## 3 Uncertainty Evaluation
\begin{array}{c}\sigma^{2}[e_{X}(x/\alpha,y)]=\big{(}\frac{g}{4\pi}\big{)}^{ 2}\big{\{}\sigma^{2}[I_{X+1}(x,y)]+\sigma^{2}[I_{X-1}(x,y)]\big{\}}+\Big{\{} \frac{E[e_{X}(x/\alpha,y)]}{g}\Big{\}}^{2}\sigma^{2}(g)\\ \hskip 14.226378 pt+2\Big{[}\frac{g}{\lambda}e_{Z}(x/\alpha,y)+\frac{g}{\lambda} \alpha\frac{\partial e_{Z}(x/\alpha,y)}{\partial\alpha}\Big{]}^{2}\sigma^{2}( \alpha)+2\big{(}\frac{g}{\lambda}\alpha\big{)}^{2}\sigma^{2}[e_{Z}(x/\alpha,y )],\end{array}  \tag{12}\]  
where the variance \(\sigma^{2}(\alpha)\) is evaluated by the following equation:  
\[\sigma^{2}(\alpha)=\sin^{2}\!\bigg{(}\frac{\theta}{2}\bigg{)}\left\{\left[ \frac{1}{2}\frac{1}{\sqrt{1-\left(\lambda/g\right)^{2}}}\frac{\lambda}{g^{2}} \right]^{2}\sigma^{2}(g)+\sigma^{2}(\epsilon)\right\}. \tag{13}\]  
The first and second terms in curly brackets of the right side of Equation (13) correspond to the uncertainties that are induced by the deviation of the nominal grating pitch to its true value and the misalignment in measurement, respectively. It should be noted from Equations (11) and (12) that, for \(e_{Z}(x,y)\), the corresponding uncertainty matrix can be obtained, while for \(e_{X}(x,y)\) it cannot, because of the change in the coordinate system after tilting the scale gating. Moreover, as shown in Equation (12), even for \(e_{X}(x/\alpha,y)\), the corresponding uncertainty matrix cannot be obtained either due to the involvement of the evaluation of the unknown \(e_{Z}(x/\alpha,y)\), \(\partial e_{Z}(x/\alpha,y)/\partial\alpha\), and \(\sigma^{2}[e_{Z}(x/\alpha,y)]\). On the other hand, it should be pointed out that the uncertainty matrix approach has a number of drawbacks in practice. As stated in [16], the information density that is contained in the uncertainty matrix is too high for users who only want a single number. In addition, the uncertainty matrix cannot easily be used for the decision of either acceptance or rejection. Evaluations of measurement uncertainty should be expressed in a manner that is informative and actionable [17]. Therefore, in this paper, a PV value is adopted to evaluate the quality of the obtained form errors. Although a PV value also has inadequacies in charactering surfaces, it is still widely recognized that a PV value remains one of the most commonly used specifications of surface form errors [18]. Next, the details are presented in evaluating a PV value and its uncertainty for the characterization of \(e_{Z}(x,y)\) and \(e_{X}(x,y)\). In this paper, the form errors \(e_{Z}(x,y)\) and \(e_{X}(x,y)\) are assumed can be characterized in terms of polynomials. This assumption is reasonable, since, as stated in [19], the low-frequency errors can be completely modeled by the set of polynomials and are of the main concern in the application. Moreover, modeling the form errors in terms of polynomials is also favorable to evaluate the associated PV values since the estimated PV values will not be biased by the high-frequency noises contained in the raw data. Without losing generality, the \(n\)-degree bivariate polynomials are used to fit the discrete form errors (\(x_{i},y_{j}\), \(E[e_{Z}(x_{i},y_{j})]\)) and (\(x_{i},y_{j}\), \(E[e_{X}(x_{i}/\alpha,y_{j})]\)) obtained at each pixel position (\(x_{i}\),\(y_{j}\)), according to Equations (9) and (10), as follows:  
\[e_{Z}(x,y)=[1,\,x,\,x^{2},\,\ldots,\,x^{n}]\left[ \begin{array}{cccc}A_{00}&A_ {01}&A_{02}&\cdots&A_{0,n}\\ A_{10}&A_{11}&\cdots&A_{1,n-1}\\ A_{20}&\cdots&A_{2,n-2}\\ \vdots\\ A_{n0}&&&0\end{array} \right]\left[ \begin{array}{c}1\\ y\\ y^{2}\\ \vdots\\ y^{n}\end{array} \right]=\mathbf{x}^{\mathrm{T}}\mathbf{A}\mathbf{y}, \tag{14}\]\[e_{X}(x/\alpha,y)=\left[1,\,x,\,x^{2},\,\ldots,\,x^{n}\right]\left[ \begin{array}{ ccccc}B^{\prime}_{00}&B^{\prime}_{01}&B^{\prime}_{02}&\cdots&B^{\prime}_{0,n}\\ B^{\prime}_{10}&B^{\prime}_{11}&\cdots&B^{\prime}_{1,n-1}\\ B^{\prime}_{20}&\cdots&B^{\prime}_{2,n-2}\\ \vdots\\ B^{\prime}_{n0}&&&0\end{array} \right]\left[ \begin{array}{c}1\\ y\\ y^{2}\\ \vdots\\ y^{n}\end{array}