Inconsistency between height-height correlation and power-spectrum functions of scale-invariant surfaces for roughness exponent α∼1

The height-height correlation function H(r) from a scale-invariant surface is compared with the corresponding power-spectrum function W(q) using a variety of mathematical scaling functions. We show that for a non-self-affine surface with the roughness exponent {α}{≥}1, one of the asymptotic scaling relations, either H(r){~}r^2α or W(q){~}q^-2α-d, can be violated. An inconsistency in the values of {α} also exists between H(r) and W(q) when {α}>0.9. The impact on the data analysis using different experimental methods is discussed.