## 3 Semi-Analytical Echo Model of Airborne SAR Altimeters
### Convolution Model  
To obtain the convolution model in (13), we can first consider a flat surface with a small scale of roughness, namely, \(z\approx 0\), for all the scatterers. Then, the average power in the delay-Doppler bin \((t_{r},k)\) can be provided by  
\[P_{FS}(t_{r},k)=\iint_{\mathbb{R}^{2}}\frac{\lambda^{2}G^{2}(\theta)\sigma_{0} }{(4\pi)^{3}R_{0}^{4}L_{p}}PTR(t_{r}-2R_{0}/c,k-Nf_{d}T_{r})\,\mathrm{d}x \mathrm{d}y \tag{14}\]for \(k=-N/2,\ldots,N/2-1\) and \(0\leq t_{r}\leq T_{r}\). We define \(p^{\prime}(t_{r})=p(-ct_{r}/2)\). It was proposed in [16] that \(P_{I}(t_{r},k)\) can be expressed as a convolution between \(p^{\prime}(t_{r})\) and \(P_{FS}(t_{r},k)\):  
\[P_{I}(t_{r},k)=\frac{c}{2}\int_{\mathbb{R}}P_{FS}(t_{r}-\tau,k)p^{\prime}(\tau )\,\mathrm{d}\tau. \tag{15}\]  
As it is difficult to determine the analytical expression of \(P_{FS}(t_{r},k)\), we can consider an ideal point target response that is a delta function in the time domain and rectangular in the Doppler domain, namely, \(PTR^{\prime}(t_{r},k)=\delta(t)I(k)\) and  
\[I(k)=\left\{ \begin{array}{ll}1,&|k|\leq 1/2,\\ 0,&|k|>1/2.\end{array} \right. \tag{16}\]  
For \(PTR^{\prime}(t_{r},k)\), \(P_{FS}(t_{r},k)\) reduces to the impulse response from the flat surface  
\[FSIR(t_{r},k)=\iint_{\mathbb{R}^{2}}\frac{\lambda^{2}G^{2}(\theta)\sigma_{0}} {(4\pi)^{3}R_{0}^{4}L_{p}}\delta(t_{r}-2R_{0}/c)I(k-Nf_{d}T_{r})\,\mathrm{d}x \mathrm{d}y \tag{17}\]  
for \(k=-N/2,\ldots,N/2-1\) and \(0\leq t_{r}\leq T_{r}\). The convolution between \(FSIR(t_{r},k)\) and the actual point target response is  
\[ \begin{split}&\tilde{P}_{FS}(t_{r},k)=\sum_{k^{\prime}=-\infty}^{+ \infty}\int_{\mathbb{R}}FSIR\big{(}t_{r}-\tau,k-k^{\prime})PTR(\tau,k^{ \prime}\big{)}\,\mathrm{d}\tau\\ &=\sum_{k^{\prime}=-\infty}^{+\infty}\iint_{\mathbb{R}^{2}}\frac {\lambda^{2}G^{2}(\theta)\sigma_{0}}{(4\pi)^{3}R_{0}^{4}L_{p}}PTR(t_{r}-2R_ {0}/c,k^{\prime})I(k-k^{\prime}-Nf_{d}T_{r})\,\mathrm{d}x\mathrm{d}y.\end{split}  \tag{18}\]  
Using staircase approximation on the point target function  
\[PTR(t_{r}-2R_{0}/c,k)\approx\sum_{k^{\prime}=-\infty}^{+\infty}PTR(t_{r}-2R _{0}/c,k^{\prime})I(k-k^{\prime}),\]  
we have \(\tilde{P}_{FS}(t_{r},k)\approx P_{FS}(t_{r},k)\). Combining (15) and (18), we can finally write \(P_{I}(t_{r},k)\) as a convolution of the height probability density, point target response, and impulse response from the flat surface, as shown in (13). The corresponding FSIR is obtained by integrating over each Doppler beam. The \(k\)-th Doppler beam for time instant \(t\), as shown in Figure 2, is defined by an integral angle \(\phi\) varying within the range \(D_{t,r,k}=[\phi_{t_{r},k},\phi_{t,k+1}]\cup\left[\phi^{\prime}{}_{t_{r},k^{ \prime}}{}_{t_{r},k+1}\right]\), leading to  
\[\phi_{t,k}=\arcsin(y/\rho(t_{r}))=\arcsin\left(\frac{f_{d}\lambda R_{0}+2v_{c }h\sin\mu}{2v_{c}\rho(t_{r})\cos\mu}\right), \tag{19}\]  
where \(-2v(\rho\cos\mu+h\sin\mu)/\lambda R_{0}\leq f_{d}\leq-2v(h\sin\mu-\rho\cos\mu )/\lambda R_{0}\); here, the parameters \(xy\) in \(f_{d}\) are omitted for brevity. Note that in the absence of antenna mis-pointing the two delay-Doppler bins exhibit symmetry along the track, resulting in equal power contributions to the DDM. However, their contributions differ when mis-pointing is considered, resulting in  
\[\phi_{t,k}=\pi-\phi^{\prime}{}_{t,k+1}\quad\text{and}\quad\phi_{t_{r},k+1}= \pi-\phi^{\prime}{}_{t_{r},k}. \tag{20}\]In SAR altimetry, the integral area of each Doppler beam varies due to the Doppler effect.