## 3 Simulations of the Profile Reconstruction Using the Existing Method and the New Method
### Comparison Between the Existing Method and the New Method  
The impact of data processing error is mainly reflected in higher-order harmonic component distortion, and the stepwise variations are more seriously and significantly affected. We define the measured profiles as two irregular profiles (\(f(x)\) and \(g(x)\)), which are composed of smooth profiles and stepwise variations. \(f_{b}(x)\) and \(g_{b}(x)\), their smooth parts, are expressed as (unit: \(\upmu\)m):  
\[f_{b}(x)=22.446\sin(2\pi\cdot\frac{0.94}{200}\cdot x)-17.905\sin(2\pi\cdot\frac {0.88}{200}\cdot x)+50,\]  
\[g_{b}(x)=-44.662\sin(2\pi\cdot\frac{0.09}{200}\cdot x)-8.451\sin(2\pi\cdot \frac{0.904}{200}\cdot x)-50.\]  
The coefficients of \(f_{b}(x)\) and \(g_{b}(x)\) are random. Adding random sized pulses and stepwise variations into \(f_{b}(x)\) and \(g_{b}(x)\) at random positions, the whole measured profiles \(f(x)\) and \(g(x)\) are shown in Figure 3. The systematic error in straightness error of the reference is defined as \(e_{ys}(x)\) (unit: \(\upmu\)m):  
\[e_{ys}(x)=5\sin(2\pi\cdot\frac{0.2}{300}\cdot x)+5\sin(2\pi\cdot\frac{2}{300} \cdot x).\]  
Figure 3: Defined profiles of the measured profiles and the straightness error.  
Figure 2: Schematic of the three-probe system for reconstruction of a pair of profiles.  
Adding a Gaussian distribution, whose standard deviation is 2\(\upmu\)m, into \(e_{ys}(x)\) as the random part of the straightness error, the straightness error of the reference, which is defined as \(e_{y}(x)\) and includes the systematic part and the random part, is shown in Figure 3. When the probe spacing between \(P_{1}\) and \(P_{2}\) is defined as 20 mm and the spacing between \(P_{3}\) and \(P_{4}\) is 18 mm, the step distance of the measuring device is 2 mm according to Section 2. Considering such straightness error as well as zero-adjustment errors whose absolute values are less than 0.1 \(\upmu\)m, the outputs of probes during the scanning process are shown in Figure 4.  
The reconstruction result calculated by Equations (12)-(14) (using measurement data of three probes) is shown in Figure 5 (the linear trends of the original profiles and reconstruction profiles are deprived and reconstruction profiles are translated to the positions, which assure their first reconstruction points on the original profiles). It can be seen that this method suffers from data processing error and some features of profiles cannot be expressed, especially for the stepwise variations of the measured profiles. This type of disadvantage is consistent with existing analyses of the GTP method. The reconstruction result calculated from the outputs of four probes and Equations (5)-(8) is shown in Figure 6. In Figure 6c, it can be seen that the influence of data processing error is effectively removed. Using the new method, the data processing error can be avoided and high lateral resolution can be retained. Considering that the straightness error can be eliminated, the exact reconstruction with no error can be achieved if the yaw error and sensor measurement error are obtained by other devices or ignored. According to Figure 6b, the error curves are parallel, which means that both reconstruction profiles are rotated the same angle compared with their original profiles. Therefore, reconstruction profiles can be used to evaluate both straightness and parallelism.  
Figure 4: Measurement data of four probes.  
Figure 5: Reconstruction result impacted by data processing error.