Estimation Method of Ideal Fractal Parameters for Multi-Scale Measurement of Polished Surface Topography

Abstract

A surface topography characterization parameter system based on fractal parameters has been established, and several estimation methods for these fractal parameters have been suggested accordingly. Since scale dependence exists in these conventional methods, it is necessary to find an estimation method for characterization parameters with uniqueness. An estimation method for ideal fractal parameters for multi-scale measurement of polished surface topography is proposed in this study. Polished surfaces of two materials, WC-Ni and 9Cr18Mo, are measured under multi-scale for frequency component analysis. This study proposes an estimation method for ideal fractal parameters based on a modified determination method for the scale-free region and the decomposition of frequency components into three classifications. The reasonable results verify the existence of ideal fractal parameters: for the WC-Ni surface, ideal fractal dimension D = 1.3 and scale coefficient G = $2.23\times {10}^{20 }\mathsf{\mu }\mathrm{m}$; for the 9Cr18Mo surface, ideal fractal dimension D = 1.2 and scale coefficient G = $3.33\times {10}^{33} \mathsf{\mu }\mathrm{m}$. Additionally, it is revealed that the scale-dependent components conform to the same regulation on the same instrument by comparing the results of two materials. The conclusions of this study are expected to support tribology research and mechanical engineering related to surface topography.

1. Introduction

The parameter system is the basis of surface topography characterization. Traditionally, a statistical parameter system is used for such characterization [1,2,3,4,5,6]. In recent decades, with the increasing demands on describing the multi-scale intrinsic of surface topography, a fractal parameter system has been established since the self-similarity of geometries between different scales could be represented by a fractal [7,8,9,10]. Generally, the fractal parameter system includes fractal dimension D, an evaluation parameter of the space occupancy by surface roughness, and scale coefficient G, a scaling constant for surface roughness [11,12,13]. In addition, several estimation methods have been proposed for these fractal parameters. Power spectrum density (PSD) function and structure function are two common methods. Among these two methods, D and G are obtained from the slopes and intercepts of the linear region in the log-log figure of the functions [14,15,16]. In practice, fractal parameter systems and corresponding estimation methods have been widely applied to characterize different surfaces, including machined metal, anisotropy graphite, magnetic disks, and textured rocks [8,9,10,17,18].

Though remarkable progress has been achieved in surface topography characterization based on fractal parameter systems, a few issues remain to be solved in this field. Firstly, it is necessary to define the scale-free region range where fractal components exist [19,20,21,22] due to the surface topography being a combination of stationary random processes in the low-frequency range and fractal processes in the high-frequency range [1,9,23]. The scale-free region has received less attention as a prerequisite for estimating fractal parameters. Furthermore, the classification of the surface topography components has not been determined. Among existing studies, three components have been mentioned: a fractal component, a random component, and a scale-dependent component, while previous researchers [12,24] generally consider two of the three components in each study. Finally, due to the ignoring of scale-free region determination and incomplete consideration of three components, the fractal dimension D and scale coefficient G change under multi-scale measurement [25,26] though they are ideally expected to be scale-independent [27,28]. Conventional methods focus on mathematical modifications to solve this scale-dependent problem [12,24], but the intrinsic and origin of the mentioned scale-dependent components are seldom discussed.

Thus, this study aims to provide an approach to obtaining ideal unique fractal parameters, filling the gap that fractal parameters are not the same due to different measurement frequencies for the same surface. This paper is organized as follows: In Section 2, the theoretical method is proposed based on the scale-free region determination method proposed in the authors’ previous study and the components’ decomposition into three classifications; in Section 3, polished surfaces of WC-Ni and 9Cr18Mo measured under different scales are used as examples, and the ideal fractal parameters of them are calculated to verify the proposed method; and in Section 4, the three components are further discussed.

2. Theoretical Methods

According to the research of Majumdar and Bhushan [8,9,10], for surface topography, there is a (−5 − 2D)) power function relationship between the PSD P(f) of fractal components and frequency f, shown in Equation (1). Equation (1) could be represented as a linear function by a logarithmic operation, shown as Equation (2). The slope (−(5 − 2D)) and intercept ((2D − 2)log(G) − log(2ln$\gamma$)) of Equation (2) represent fractal dimension D and scale coefficient G, respectively [14,15,16]. Thus, fractal parameters could be estimated from the linear log-log PSD of fractal components.

$$P(f)=\frac{G^{2D-2}}{2\ln \gamma }\frac{1}{f^{5-2D}}$$

(1)

$$\log (P(f))=-(5-2D)\log (f)+(2D-2)\log (G)-\log (2\ln \gamma )$$

(2)

However, the log-log PSD of actual surface topography, which is represented by measured height data, could be divided into two parts (Figure 1a): a linear region at high frequencies and a constant region at low frequencies. The fractal characterizations only exist in this linear region, which is defined as the scale-free region. Thus, the determination of the scale-free region range should be one of the keys to estimating fractal parameters.

Figure 1b shows that the linear functions’ offset changed with the measurement scales. Such an offset represents the scale dependence in fractal parameters. Thus, another key to estimating fractal parameters is the elimination of such scale dependence.

In conclusion, the estimation method for ideal unique fractal parameters is proposed in this section. The main steps of this method include scale-free region determination, the decomposition of three components, and scale dependence elimination.

Additionally, two concepts of frequency need to be stated here. The natural frequency f represents the component characteristics of surface topography. Another is measurement frequency $f_s$, which represents the scale. The natural frequency f is the reciprocal of the characteristic lateral length l of surface asperity. In comparison, the measurement frequency $f_s$ is the reciprocal of the measurement interval $∆l_s$, shown in Equation (3).

$$f={\gamma }^n=\frac{1}{l}$$

(3)

$$f_S=\frac{1}{\Delta l_S}$$

(4)

2.1. Scale-Free Region Determination

A determination method was proposed in the authors’ previous study [29]. The Weierstrass–Mandelbrot (W–M) function (Equation (5)) is rewritten into an equal amplitude variant (Equation (6)), which modifies the amplitude $G^{D-1}/{\gamma }^{\left(2-D\right)n}$ into an adjustment parameter $k_n$, setting spectrum line amplitudes in PSD as 1.

$$z(x)=G^{D-1}{\displaystyle \sum _{n=n_0}^{n=n_1}\frac{\cos (2\pi {\gamma }^{n}x+{\phi }_n)}{{\gamma }^{(2-D)n}}}$$

(5)

$$z_1(x)={\displaystyle \sum _{n=n_0}^{n=n_1}{k}_n\cos (2\pi {\gamma }^{n}x+{\phi }_n)}$$

(6)

$$z_1(x)={\displaystyle {\int }_{n_0}^{n_1}{k}_n\cos (2\pi {\gamma }^{n}x+{\phi }_n)}\mathrm{d}n$$

(7)

In the previous method, the frequency index n is an integer, so that only discrete spectrum lines at specific frequencies ${\gamma }^n$ could be extracted. This study must consider continuous PSD information to analyze its components. So that the frequency index n is extended to a non-integer, modifying the discrete sum of Equation (6) into the continuous integration of Equation (7). Equation (7) could be applied as a band-pass filter, shown in Figure 2.

Figure 2a shows the introduction of the non-integer frequency index. By gradually subdividing the values of n, more spectrum lines are extracted, filling the blanks between discrete lines. The spectrum could be considered a continuous curve when subdivided to a certain extent. All components in the scale-free region could be extracted by modifying the amplitudes of such continuous curves as one and multiplying them with surface PSD. Figure 2b shows the resulting PSD and its log-log plot. The region where the PSD could be fitted as a linear function represents the scale-free region, and the frequency of the highest peak (marked as orange) represents the boundary of this scale-free region. Though the modified W–M function becomes continuous in the frequency domain, its characteristic in the spatial domain is still continuous but not differentiable.

2.2. Components Decomposition

After the scale-free region determination and inside component extraction, these components are investigated to determine the scale dependence under multi-scale measurement of surface topography. The polished surface of the 9Cr18Mo stainless steel is used as an example here. The surface is measured at intervals of 0.8$\mathsf{\mu }\mathrm{m}$ and 0.16$\mathsf{\mu }\mathrm{m}$ on a ZYGO NexView white light interferometer (Middlefield, CT, USA). Detailed information is given in Section 3.

Under single-scale, PSD in the scale-free region (Figure 3a) could be decomposed into two parts: the linear components and the fluctuation components, shown in Figure 3b. Though the fractal component is a linear function according to Equation (2), there are other components in this linear function in addition to the fractal component in Figure 3b. Figure 3c shows the PSD surface topography of the 9Cr18Mo under two measurement scales. It can be seen from Figure 3d that there is an offset between the linear components under different scales, while the fluctuation components are approximately the same. The detailed average values and variances are shown in

Table 1. Detailed parameters under the two scales.

$\mathbf{Measurement} \mathbf{Interval} \mathbf{∆}{\mathit{l}}_{\mathit{s}}$$/\mathsf{\mu }\mathbf{m}$	Slopes of the Linear Function	$\mathbf{Intercepts} \mathbf{of} \mathbf{the} \mathbf{Linear} \mathbf{Function}/\mathsf{\mu }\mathbf{m}$	$\mathbf{The} \mathbf{Average} \mathbf{Value} \mathbf{of} \mathbf{Fluctuation}/\mathsf{\mu }\mathbf{m}$	$\mathbf{Variance} \mathbf{of} \mathbf{Fluctuation}/\mathsf{\mu }\mathbf{m}$
0.8	−4.0748	18.6051	0	0.3001
0.16	−3.0414	14.8144	0	0.2815

, showing that the two fluctuations conform to approximately the same distribution.

Thus, if the fractal components are supposed to be unique, they should be mixed with scale-dependent offset components.

So, the PSD of an actual surface topography under different measurement scales could be decomposed into three components: the unique fractal components with a linear function, the fluctuation components with approximately the same parameters under every scale, and scale-dependent offset components. Therefore, the PSD of measured surface topography is the sum of these three components, shown in Equation (8).

$$\log P=\log P_{fractal}+\log P_{fluctuation}+\log P_{offset}$$

(8)

2.3. Scale-Dependence Elimination

Figure 3d shows that though linear functions could be fitted and relative offsets between them could be obtained, the ideal fractal linear function and the absolute offsets could not be decomposed through their mixed functions under each scale. The scale-dependent components could not be recognized and eliminated through data under a single scale. Measurement data collected under multi-scales must be analyzed comprehensively to obtain unique fractal parameters. Because the fluctuation components are scale-independent, they could be subtracted from the log-log PSD first, obtaining a smooth linear function that is the combination of fractal components and offset components, shown in Equation (9).

$$\log P^{\prime }=\log P-\log P_{fluctuation}=\log P_{fractal}+\log P_{offset}$$

(9)

For this linear function $\log P^{\prime }$, its slope K($f_s$) (changed with a measurement scale $f_s$) and intercept I($f_s$) could be assumed as the sum of two parts: a scale-dependent function E($f_s$) or S($f_s$) representing the offset components and a constant part (c and b) representing the ideal unique fractal components, shown in Equations (10) and (11).

$$K(f_S)=E(f_S)+c$$

(10)

$$I(f_S)=S(f_S)+b$$

(11)

Based on such an assumption, the $\log P^{\prime }$ could be written as Equation (12).

$$\log P^{\prime }=\log P_{fractal}+\log P_{offset}=K(f_S)\log (f)+I\left(f_S\right)=[E(f_S)+c]\log (f)+[S(f_S)+b]$$

(12)

The combination functions K($f_s$) and I($f_s$) could be fitted with slopes and intercepts obtained under multi-scales. If the ideal unique fractal parameters exist, the constants c and b will be obtained from the fitted K($f_s$) and I($f_s$). According to Equation (2), the ideal fractal dimension D and scale coefficient G could be obtained through Equations (13) and (14). D is calculated from the slopes of Equation (2), shown as Equation (13). G could be calculated with the obtained D and any point (log(f), log(P)) in Equation (2). Herein, the point where log(f) = 0 is selected to simplify the calculation, shown as Equation (14).

$$D=\frac{c}{2}+2.5$$

(13)

$$G={10}^{\frac{\log 10(2\ln 1.5)+b}{2D-2}}$$

(14)

3. Verification of the Experiment

3.1. Experiment Design

The method proposed in Section 2 is verified through two surface topography measurement experiments. The surfaces of WC-Ni hard alloy and 9Cr18Mo stainless steel, which are common materials for tribo-pair in seals and bearings, were selected as the experiment samples to verify if unique fractal parameters could be obtained. The different surfaces are conducted on the same instrument as the verification experiment, aiming to prove whether the regular offset components are invariant. The surface topography is measured by the ZYGO NexView white-light interferometer.

Each surface is machined in three steps, a common machining process for mechanical seal surfaces.

The detailed machining information is listed in

Table 2. Machining information of experiment sample surface of WC-Ni.

	Step1: Rough Lapping	Step2: Fine Grinding	Step3: Polishing
Abradant	W20 boron carbide/oil	W3.5 artificial diamond/water	W7 artificial diamond/water
Lapping duration	20 min	15 min	7 min
Grinding miller	Cast iron plate	Brass plate	Nylon cloth

and

Table 3. Machining information of additional experiment sample surface of 9Cr18Mo.

	Step 1: Original Grinding	Step 2: Rough Lapping	Step 3: Polishing
Abradant	10# abrasive sand	W20 boron nitride	Polishing lint
Lapping duration	N/A	20 min	5 min
Grinding miller	Cast iron plate	Cast iron plate	Nylon cloth

.

The measurement process is shown in Figure 4a; 1024 × 1024 data points are obtained from this process. Then, the obtained surface data could be divided into 1024 roughness profiles, as shown in Figure 4b. The PSD of these profiles will then be applied to calculate the fractal parameters.

The sampling point numbers of the measurement instrument are fixed at 1024 × 1024. The sample surface is measured under four different scales listed in

Table 4. Measurement information of experiment sample surface.

Times of Scopes	$\mathbf{Measurement} \mathbf{Length} {\mathit{l}}_{\mathit{s}}$$/\mathsf{\mu }\mathbf{m}$	$\mathbf{Measurement} \mathbf{Interval} \mathbf{∆}{\mathit{l}}_{\mathit{s}}$$/\mathsf{\mu }\mathbf{m}$	$\mathbf{Measurement} \mathbf{Frequency} {\mathit{f}}_{\mathit{s}}$$/{\mathbf{m}}^{-1}$
100$\times$	84	0.08	$1.25\times {10}^7$
50$\times$	167	0.16	$6.25\times {10}^6$
25$\times$	336	0.33	$3.03\times {10}^6$
10$\times$	833	0.81	$1.23\times {10}^6$

. The measurement frequencies are determined by the setting times of the scopes of the instrument. Table 4 gives the corresponding relation between the times of scopes and the measurement frequencies. According to Equation (4), the measurement frequency $f_s$ is the reciprocal of the measurement interval $∆l_s$.

The components in the scale-free region are extracted according to the method proposed in Section 2. The slopes and intercepts of 1024 roughness profiles are calculated under each scale, and the statistical average results are used to fit the functions K($f_s$) and I($f_s$), with the goal of determining the ideal fractal parameters.

3.2. Experiment Results

3.2.1. Results of WC-Ni Surface

The multi-scale measurement data from roughness profiles of the WC-Ni surface is analyzed by the steps of the proposed method, as shown in Figure 5. First, the scale-free region range is determined as 127,830 ${\mathrm{m}}^{-1}$ in Figure 5a; then, the fluctuation components are filtered, resulting in linear functions, which are combinations of fractal components and offset components in Figure 5b.

Since there are 1024 profiles measured under each scale $f_s$, these 1024 slopes and intercepts will be calculated to fit the functions $K\left(f_s\right)$ and I($f_s$). The results of 1024 slopes and 1024 intercepts under each scale (time of scope) are shown in Figure 6 and Figure 7, respectively. The average values of 1024 slopes and 1024 intercepts under each scale are listed in

Table 5. Average slopes and intercepts under different measurement frequencies (WC-Ni).

Times of Scopes	$\mathbf{Measurement} \mathbf{Frequency} {\mathit{f}}_{\mathit{s}}$$/{\mathbf{m}}^{-1}$	Average Slope	$\mathbf{Average} \mathbf{Intercept}/\mathsf{\mu }\mathbf{m}$
100$\times$	$1.25\times {10}^7$	−2.66	13.15
50$\times$	$6.25\times {10}^6$	−3.07	14.95
25$\times$	$3.03\times {10}^6$	−3.47	16.50
10$\times$	$1.23\times {10}^6$	−4.07	18.40

.

The multi-scale functions K($f_s$) and I($f_s$) of slopes and intercepts could be fitted with their average values according to Equation (12), as shown in Figure 8. The slopes and intercepts tend to converge to a certain value with increasing measurement frequency. When the measurement frequency approaches infinite, the measurement result could be regarded as the actual surface topography, which is the ideal fractal characteristic.

According to Figure 8, the fitted functions could be written as Equations (15) and (16), the goodness of fitting $R^2>0.99$. In Equations (15) and (16), the exponential functions E($f_s$) and S($f_s$) represent the offset components that changed with scales, while the existence of constants c and b represents the ideal fractal components.

$$K(f_S)=E(f_S)+c=-1.871e^{-2\times {10}^{-7}{f}_S}-2.4$$

(15)

$$I(f_S)=S(f_S)+b=7.431e^{-2\times {10}^{-7}{f}_S}+12.3$$

(16)

Based on the assumption in Section 2.3, the proposed method is verified through the existence of the constants c and b; thus, the fractal dimension D and the scale coefficient G could be obtained with uniqueness according to Equations (13) and (14). For this experiment, the results were D = 1.3 and G = $2.23\times {10}^{20}\mathsf{\mu }\mathrm{m}$. Such D and G are regarded as the ideal fractal parameters for this surface.

3.2.2. Results of 9Cr18Mo Surface

For this additional experiment, the surface topography of 9Cr18Mo is measured under three scope magnifications: 100$\times$, 50$\times$, and 10$\times$. The process is the same as that of the WC-Ni surface. The results of 1024 slopes and 1024 intercepts under each scale are shown in Figure 9 and Figure 10, respectively. The norm plots of them are also presented. Linearity is revealed from their norm plots, which represent a Gaussian random characteristic. The average values of 1024 slopes and 1024 intercepts under each scale are listed in

Table 6. Average slopes and intercepts under different measurement frequencies (9Cr18Mo).

Times of Scopes	$\mathbf{Measurement} \mathbf{Frequency} {\mathit{f}}_{\mathit{s}}$$/{\mathbf{m}}^{-1}$	Average Slope	$\mathbf{Average} \mathbf{Intercept}/\mathsf{\mu }\mathbf{m}$
100$\times$	$1.25\times {10}^7$	−2.84	14.39
50$\times$	$6.25\times {10}^6$	−3.22	15.72
10$\times$	$1.23\times {10}^6$	−4.34	20.01

.

The average values of slopes and intercepts could be fitted according to Equations (17) and (18), as shown in Figure 11.

According to Figure 11, the fitted functions could be written as Equations (17) and (18), the goodness of fitting $R^2>0.99$. In Equations (17) and (18), the exponential functions E($f_s$) and S($f_s$) represent the offset components that change with scales, while the constants c and b represent the unique fractal components.

$$K(f_S)=E(f_S)+c=-1.935e^{-2\times {10}^{-7}{f}_S}-2.6$$

(17)

$$I(f_S)=S(f_S)+b=7.532e^{-2\times {10}^{-7}f}+13.5$$

(18)

The proposed method is verified through the existence of the constants c and b, the fractal dimension D, and the scale coefficient G, which could be obtained with uniqueness. For this experiment, the resulted ideals were D = 1.2 and G = $3.33\times {10}^{33}\mathsf{\mu }\mathrm{m}$. Such D and G are regarded as the ideal fractal parameters for this surface.

4. Discussion

The sections above propose methods to analyze the fractal components and obtain ideal fractal parameters. Here, the characteristics of fluctuation and offset components in the experiment results and the scale-independent fractal parameters are discussed further.

4.1. Fluctuation Components

According to Equation (9), the fluctuation components could be obtained by subtracting the fitted linear function shown in Equation (19).

$$\log P_{fluctuation}=\log P-\log P^{\prime }=\log P-\log P_{fractal}-\log P_{offset}$$

(19)

The fluctuation components of the WC-Ni surface and the 9Cr18Mo surface under multi-scales are obtained according to Equation (19), as shown in Figure 12.

Table 7. Detailed values of fluctuation components in WC-Ni surfaces.

Times of Scopes	$\mathbf{Measurement} \mathbf{Frequency} {\mathit{f}}_{\mathit{s}}$$/{\mathbf{m}}^{-1}$	$\mathbf{The} \mathbf{Average} \mathbf{Value} \mathbf{of} \mathbf{Fluctuation}/\mathsf{\mu }\mathbf{m}$	$\mathbf{Variance} \mathbf{of} \mathbf{Fluctuation}/\mathsf{\mu }\mathbf{m}$
10$\times$	$1.23\times {10}^6$	0	0.3688
25$\times$	$3.03\times {10}^6$	0	0.3280
50$\times$	$6.25\times {10}^6$	0	0.2750
100$\times$	$1.25\times {10}^7$	0	0.2985

and

Table 8. Detailed values of fluctuation components on 9Cr18Mo surfaces.

Times of Scopes	$\mathbf{Measurement} \mathbf{Frequency} {\mathit{f}}_{\mathit{s}}$$/{\mathbf{m}}^{-1}$	$\mathbf{The} \mathbf{Average} \mathbf{Value} \mathbf{of} \mathbf{Fluctuation}/\mathsf{\mu }\mathbf{m}$	$\mathbf{Variance} \mathbf{of} \mathbf{Fluctuation}/\mathsf{\mu }\mathbf{m}$
10$\times$	$1.23\times {10}^6$	0	0.3001
50$\times$	$6.25\times {10}^6$	0	0.2815
100$\times$	$1.25\times {10}^7$	0	0.3440

list the detailed average values and variances of these fluctuation components. It could be revealed that the fluctuation components under each scale have approximately the same values, so the fluctuation components could be regarded as scale-independent.

4.2. Offset Components

The same instrument measures the surfaces of WC-Ni and 9Cr18Mo. According to Equations (15) to (18), the regular pattern of the multi-scale fitted slopes and intercepts function is approximately the same trend for the two surfaces, as shown in Figure 13.

The similarity could be evaluated through the relative errors between the exponential parts of the two surfaces, as shown in Equations (20) and (21). The relative errors in Figure 14a,b are 3.33% and 1.34%, respectively.

$$e_K(f_S)=\frac{E_{WC-Ni}(f_S)-E_{9Cr18Mo}(f_S)}{E_{9Cr18Mo}(f_S)}$$

(20)

$$e_I(f_S)=\frac{S_{WC-Ni}(f_S)-S_{9Cr18Mo}(f_S)}{S_{9Cr18Mo}(f_S)}$$

(21)

Thus, it could be concluded that, on the same instrument, the same offset functions E($f_s$) and S($f_s$) are shared by the fitted functions K($f_s$) and I($f_s$) while their constants c and b are different. In other words, it could be revealed that the offset function is caused by the measurement instrument’s characteristics when noteworthy fractal characteristics exist. On this basis, a practical inspiration is that the offset function of an instrument can be calibrated before the formal measurement experiment and subtracted from the results to obtain the fractal components and parameters independent of the scale.

4.3. Fractal Components

In Section 3.2, the functions of offset components E($f_s$) and S($f_s$) (Equations (15) to (18)) have been calibrated. In this section, the calibrated function values of offset components are eliminated from the original slopes and intercepts values under each scale (see Equations (22) and (23)), and fractal parameters are estimated for each scale.

$$c_{es}=K_{origin}(f_S)-E(f_S)$$

(22)

$$b_{es}=I_{origin}(f_S)-S(f_S)$$

(23)

By checking the differences between the ideal parameters and the estimated parameters, the scale independence of the fractal parameters was verified. Taking the data of the measured surface of WC-Ni as an example, the results of the fractal dimension and scale coefficient, under each scale, before and after eliminating the influence of offset, are shown in Figure 14.

The comparison results and their errors between the ideal fractal parameters obtained in Section 3.2.1 (D = 1.3, G = $2.23\times {10}^{20}\mathsf{\mu }\mathrm{m}$) are listed in

Table 9. Fractal dimension D obtained by different methods and their relative errors between the ideal D.

$\mathbf{Measurement} \mathbf{Frequency} {\mathit{f}}_{\mathit{s}}$$/{\mathbf{m}}^{-1}$	Ideal Fractal Dimension	Estimated Fractal Dimension	Relative Errors	Original Fractal Dimension	Relative Errors
$1.23\times {10}^6$	1.3	1.1963	7.98%	0.465	64.23%
$3.03\times {10}^6$		1.2752	1.91%	0.765	41.15%
$6.25\times {10}^6$		1.2380	4.77%	0.97	25.38%
$1.25\times {10}^7$		1.2468	4.09%	1.17	10%

and

Table 10. Scale coefficient logG obtained by different methods and their relative errors between the ideal logG.

$\mathbf{Measurement} \mathbf{Frequency} {\mathit{f}}_{\mathit{s}}$	$\mathbf{Ideal} \mathbf{Scale} \mathbf{Coefficient}/\mathsf{\mu }\mathbf{m}$	$\mathbf{Estimated} \mathbf{Cale} \mathbf{Coefficient}/\mathsf{\mu }\mathbf{m}$	Relative Errors	$\mathbf{Original} \mathbf{Cale} \mathbf{Coefficient}/\mathsf{\mu }\mathbf{m}$	Relative Errors
$1.23\times {10}^6$	20.3483	20.8309	2.37%	30.5150	49.96%
$3.03\times {10}^6$		20.5919	1.20%	27.3483	34.40%
$6.25\times {10}^6$		21.2166	4.27%	24.7650	21.71%
$1.25\times {10}^7$		20.7483	1.97%	21.7650	6.96%

. Since the errors of G before elimination are numerous, logarithmic G is applied here to calculate errors. The proposed method reveals that the scale-dependent errors are remarkably reduced. Thus, this result shows that the proposed method could be applied generally to analyze surfaces with fractal characteristics.

5. Conclusions

In this paper, an estimation method for ideal fractal parameters for multi-scale measurement of polished surface topography is launched. The detailed conclusions are listed as follows:

(1)

The method to obtain ideal fractal parameters of the polishing surface measured under multi-scales is proposed: firstly, determining the scale-free region through a modified W–M function with a non-integer frequency index; then decomposing the surface PSD into three components: fractal, fluctuation, and offset; and finally, obtaining the fractal components by filtering the fluctuation components and eliminating the scale-dependent influence of the offset components.

(2)

Experiments are designed to verify the proposed method. Polished WC-Ni and 9Cr18Mo surfaces are measured under multiscales by the same instrument. The results show that ideal fractal parameters were obtained for the two surfaces: for the WC-Ni surface, ideal fractal dimension D = 1.3 and scale coefficient G = $2.23\times {10}^{20}$; for the 9Cr18Mo surface, ideal fractal dimension D = 1.2 and scale coefficient G = $3.33\times {10}^{33}$.

(3)

Experiment results show that the offset components have the same regular pattern for different surfaces on the applied measurement instrument in this study. Thus, scale-independent fractal parameters were obtained by eliminating the calibration error of the offset components under each scale. The maximum relative error between the obtained fractal parameters and ideal parameters is 7.98%. The experiment results also reveal that the fluctuation components are scale-independent.