## 3 Semi-Analytical Echo Model of Airborne SAR Altimeters
### Semi-Analytical Expression of \(FSIR(t_{r},k)\)  
Using the change of variable \(x=(2h/c)\sqrt{1+\varepsilon^{2}}\) and the property of the Dirac distribution \(h(t)=\int_{c_{1}}^{+\infty}\delta(x-t)h(x)\mathrm{d}x\) for \(t\geq c_{1}\), integrating with respect to \(x\) leads to  
\[FSIR(t_{r},k)=\frac{P_{u}}{2\pi}\left(1+\frac{ct}{2h}\right)^{-3}\!\!\left(t \right)\!\!\times\!\left\{\int_{D_{t,k}}F[\tilde{\phi}-\phi,\varepsilon, \xi,\mu]\,\mathrm{d}\phi\right\} \tag{24}\]  
with  
\[F[\tilde{\phi}-\phi,\varepsilon,\xi,\mu]=\exp\left\{-4\!\left[1-\cos^{2}( \xi)/\left(1+\varepsilon^{2}\right)\right]/\gamma+b\!\left(1-\sin^{2}(\tilde{ \phi}-\phi)\right)+a\cos(\tilde{\phi}-\phi)\right\}\,\]  
\[\varepsilon=\rho/h,\]  
\[P_{u}=\frac{\lambda^{2}G_{0}^{2}c\sigma^{0}}{4(4\pi)^{2}L_{p}h^{3}},\]  
\[a=\frac{4\varepsilon\sin(2\xi)}{\gamma(1+\varepsilon^{2})},\]  
Figure 2: Circles of propagation and Doppler beams in SAR altimetry.  
\[b=\frac{4\epsilon^{2}\sin^{2}(\xi)}{\gamma(1+\epsilon^{2})}, \tag{25}\]  
where \(J(\cdot)\) is the step function and \(D_{t,k}\) and \(\varphi_{t,k}\) are both provided in (19). Performing a variable conversion \(w=\tilde{\phi}-\phi\) in (24) results in  
\[\begin{split}\text{FSIR}(t_{r},k)&=\frac{p_{u}}{2 \pi}\big{(}1+\frac{ct}{2h}\big{)}^{-3}J(t)\times\exp\big{\{}-4\big{[}1-\cos^{2} (\xi)/\big{(}1+\epsilon^{2}\big{)}\big{]}/\gamma+b/2\big{\}}\\ &\times\Big{[}f(\tilde{\phi}-\phi_{t,k+1},\tilde{\phi}-\phi_{t,k} )+f\Big{(}\tilde{\phi}-\phi^{\prime}_{t,r,k+1},\tilde{\phi}-\phi^{\prime}_{t,r, k}\Big{)}\Big{]}\end{split} \tag{26}\]  
with  
\[f(w_{1},w_{2})=\int_{u_{1}}^{u_{2}}\exp\bigg{[}a\cos(w)+\frac{b}{2}\cos(2w) \bigg{]}\,\text{d}w, \tag{27}\]  
where \(f(\cdot)\) can be expanded using Bessel functions as follows:  
\[\begin{split}\exp[a\cos(w)]&=I_{0}(a)+2\sum_{k=1}^ {\infty}I_{k}(a)\cos(kw)\\ \exp\bigg{[}\frac{b}{2}\cos(2w)\bigg{]}&=I_{0}\bigg{(} \frac{b}{2}\bigg{)}+2\sum_{k=1}^{\infty}I_{k}\bigg{(}\frac{b}{2}\bigg{)}\cos( 2 kw),\end{split} \tag{28}\]  
where \(I_{k}\) denotes the first-kind modified Bessel function of order \(k\). Note that if any of these terms are approximated, the higher-order expansions of \(\exp\Big{[}\frac{b}{2}\cos(2w)\Big{]}\) can lead to an unacceptable increase in model error when the mis-pointing angle exceeds 1 degree. In this paper, the complete analytical form of the Bessel function approximation is obtained without neglecting any terms. Subsequently, upon substituting (28) into (27) and simplifying, we have  
\[\begin{split} f(w_{1},w_{2})&=\sum_{k=0}^{\infty} \sum_{j=0}^{\infty}C_{k}C_{j}I_{k}(a)I_{j}\bigg{(}\frac{b}{2}\bigg{)}/2\times \{w_{2}[\text{sinc}((k-2j)w_{2})-\text{sinc}((k+2j)w_{2})] \tag{29}\]  
where \(C_{k},C_{j}=\left\{\begin{array}{l}1,k=0\\ 2,k\geq 1\end{array}\right..\) The infinite sum provided in (29) can be truncated to reduce computational complexity by maintaining a finite number \(m\) of elements based on the accuracy requirement, as follows:  
\[\begin{split} f_{1}(w_{1},w_{2})&=\sum_{k=0}^{m}\sum_{j=0}^ {m}C_{k}C_{j}I_{k}(a)I_{j}\bigg{(}\frac{b}{2}\bigg{)}/2\times\{w_{2}[\text{sinc }((k-2j)w_{2})-\text{sinc}((k+2j)w_{2})]\\ &\quad-w_{1}[\text{sinc}((k-2j)w_{1})-\text{sinc}((k+2j)w_{1})]\}. \end{split} \tag{30}\]  
Finally, the analytical expression of FSIR is provided by  
\[\begin{split}\text{FSIR}(t_{r},k)&=\frac{P_{u}}{2 \pi}\bigg{(}1+\frac{ct}{2h}\bigg{)}^{-3}J(t)\times\exp\Big{\{}-4\Big{[}1-\cos^ {2}(\xi)/\Big{(}1+\epsilon^{2}\Big{)}\Big{]}/\gamma+b/2\Big{\}}\\ &\times\Big{[}f_{1}(\tilde{\phi}-\phi_{t,r,k+1},\tilde{\phi}-\phi_ {t,r,k})+f_{1}\Big{(}\tilde{\phi}-\phi^{\prime}_{t,r,k+1},\tilde{\phi}-\phi^{ \prime}_{t,r,k}\Big{)}\Big{]}.\end{split} \tag{31}\]