When measuring the roughness of rough surfaces, the limited sizes of scanned areas lead to its systematic underestimation. Levelling by polynomials and other filtering used in real-world processing of atomic force microscopy data increases this bias considerably. Here a framework is developed providing explicit expressions for the bias of squared mean square roughness in the case of levelling by fitting a model background function using linear least squares. The framework is then applied to polynomial levelling, for both one-dimensional and two-dimensional data processing, and basic models of surface autocorrelation function, Gaussian and exponential. Several other common scenarios are covered as well, including median levelling, intermediate Gaussian--exponential autocorrelation model and frequency space filtering. Application of the results to other quantities, such as Rq, Sq, Ra and~Sa is discussed. The results are summarized in overview plots covering a range of autocorrelation functions and polynomial degrees, which allow graphical estimation of the bias.

中文翻译：

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源自背景减法的粗糙度测量偏差的估计

在测量粗糙表面的粗糙度时，扫描区域的有限尺寸导致其系统性低估。在原子力显微镜数据的实际处理中使用多项式和其他过滤进行调平，大大增加了这种偏差。这里开发了一个框架，通过使用线性最小二乘法拟合模型背景函数，在平整情况下提供均方粗糙度偏差的显式表达式。然后将该框架应用于多项式调平，用于一维和二维数据处理，以及表面自相关函数、高斯和指数的基本模型。还涵盖了其他几种常见场景，包括中值调平、中间高斯指数自相关模型和频率空间滤波。讨论了将结果应用于其他量，例如 Rq、Sq、Ra 和~Sa。结果总结在涵盖一系列自相关函数和多项式次数的概览图中，允许对偏差进行图形估计。