## 5 Evaluation of the Fusion Effect of Semi-Physical Simulation and Field Data
Then, the joint probability density of \(Y_{i}^{B}=\left\{x_{1i}^{B},x_{2i}^{B},\cdots,x_{N_{B}i}^{B}\right\}\) is:  
\[p_{Y_{i}^{B}}\Big{(}x_{1i}^{B},x_{2i}^{B},\cdots,x_{N_{B}i}^{B}\mid\mu_{i} \Big{)}=\left(\frac{1}{\sqrt{2\pi}\sigma_{i}^{2}}\right)^{N_{B}}\exp\left\{- \frac{1}{2\sigma_{i}^{2}}{\sum_{j=1}^{N_{B}}{{(x_{j}^{B}-\mu_{i})}^{2}}}\right\} \tag{10}\]Assuming that the data \(Y_{i}^{A}=\left\{x_{1i}^{A},x_{2i}^{A},\cdots,x_{N_{Ai}}^{A}\right\}\) of the \(i\)-th dimension of the semi-physical simulation data source obey a normal distribution \(N(\theta_{i},\tau_{i}^{2})\), then take \(\mu_{i}\) as the prior distribution of \(N(\theta_{i},\tau_{i}^{2})\), that is:  
\[\pi(\mu_{i})=\frac{1}{\sqrt{2\pi\tau_{i}}}\exp\left\{-\frac{(\mu_{i}-\theta_{i} )^{2}}{2\tau_{i}^{2}}\right\},-\infty<\mu_{i}<+\infty \tag{11}\]  
where \(\theta_{i}\) and \(\tau_{i}^{2}\) are obtained by moment estimation using the data \(Y_{i}^{A}=\left\{x_{1i}^{A},x_{2i}^{A},\cdots,x_{N_{Ai}}^{A}\right\}\) of the \(i\)-th dimension of the semi-physical simulation data source. Then, \(\hat{\theta}_{i}=\overline{y}_{i}^{A}=\frac{1}{N_{A}}\sum\limits_{j=1}^{N_{A} }x_{ji}^{A}\), \(\hat{\tau}_{i}^{2}=s^{2}=\frac{1}{N_{A}-1}\sum\limits_{j=1}^{N_{A}}\left(x_{ ji}^{A}-\overline{y}_{i}^{A}\right)^{2}\). Thus, the posterior density of \(\mu_{i}\) is:  
\[ \begin{split} cch(\mu_{i}\Big{|}Y_{i}^{B})&\propto \pi(\mu_{i})p_{Y_{i}^{B}}\big{(}x_{1i}^{B},x_{2i}^{B},\cdots,x_{N_{Bi}}^{B}| \mu_{i}\big{)}\\ &\propto(2\pi)^{\frac{-(N_{Bi}-1)}{2}}\tau^{-1}\sigma_{i}^{-N_{B }}\exp\left\{-\frac{1}{2}\left[\frac{\sum\limits_{j=1}^{N_{B}}\left(x_{ji}^{B} -\mu_{i}\right)}{\sigma_{i}^{2}}+\frac{(\mu_{i}-\theta_{i})^{2}}{\tau_{i}^{2} }\right]\right\}\\ &\propto\exp\left\{-\frac{1}{2}\left[\frac{N_{B}\mu_{i}^{2}-2N_{ B}\mu_{i}\overline{x}_{i}^{B}+\frac{\mu_{i}^{2}-2\mu_{i}\theta_{i}}{\tau_{i}^{2} }}{\sigma_{i}^{2}}+\frac{\mu_{i}^{2}-2\mu_{i}\theta_{i}+\theta_{i}^{2}}{\tau_ {i}^{2}}\right]\right\}\\ &\propto\exp\left\{-\frac{1}{2}\left[\frac{\mu_{i}^{2}-2\mu_{i} \overline{x}_{i}^{B}}{\sigma_{i}^{2}}+\frac{\mu_{i}^{2}-2\mu_{i}\theta_{i}}{ \tau_{i}^{2}}\right]\right\}\\ &\propto\exp\left\{-\frac{\frac{1}{\sigma_{i}^{2}}+\frac{1}{\tau _{i}^{2}}}{2}\left(\mu_{i}-\frac{\overline{x}_{i}^{2}}{\sigma_{i}^{2}}+\frac{ \theta_{i}}{\tau_{i}^{2}}\right)^{2}\right\}\\ \end{split}  \tag{12}\]  
It can be seen from the posterior density that the posterior distribution of the unknown parameters is also a normal distribution, with a mean of \(\hat{\mu}_{i}=\frac{\frac{\overline{x}_{i}^{B}}{\sigma_{i}^{2}}+\frac{\theta_ {i}}{\tau_{i}^{2}}}{\frac{\overline{x}_{i}^{2}}{\sigma_{i}^{2}}+\frac{1}{\tau _{i}^{2}}}=\frac{\overline{x}_{i}^{B}\sigma_{i}^{-2}+\theta_{i}\overline{\tau }_{i}^{-2}}{\sigma_{i}^{2}+\tau_{i}^{-2}}\) and a variance of \(\frac{1}{\frac{\overline{\sigma}_{i}^{2}}{\sigma_{i}^{2}}+\frac{1}{\tau_{i}^ {2}}}\). In the above formula \(\overline{x}_{i}^{B}=\frac{1}{N_{Bi}}\sum\limits_{j=1}^{N_{B}}x_{ji}^{B}\). Similarly, the posterior probability density function (PDF) and posterior cumulative distribution function (CDF) in other dimensions of the field test data source can be obtained. To accurately illustrate the pros and cons of the fusion effect, we define a quantitative analysis of the area metrics. The area measurement method was first proposed by American scholars [PERSON] and [PERSON] [38; 39] in 2008. It is a confirmation measurement method based on probability distribution distance and is mainly used in the field of modeling and simulation. As shown in Figure 5, all the real landing point data of an aircraft must come from the same data population, obey a specific distribution, and have a unique CDF. By calculating the area between the overall cumulative distribution function of the aircraft landing point data and the empirical CDF of the experimental observation fusion (the area of the shaded part in Figure 5), the fusion effect of the external field data and the semi-physical simulation data before and after expansion is quantified and evaluated. **Definition 3**.