Uncertainties Associated with Arithmetic Map Operations in GIS

Jorge K Yamamoto,Guilherme J Rampazzo,Claudio B.B Leite,Antônio T Kikuda

doi:10.1590/0001-3765201820170589

Jorge K Yamamoto, Guilherme J Rampazzo + Show 2 more

Open Access

https://doi.org/10.1590/0001-3765201820170589

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Abstract

Arithmetic map operations are very common procedures used in GIS to combine raster maps resulting in a new and improved raster map. It is essential that this new map be accompanied by an assessment of uncertainty. This paper shows how we can calculate the uncertainty of the resulting map after performing some arithmetic operation. Actually, the propagation of uncertainty depends on a reliable measurement of the local accuracy and local covariance, as well. In this sense, the use of the interpolation variance is proposed because it takes into account both data configuration and data values. Taylor series expansion is used to derive the mean and variance of the function defined by an arithmetic operation. We show exact results for means and variances for arithmetic operations involving addition, subtraction and multiplication and that it is possible to get approximate mean and variance for the quotient of raster maps.

Highlights

A map resulting from interpolation of field data must have some assessment of uncertainty (e.g. Heuvelink et al 1989, Crosetto et al 2000, Atkinson and Foody 2002)
Propagation of uncertainty coming from arithmetic operations between random variables is well known in statistics (Mood et al 1974)
Heuvelink et al (1989) and Heuvelink (1998) have established the method of calculating mean and variances of the output raster map from several input maps based on Taylor method

Summary

Introduction

A map resulting from interpolation of field data must have some assessment of uncertainty (e.g. Heuvelink et al 1989, Crosetto et al 2000, Atkinson and Foody 2002). When estimates are derived from field data, they have associated uncertainties caused by spatial variation of continuous variables, interactions among them and the effect of neighbor data (Wang et al 2005). Kriging is used to predict values of the variable of interest at unsampled locations, because it provides an assessment of uncertainty as given by the kriging variance. The kriging variance does not depend on real local data values (Atkinson and Foody 2002) and this is not a reliable measure of local accuracy. According to Wang et al (2005), local estimates are strongly associated with neighbor data.

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