Uncertainties Associated with Arithmetic Map Operations in GIS

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.

Quantifying mean grades and uncertainties from the ratio of service variables: accumulation to thickness. Applied to a phosphate deposit in Southern Mato Grosso, Brazil

“…Thus, we have the same equation as (4):Actually, this is an advantage of the Taylor method because it allows computation of the variance of the function without knowing the shapes of input distributions (Maskey and Guinot, 2003). Yamamoto et al (2018) developed mathematical expressions for calculating mean and variance for arithmetically combined variables. For the ratio function the mean or the mathematical expectation around a point θ=( μ x , μ y ) and the variance are calculated as:…”

“…Where σ xy is the covariance between x and y. Details of the mathematical development of Equations (5) and (6) can be found in Yamamoto et al (2018).…”

“…Moreover, Equation (5) is still an approach because there is no exact formula for the mean of the ratio of x to y, since the function f (x,y)=x⁄y is infinitely differentiable with respect to y. Yamamoto et al (2018) also provided a formula for the mean after the third order Taylor expansion.…”

“…The first three terms on the right side of Equation (6) are related to the first order expansion (equation 4). Yamamoto et al (2018) showed that this equation is a much better approach than just keeping the first order terms.…”

“…The mean grade and its variance can be calculated by mathematical expressions. Herein, the equations developed for the calculation of mean grade and variance by Yamamoto et al (2018) are tested in a real database of research in a stratiform phosphate mineral deposit located in the southern state of Mato Grosso.…”