Geographical Modeling: from Characteristic Scale to Scaling

TL;DR

This paper discusses the challenge of modeling scale-free geographical phenomena and proposes scaling analysis as a method to characterize such phenomena when characteristic scales are absent.

Contribution

It introduces the concept of using scaling analysis and power exponents to model and understand scale-free geographical systems.

Findings

Scaling analysis effectively characterizes scale-free phenomena.

Power exponents follow a scaleful distribution.

Scaling methods enable mathematical modeling without characteristic scales.

Abstract

Geographical research was successfully quantified through the quantitative revolution of geography. However, the succeeding theorization of geography encountered insurmountable difficulties. The largest obstacle of geography's theorization lies in scale-free distributions of geographical phenomena which exist everywhere. The first paradigm of scientific research is mathematical theory. The key of a quantitative measurement and mathematical modeling is to find a valid characteristic scale. Unfortunately, for many geographical systems, there is no characteristic scale. In this case, the method of scaling should be employed to make a spatial measurement and carry out mathematical modeling. The basic idea of scaling is to find a power exponent using the double logarithmic linear relation between a variable scale and the corresponding measurement results. The exponent is a characteristic parameter which follows a scaleful distribution and can be used to characterize the scale-free phenomena. The importance of the scaling analysis in geography is becoming more and more evident for scientists.