## Appendix A  
Appendix A shows the derivation of the partial derivative of the worst-case integrity risk to _URA_. The derivative of worst-case integrity risk with respect to _URA_ can be analytically derived as  
\[ \begin{array}{rl}\frac{\partial(WIR)}{\partial URA}=&P_{\text{sat}}(1-P_{ \text{sat}})^{n-1}\sum\limits_{i=1}^{n}\left\{c_{1}P(T,n-m,\lambda_{i})+c_{2} \left[Q\big{(}z_{i}^{+}\big{)}+Q\big{(}z_{i}^{-}\big{)}\right]\right\}\\ &+2c_{3}P(T,n-m)\big{(}1-P_{\text{sat}}\big{)}^{n}\end{array}  \tag{11}\]  
where  
\[c_{1}=\frac{\partial\big{[}Q\big{(}z_{i}^{+}\big{)}+Q\big{(}z_{i}^{-}\big{)} \big{]}}{\partial URA},c_{2}=\frac{\partial\big{[}P(T,n-m,\lambda_{i})\big{]}} {\partial URA},c_{3}=\frac{\partial\big{[}Q(z_{0})\big{]}}{\partial URA} \tag{12}\]  
The probability density function (PDF) of the standard normal distribution can be expressed as  
\[q(x)=\frac{1}{\sqrt{2\pi}}e^{-\frac{x^{2}}{2}} \tag{13}\]  
then  
\[Q(z)=\int_{z}^{\infty}q(x)dx \tag{14}\]  
According to the law of derivation,  
\[ \begin{array}{rl}c_{1}&=\frac{\partial\big{[}Q\big{(}z_{i}^{+}\big{)}+Q \big{(}z_{i}^{-}\big{)}\big{]}}{\partial URA}=\left\{\frac{\partial\Big{[} \int_{z_{i}^{+}}^{\infty}q(x)dx\Big{]}}{\partial\sigma}+\frac{\partial\Big{[} \int_{z_{i}^{-}}^{\infty}q(x)dx\Big{]}}{\partial\sigma}\right\}\times\frac{ \partial\sigma}{\partial URA}\\ &=-\bigg{[}q\big{(}z_{i}^{+}\big{)}\frac{\partial z_{i}^{+}}{\partial\sigma}+ q\big{(}z_{i}^{-}\big{)}\frac{\partial z_{i}^{-}}{\partial\sigma}\bigg{]}\times \frac{\partial\sigma}{\partial URA}\end{array}  \tag{15}\]  
\[ \begin{array}{rl}c_{3}&=\frac{\partial\big{[}Q\big{(}z_{0}\big{)}\big{]}}{ \partial URA}=\frac{\partial\Big{[}\int_{z_{0}}^{\infty}q(x)dx\Big{]}}{ \partial\sigma}\times\frac{\partial\sigma}{\partial URA}=-q(z_{0})\frac{ \partial z_{0}}{\partial\sigma}\times\frac{\partial\sigma}{\partial URA}\end{array} \]  
The \(z_{0}\), \(z_{i}^{+}\) and \(z_{i}^{-}\) are given in (5) by  
\[z_{i}^{+}=\frac{l+v_{i_{id}}}{\sigma_{q}},z_{i}^{-}=\frac{l-v_{i_{id}}}{\sigma_ {q}},z_{0}=\frac{l}{\sigma_{q}} \tag{16}\]  
Therefore,  
\[\frac{\partial z_{0}}{\partial\sigma}=-\frac{z_{0}}{\sigma},\frac{\partial z_ {i}^{+}}{\partial\sigma}=-\frac{z_{i}^{+}}{\sigma},\frac{\partial z_{i}^{-}}{ \partial\sigma}=-\frac{z_{i}^{-}}{\sigma} \tag{17}\]  
The \(\frac{\partial\sigma}{\partial URA}\) is determined by  
\[\frac{\partial\sigma}{\partial URA}=\frac{\partial\Big{(}\sqrt{URA^{2}+\sigma _{\text{trop}}^{2}+\sigma_{\text{user}}^{2}}\Big{)}}{\partial URA}=\frac{URA}{\sigma} \tag{18}\]  
Combining (15), (17) and (18),  
\[ \begin{array}{rl}c_{1}=\bigg{[}q\big{(}z_{i}^{+}\big{)}\frac{z_{i}^{+}}{ \sigma_{q}^{2}}+q\big{(}z_{i}^{-}\big{)}\frac{z_{i}^{-}}{\sigma_{q}^{2}}\bigg{]} \times URA=\frac{URA}{\sqrt{2\pi\sigma^{2}}}\bigg{[}z_{i}^{+}e^{-\frac{(z_{i}^ {+})^{2}}{2}}+z_{i}^{-}e^{-\frac{(z_{i}^{-})^{2}}{2}}\bigg{]}\\ c_{3}=q\big{(}z_{0}\big{)}\frac{z_{0}\times URA}{\sigma^{2}}=\frac{z_{0} \times URA}{\sqrt{2\pi\sigma^{2}}}e^{-\frac{x^{2}}{2}}\end{array}  \tag{19}\]  
According to the derivative rule of compound function,  
\[c_{2}=\frac{\partial\big{[}P(T,n-m,\lambda_{i})\big{]}}{\partial URA}=\frac{ \partial\big{[}P(T,n-m,\lambda_{i})\big{]}}{\partial\lambda_{i}}\times\frac{ \partial\lambda_{i}}{\partial\sigma}\times\frac{\partial\sigma}{\partial URA} \tag{20}\]where \(\lambda_{i}\) is given by \(\lambda_{i}=(b_{i}/\sigma)^{2}S_{i,i}\). Therefore,  
\[\frac{\partial\lambda_{i}}{\partial\sigma}=\frac{-2b_{i}^{2}S_{i,i}}{\sigma^{3}} \tag{11}\]  
The PDF of the non-central Chi-square distribution can be expressed as  
\[f(x,n-m,\lambda_{i})=\frac{1}{2}e^{-(x+\lambda_{i})/2}\bigg{(}\frac{x}{ \lambda_{i}}\bigg{)}^{(n-m)/4-1/2}I_{(n-m)/2-1}\bigg{(}\sqrt{\lambda_{i}x} \bigg{)} \tag{12}\]  
where \(I_{(n-m)/2-1}\big{(}\sqrt{\lambda_{i}x}\big{)}\) is the modified Bessel function of the first kind of \((n-m)/2-1\) order.