and geophysical applications, on an equal footing to the three spatial coordinate dimensions. The utility of adopting a four spatial dimension approach in geoscience is that it forces length scale to be explicitly specified in descriptions where it is routinely omitted, for example, measurements of structural orientation where a scale parameter greatly simplifies later mapmaking and model building. Taken together with geological time, it is convenient to define geological and geophysical structure in five spacetime dimensions.

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Release of data relating to this study is limited to the materials presented in this paper. Figure 3 outcrop image is courtesy Alan J. Kaufman. Reviews by three anonymous referees are acknowledged.

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Stewart, S.A. Length Scale as a Fourth Spatial Dimension in Geology and Geophysics. Pure Appl. Geophys. 179, 2817–2830 (2022). https://doi.org/10.1007/s00024-022-03086-w

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