On the relation between UHI intensity and city proximity

TL;DR

This paper analytically demonstrates that the exponential decay of UHI intensity with distance, observed empirically, is mathematically equivalent to a Gaussian function when considering the proper distance definition, clarifying previous apparent contradictions.

Contribution

It provides an explicit derivation showing the equivalence between exponential decay and Gaussian functions for UHI intensity, based on belt-based distance definitions.

Findings

Exponential decay of UHI is equivalent to a Gaussian function under certain distance definitions.

The apparent contradiction in previous studies is due to different distance measures used.

The analysis clarifies the mathematical relationship between empirical UHI decay and theoretical models.

Abstract

Recently, D. Zhou et al. 2015 studied empirically the land surface temperature of 32 Chinese cities and reported an exponentially decaying residual effect of the urban heat island (UHI) around the cities. Here we show analytically that such a form is equivalent to a previously proposed two-dimensional Gaussian function. The reason for this seeming contradiction is the way how the distance from the considered city is defined. While in the former, consecutive equal area belts around the city are used, in the latter it is the euclidean distance. In simplified terms, the definition of the belts implies a transformation of the independent variable $\rho\sim d^{\alpha}$ with $\alpha\approx 1/2$, where $\rho$ is the euclidean distance from the center and $d$ is the index of the belt. Since the belts have equal area, outer ones become more narrow. As a consequence, the Gaussian function $\sim \mathrm{e}^{-\rho^2}$ becomes an exponential cone $\sim \mathrm{e}^{-d}$. This paper provides an explicit derivation of the equivalence.