# Error Analysis of Integrated Absorbance for TDLAS in a Nonuniform Flow Field
## 2 Materials and Methods
### Error Analysis
#### 2.2.3 Error Analysis under a Nonuniform Flow Field  
In a nonuniform flow field, the IA error is affected by multiple physical quantities, which is the difficulty of error analysis. Based on the analysis in the previous section, we can find that it may be simple to analyze the error from the perspective of line-shape parameters. At this time, we only need to consider three parameters (i.e., the line damping parameter \(y_{i}\), normalization coefficient \(x^{\prime}_{i}\), and \(A_{i}\)). Focusing on the absorption signal of the general nonuniform flow field, the effects of the line-shape parameters on the IA error are analyzed with the line-shape fitting method. For the convenience of analysis, the number of quasi-uniform flow field areas is set at two. The corresponding physical background is a flow field with a boundary layer. By substituting Equations (12) and (13) into Equations (14) and (16) yields, we have,  
\[\chi^{2}(\beta=(\
u_{f},A_{f},x^{\prime}_{f},y_{f}))=\sum_{j=1}^{j=n}\Biggl{[} \begin{array}{c}A_{f}\frac{1}{\sqrt{\pi}}x^{\prime}_{f}K(x^{\prime}_{f}v^{ \prime},y_{f})-\\ (A_{1}\frac{1}{\sqrt{\pi}}x^{\prime}_{1}K(x^{\prime}_{1}v^{\prime},y_{1})+A_{ 2}\frac{1}{\sqrt{\pi}}x^{\prime}_{2}K(x^{\prime}_{2}v^{\prime},y_{2}))\end{array} \Biggr{]}^{2}. \tag{17}\]  
\[R=\frac{\int_{-\infty}^{+\infty}}{\begin{array}{c}A_{f}\frac{1}{\sqrt{\pi}} x^{\prime}_{f}K(x^{\prime}_{f}v^{\prime},y_{f})-\\ (A_{1}\frac{1}{\sqrt{\pi}}x^{\prime}_{1}K(x^{\prime}_{1}v^{\prime},y_{1})+A_{ 2}\frac{1}{\sqrt{\pi}}x^{\prime}_{2}K(x^{\prime}_{2}v^{\prime},y_{2}))dv_{j} \\ \int_{-\infty}^{+\infty}A_{1}\frac{1}{\sqrt{\pi}}x^{\prime}_{1}K(x^{\prime}_{ 1}v^{\prime},y_{1})+A_{2}\frac{1}{\sqrt{\pi}}x^{\prime}_{2}K(x^{\prime}_{2}v^ {\prime},y_{2})dv_{j}.\end{array}} \tag{18}\]  
An analysis of Equations (17) and (18) reveals that when \(x^{\prime}_{i}\) and \(y_{i}\) are unchanged and \(A_{i}\) changes by a multiple, the minimum \(\chi^{2}(\beta)\) is obtained by multiplying the fitting parameter \(A_{f}\) by the same multiple. Therefore, the error \(R\) does not change. Similarly, because \(x^{\prime}_{i}\) is a line-shape normalization coefficient, when \(A_{i}\) and \(y_{i}\) are unchanged and \(x^{\prime}_{i}\) changes by a multiple, \(x^{\prime}_{f}\) obtained by the fitting changes by the same multiple and \(A_{f}\) obtained in the fitting does not change. Therefore, the error \(R\) does not change. It is easily seen that the above analysis is also applicable for \(n>2\).  
Therefore, when the number \(n\) of quasi-uniform flow field areas is determined, the error only depends on \(y_{i}\) of each quasi-uniform flow field area, the relative ratio \({x^{\prime}}_{1}:{x^{\prime}}_{2}:{x^{\prime}}_{3}:\dots:{x^{\prime}}_{i}\) of each \({x^{\prime}}_{i}\), and the relative ratio \(A_{1}:A_{2}:A_{3}:\dots:A_{i}\) of each \(A_{i}\). For the flow field comprising \(n\) quasi-uniform flow field areas, the IA error distribution can be calculated from the line damping parameter range, normalized coefficient ratio range, and IA ratio range of each flow field area.