Time-Geography in Four Dimensions: Potential Path Volumes around 3D Trajectories

Abstract

GIScience 2016 Short Paper Proceedings Time-Geography in Four Dimensions: Potential Path Volumes around 3D Trajectories U. Demsar 1 , J. A. Long 1 School of Geography & Geosciences, University of St Andrews, Scotland, UK Email: {urska.demsar; jed.long}@st-andrews.ac.uk Abstract An upcoming increase in availability and accuracy of 3D positioning requires development of new analytical approaches that will incorporate the third positional dimension, the elevation and model space and time as a 4D concept. In this paper we propose the extension of time geography into four dimensions. We generalise the time geography concept of a Potential Path Area into a Potential Path Volume around a 3D trajectory and present its mathematical definition. The algorithm for calculating PPVs around 3D trajectories is currently being implemented and will be tested on simulated data and real 3D data from movement ecology. 1. Introduction Movement data are collected in the form of trajectories, which are sequences of locations collected at certain times. These data are collected using trackers, such as GPS devices, which are capable of recording location in three dimensions. However, typically only the two geographical dimensions (either longitude/latitude or easting/northing) are used for analysis and the third dimension, elevation, is neglected (Belant et al. 2012). This is first because the accuracy of GPS elevation measurements is poor, but also because including the third dimension into any kind of geometrical calculations necessary for correct mathematical analysis introduces higher complexity. The upcoming deployment of the European Galileo and the Chinese COMPASS/Beidou systems are expected to improve elevation accuracy (Li et al. 2015). Therefore an upcoming critical gap is for analytical methods for 3D movement. In this paper we propose a generalisation of a well-established movement analysis framework, time geography (Hagerstrand 1970) to consider location in three dimensions. This framework originally operates in the conceptual space of a Space-Time Cube (STC), which consists of a 2D geographic plane and time as the third axis. We propose to extend this concept into a Space-Time HyperCube (STHC), which we define as a 4D space, consisting of a 3D geographic space and fourth dimension - time. By applying the conceptual extension from STC to STHC, we can mathematically generalise time geography into four dimensions. We focus on one particular time geography concept: the Potential Path Area (PPA), a popular accessibility measure in transport (Patterson and Farber 2015). A PPA is the projection of a Space-Time Prism (STP) onto the geographical plane (Miller 2005). The STP is an accessibility volume within the STC, which represents all the paths which the object could have traversed between two observed positions, P i and P i+1 (Figure 1a). In the 2D case, the PPA is an ellipse. If this ellipse is calculated around each segment of a trajectory, their union can be used to delineate the range of a moving object (Long and Nelson 2012). To extend this principle into four dimensions, we propose to generalise the PPA ellipse into an ellipsoid located within the three dimensional geographic space. This ellipsoid is the projection of the four-dimensional Space-Time Prism (the 4D accessibility volume between the two observed positions) onto the 3D base space of the STHC. We call this ellipsoid the Potential Path Volume (PPV, Figure 1b) and propose that it can be used in the same way as the Potential Path Area for trajectories where location is measured in three dimensions.