Analytical Estimate of Digital Elevation Model Accuracy (Mean Square Error) as a Function of Cell Size and Dataset Elevation Error

Modern digital elevation models (DEMs) are generated with increasingly small cell sizes and high accuracy, yet no closed-form relationship links these variables. Previous work relied on empirical formulations validated experimentally. This paper derives an analytical relationship for the traditional elevation mean square error (MSE) accuracy metric and highlights limitations of empirical approaches. Common practice wrongly dismisses the effect of the choice of interpolation method on DEM accuracy. Accuracy reports may even fail to mention which interpolation method was used. The analysis was limited to two local interpolation methods: nearest neighbor and bilinear interpolation. In those cases we show that the model accuracy is bounded by three terms: one depending only on the dataset elevation MSE; mixed monomials depending on both the dataset elevation MSE and powers of the cell size h; and a term independent of the dataset elevation MSE and proportional to a power of h. For a given interpolation method, the exponents of h are constant, whereas the coefficients can be estimated from local terrain characteristics at the control points. For bilinear interpolation, the exponents are twice those for nearest-neighbor interpolation. The availability of this analytical relationship makes it possible to determine the cell size or instrument accuracy required to achieve a specified target DEM accuracy. Because the result depends strongly on the interpolation method, the recommended interpolation method should be routinely and explicitly mentioned in the DEM producer’s accuracy report