Making direction a first‐class citizen of Tobler's first law of geography

Abstract

Waldo Tobler frequently reminded us that the law named after him was nothing more than calling for exceptions. This article discusses one of these exceptions. Spatial relations between points are frequently modeled as vectors in which both distance and direction are of equal prominence. However, in Tobler's first law of geography, such a relation is described only from the perspective of distance by relating the decreasing similarity of observations in some attribute space to their increasing distance in geographic space. Although anisotropic versions of many geographic analysis techniques, such as directional semivariograms, anisotropy clustering, and anisotropic point pattern analysis, have been developed over the years, direction remains on the level of an afterthought. We argue that, compared to distance, directional information is still under‐explored and anisotropic techniques are substantially less frequently applied in everyday GIS analysis. Commonly, when classical spatial autocorrelation indicators, such as Moran's I, are used to understand a spatial pattern, the weight matrix is only built from distance, without direction being considered. Similarly, GIS operations, such as buffering, do not take direction into account either, with distance in all directions being treated equally. In reality, meanwhile, particularly in urban structures and when processes are driven by the underlying physical geography, direction plays an essential role. In this article we ask whether the development of early GIS, data (sample) sparsity, and Tobler's law lead to a theory‐induced blindness for the role of direction. If so, is it possible to envision direction becoming a first‐class citizen of equal importance to distance instead of being an afterthought only considered when the deviation from a perfect circle becomes too obvious to be ignored?

A Review of Location Encoding for GeoAI: Methods and Applications

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“…Property 2.2 is a reflection of the Generalized First Law of Geography (Zhu et al 2019b) which includes direction into the consideration of similarities. In this paper, we call a location encoder an isotropic location encoder if it only preserves the spatial proximity but ignores the variance of location embeddings when direction changes.…”

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“…Nevertheless, no matter what kinds of contexts are being faced, enhancing the estimation accuracy and reliability is a common goal that most SI methods pursue, and so does the typical SI method—inverse distance weighting (IDW) 1 , 5 , 15 – 21 . In general, the interpolation accuracy of the conventional IDW or its variants could be improved by choosing a set of appropriate parameters such as the search model of local samples or observed data 3 , 22 – 24 , the type of distance metric 19 , 25 , 26 , and the exponent imposed on the distance 7 , 22 , 23 , 27 , 28 .…”

Generation of Spatial Profiles and Mapping of Volcanic Ash Distribution

“…This law provides the foundation of the fundamental concepts in spatial dependence and spatial autocorrelation, and is utilized specifically in spatial interpolation techniques. Spatial autocorrelation (Zhu et al, 2019) is a key concept that is used to analyse the degree of dependency among observations (samples) in a given geographic space. Distance between neighbours, lengths of shared borders, and orientation are just some of the measurements used in conjunction, when modelling a given field, to estimate the unknowns.…”