From Zipf's law to hypsometry: seeking the ‘signature’ of elevation distribution

Geographic Information Systems (GIS) users now have multiple options for using elevation data ranging from submeter to kilometers, owing to the rapid development and extensive use of geographic information technologies, such as light detection and ranging (lidar). However, such data are often provided ‘as is,’ creating a need for error propagation and validation of digital elevation data, which is the motivation for this research. We start by seeking the ‘signature’ of elevation distributions. Zipf's law and hypsometry both analyze the distribution of data by aggregation and ranking, the former for discrete data and the latter for continuous data. The objectives of this study are (1) to adopt Zipf's law for discrete nominal or ordinal data and apply it to continuous interval or ratio data; (2) to propose a uniform, parametric model of the elevation distribution of inland water basins for characterizing the overall topographic landscape; (3) to open discussion on another possible statistical method to generalize physical phenomena for geographers and other researchers; and (4) to explore new approaches for error propagation and validate digital elevation data from lidar and other sources. Combining Zipf's law and hypsometry, this article proposes a quadratic polynomial fitting of the log(value)–log(frequency) plot to study the elevation distribution of inland water basins, thus providing a holistic description of the topographic landscape of an inland water basin. Based on several experimental designs, we conclude that the method is scale independent, and it can be applied to different hierarchical levels of water basins. The vertical resolution of elevation is more sensitive than the horizontal resolution. However, the method cannot be applied to arbitrary regions or basins with outflow to the ocean, and a value shift is suggested for using the method in near sea level inland basins. This method introduces additional statistical regularity based on empirical observations of elevation data of inland water basins. It extends Zipf's law from the nominal/ordinal scale to the interval/ratio scale and extends hypsometry from a nonparametric histogram to a parametric quadratic polynomial. Future research that will specifically tackle issues in applications, such as lidar data validation and archeological site prediction models, is also discussed.