Accuracy of roughness exponent measurement methods

Jan Øystein Haavig Bakke and Alex Hansen

Phys. Rev. E 76, 031136 – Published 27 September 2007

Abstract

We test methods for measuring and characterizing rough profiles with emphasis on measurements of the self-affine roughness exponent, and describe a simple test to separate between roughness exponents originating from long range correlations in the signs of the profile, and roughness exponents originating from Lévy distributions of jumps. Based on tests on profiles with known roughness exponents we find that the power spectrum density analysis and the averaged wavelet coefficients method give the best estimates for roughness exponents in the range $0.1$ – $0.9$ . The error bars are found to be less than $0.03$ for profile lengths larger than $256$ , and there is no systematic bias in the estimates. We present quantitative estimates of the error bars and the systematic error and their dependence on the value of the roughness exponent and the profile length. We also quantify how power-law noise can modify the measured roughness exponent for measurement methods different from the power spectrum density analysis and the second order correlation function method.

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Received 14 June 2007

DOI:https://doi.org/10.1103/PhysRevE.76.031136

Authors & Affiliations

Jan Øystein Haavig Bakke^* and Alex Hansen^†

Department of Physics, Norwegian University of Science and Technology, N–7491 Trondheim, Norway

^*Jan.Bakke@ntnu.no
^†Alex.Hansen@ntnu.no

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Vol. 76, Iss. 3 — September 2007

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Images

Figure 1
(Color online) If a surface of a given linear size $L_{1}$ is with $L_{2} = λ L_{1}$ in the horizontal direction, the surface must be scaled with $h (λ x) = λ^{- ζ} h (x)$ in the vertical direction to be statistically similar. $ζ$ is the roughness, or Hurst exponent, measuring the degree of anisotropy. $ζ = 1$ gives self-similar scaling. In this figure the part of the black (upper) profile, which is inside the square, is rescaled with the scaling relation above as the red (lower) profile.Reuse & Permissions
Figure 2
(Color online) Sample profiles from the Voss algorithm (a) and the wavelet algorithm (b). The profiles are from top to bottom for roughnesses $0.2$ , $0.5$ , and $0.8$ . To make the scale of the fluctuations the same for samples with different $ζ$ the samples are rescaled with $W_{ζ} (L) ∕ W_{0.5} (L)$ .Reuse & Permissions
Figure 3
(Color online) Measurements of the roughness exponent using intrinsic window methods. From the top: The local max-min method [Eq. (4)], the variable bandwidth method [Eq. (2)], and the detrended fluctuation analysis [Eq. (3)]. The straight lines show $ζ = 0.6$ .Reuse & Permissions
Figure 4
(Color online) Measurements of the roughness exponent using the power spectrum density analysis [Eq. (6)]. The straight line shows $ζ = 0.6$ .Reuse & Permissions
Figure 5
(Color online) Measurements of the roughness exponent using the averaged wavelet coefficients method [Eq. (7)]. The straight line shows $ζ = 0.6$ .Reuse & Permissions
Figure 6
(Color online) Measurements of the roughness exponent using second order correlation function [Eq. (5)]. The straight line shows $ζ = 0.6$ .Reuse & Permissions
Figure 7
(Color online) Measurements of the roughness exponent using the scaling of $σ (l)$ for the distribution $p (Δ H, l)$ The straight line shows $ζ = 0.6$ .Reuse & Permissions
Figure 8
(Color online) $ζ_{m e a s u r e d} - ζ_{t r u e}$ for profiles generated by the Voss algorithm. (a) The detrended fluctuation analysis $(\times)$ , the variable bandwidth $(+)$ , the max-min $(엯)$ , and the standard deviation of $Δ h (l)$ $(●)$ . (b) Second order correlation function $(●)$ , power spectrum density analysis $(\times)$ , and averaged wavelet coefficients $(엯)$ . Error bars are only shown where they are larger than the symbol size.Reuse & Permissions
Figure 9
(Color online) $ζ_{m e a s u r e d} - ζ_{t r u e}$ for profiles generated by the wavelet algorithm. (a) The detrended fluctuation analysis $(\times)$ , the variable bandwidth $(+)$ , the max-min $(엯)$ , and the standard deviation of $Δ h (l)$ $(●)$ . (b) Second order correlation function $(●)$ , power spectrum density analysis $(\times)$ , and averaged wavelet coefficients $(엯)$ . Error bars are only shown where they are larger than the symbol size.Reuse & Permissions
Figure 10
(Color online) $R_{k} ∕ R_{k}^{G}$ for profiles of length 512 made by (a) the Voss algorithm and (b) the wavelet method for $ζ = 0.7$ .Reuse & Permissions
Figure 11
(Color online) $p (Δ h, l)$ for profiles of length $512$ made by (a) the Voss algorithm and (b) the wavelet method for $ζ = 0.7$ . Three different length scales are presented here: $l = 64 (\times)$ , $l = 128$ $(+)$ , and $l = 256$ $(엯)$ The solid lines are Gaussian fits of the $l = 128$ data.Reuse & Permissions
Figure 12
(Color online) $ζ_{m e a s u r e d} - ζ_{t r u e}$ for $h_{0} (x)$ for profiles generated with the Voss algorithm $L = 512$ . (a) The detrended fluctuation analysis $(\times)$ , the variable bandwidth $(+)$ , the max-min $(엯)$ , and the standard deviation of $Δ h (l)$ $(●)$ . (b) Second order correlation function $(●)$ , power spectrum density analysis $(\times)$ , and averaged wavelet coefficients $(엯)$ . Error bars are only shown where they are larger than the symbol size.Reuse & Permissions
Figure 13
(Color online) Systematic errors in roughness exponent measurements versus system size for $ζ_{t r u e} = 0.1$ . (a) The detrended fluctuation analysis $(\times)$ , the variable bandwidth $(+)$ , the max-min $(엯)$ , and the standard deviation of $Δ h (l)$ $(●)$ . (b) Second order correlation function $(●)$ , power spectrum density analysis $(\times)$ , and averaged wavelet coefficients $(엯)$ . Error bars are only shown where they are larger than the symbol size.Reuse & Permissions
Figure 14
(Color online) Systematic errors in roughness exponent measurements versus system size for $ζ_{t r u e} = 0.2$ . (a) The detrended fluctuation analysis $(\times)$ , the variable bandwidth $(+)$ , the max-min $(엯)$ , and the standard deviation of $Δ h (l)$ $(●)$ . (b) Second order correlation function $(●)$ , power spectrum density analysis $(\times)$ , and averaged wavelet coefficients $(엯)$ . Error bars are only shown where they are larger than the symbol size.Reuse & Permissions
Figure 15
(Color online) Systematic errors in roughness exponent measurements versus system size for $ζ_{t r u e} = 0.3$ . (a) The detrended fluctuation analysis $(\times)$ , the variable bandwidth $(+)$ , the max-min $(엯)$ , and the standard deviation of $Δ h (l)$ $(●)$ . (b) Second order correlation function $(●)$ , power spectrum density analysis $(\times)$ , and averaged wavelet coefficients $(엯)$ . Error bars are only shown where they are larger than the symbol size.Reuse & Permissions
Figure 16
(Color online) Systematic errors in roughness exponent measurements versus system size for $ζ_{t r u e} = 0.4$ . (a) The detrended fluctuation analysis $(\times)$ , the variable bandwidth $(+)$ , the max-min $(엯)$ , and the standard deviation of $Δ h (l)$ $(●)$ . (b) Second order correlation function $(●)$ , power spectrum density analysis $(\times)$ , and averaged wavelet coefficients $(엯)$ . Error bars are only shown where they are larger than the symbol size.Reuse & Permissions
Figure 17
(Color online) Systematic errors in roughness exponent measurements versus system size for $ζ_{t r u e} = 0.5$ . (a) The detrended fluctuation analysis $(\times)$ , the variable bandwidth $(+)$ , the max-min $(엯)$ , and the standard deviation of $Δ h (l)$ $(●)$ . (b) Second order correlation function $(●)$ , power spectrum density analysis $(\times)$ , and averaged wavelet coefficients $(엯)$ . Error bars are only shown where they are larger than the symbol size.Reuse & Permissions
Figure 18
(Color online) Systematic errors in roughness exponent measurements versus system size for $ζ_{t r u e} = 0.6$ . (a) The detrended fluctuation analysis $(\times)$ , the variable bandwidth $(+)$ , the max-min $(엯)$ , and the standard deviation of $Δ h (l)$ $(●)$ . (b) Second order correlation function $(●)$ , power spectrum density analysis $(\times)$ , and averaged wavelet coefficients $(엯)$ . Error bars are only shown where they are larger than the symbol size.Reuse & Permissions
Figure 19
(Color online) Systematic errors in roughness exponent measurements versus system size for $ζ_{t r u e} = 0.7$ . (a) The detrended fluctuation analysis $(\times)$ , the variable bandwidth $(+)$ , the max-min $(엯)$ , and the standard deviation of $Δ h (l)$ $(●)$ . (b) Second order correlation function $(●)$ , power spectrum density analysis $(\times)$ , and averaged wavelet coefficients $(엯)$ . Error bars are only shown where they are larger than the symbol size.Reuse & Permissions
Figure 20
(Color online) Systematic errors in roughness exponent measurements versus system size for $ζ_{t r u e} = 0.8$ . (a) The detrended fluctuation analysis $(\times)$ , the variable bandwidth $(+)$ , the max-min $(엯)$ , and the standard deviation of $Δ h (l)$ $(●)$ . (b) Second order correlation function $(●)$ , power spectrum density analysis $(\times)$ , and averaged wavelet coefficients $(엯)$ . Error bars are only shown where they are larger than the symbol size.Reuse & Permissions
Figure 21
(Color online) Systematic errors in roughness exponent measurements versus system size for $ζ_{t r u e} = 0.9$ . (a) The detrended fluctuation analysis $(\times)$ , the variable bandwidth $(+)$ , the max-min $(엯)$ , and the standard deviation of $Δ h (l)$ $(●)$ . (b) Second order correlation function $(●)$ , power spectrum density analysis $(\times)$ , and averaged wavelet coefficients $(엯)$ . Error bars are only shown where they are larger than the symbol size.Reuse & Permissions
Figure 22
(Color online) Two sample profiles of length $1024$ , the one with solid lines is made with the Voss algorithm with $ζ_{t r u e} = 0.7$ , and the dashed one is a random walk with Lévy-distributed jumps with $α = 1.5$ .Reuse & Permissions
Figure 23
(Color online) Roughness exponent measured with detrended fluctuation analysis for a profile made with the Voss algorithm with $ζ_{t r u e} = 0.7$ (top) and from a random walk with Lévy-distributed jumps with $α = 1.5$ (bottom). Both the straight dashed lines in the figure show $ζ = 0.7$ . 100 samples of length for each type of profile were used in the calculations of the roughness exponent.Reuse & Permissions
Figure 24
(Color online) Roughness exponent measured with PSD for a profile made with the Voss algorithm with $ζ_{t r u e} = 0.7$ (bottom) and from a random walk with Lévy-distributed jumps with $α = 1.5$ (top). The corresponding straight lines are for $ζ = 0.5$ (top) and $ζ = 0.7$ (bottom). 100 samples of length for each type of profile were used in the calculations of the roughness exponent.Reuse & Permissions
Figure 25
(Color online) Roughness measurements on random walks with power-law distributed jumps. The detrended fluctuation analysis $(\times)$ , the variable bandwidth method $(+)$ , and the power spectrum density analysis $(엯)$ . The solid line is the roughness exponent for a Lévy flight $ζ_{L F} = 1 ∕ α$ .Reuse & Permissions
Figure 26
(Color online) $R_{k} ∕ R_{k}^{G}$ for random walks of length $1024$ with a power-law step size distribution with exponent (a) $1.5$ and (b) $3.0$ .Reuse & Permissions

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