Considering the complexity of the range model, it is difficult to find a closed-form solution for the stationary point.^{14} To facilitate the derivation of the following imaging algorithm, we expand the signal phase by a power series with respect to $f\tau $, as follows: Display Formula
$S(f\tau ,fa;r0)=\sigma 0\omega a[ta(fa,f\tau )]\xb7\omega r(f\tau )\xb7exp{\Phi azi(fa;r0)}\xb7exp{\Phi RCM(f\tau ,fa;r0)}\xb7exp{\Phi rg2(f\tau ,fa;r0)}\xb7exp{\Phi HOP(f\tau ,fa;r0)},$(10)
where Display Formula$\Phi azi(fa;r0)=\u2212j4\pi \lambda [r0\u2212\lambda 2(fa\u2212fd)28\alpha +\lambda 3(fa\u2212fd)324\beta \u2212\lambda 4(fa\u2212fd)464\gamma ],$(11)
Display Formula$\Phi RCM(f\tau ,fa;r0)=\u2212j4\pi c{r0+\lambda 2(fa+fd)(fa\u2212fd)8\alpha \u2212\lambda 3(2fa+fd)(fa\u2212fd)224\beta +\lambda 4(3fa+fd)(fa\u2212fd)364\gamma}\xb7f\tau ,$(12)
Display Formula$\Phi rg2(f\tau ,fa;r0)=\u2212j\pi f\tau 2Kr+j\pi [\lambda fa22fc2\alpha \u2212\lambda 2fa2(fa\u2212fd)2fc2\beta +3\lambda 3fa2(fa\u2212fd)28fc2\gamma ]\xb7f\tau 2,$(13)
Display Formula$\Phi HOP(f\tau ,fa;r0)=\u2212j4\pi (1\lambda +f\tau c)R[ta(fa,f\tau ;r0)]\u2212j2\pi fata(fa,f\tau ;r0)\u2212\Phi azi(fa;r0)\u2212\Phi RCM(f\tau ,fa;r0)\u2212\Phi rg2(f\tau ,fa;r0)\u2212j\pi f\tau 2Kr,$(14)
where $fc$ is the carrier frequency and $\alpha $, $\beta $, and $\gamma $ are the Doppler parameters of the echo signal, given by Display Formula$\alpha =r0v02\u2009sin\u2009\phi 02,$(15)
Display Formula$\beta =\u22123\Delta a3r022v06\u2009sin\u2009\phi 06\u22123r0\u2009cos\u2009\phi 02v03\u2009sin\u2009\phi 04,$(16)
Display Formula$\gamma =\u22122\Delta a4r03v08\u2009sin\u2009\phi 08+9\Delta a32r032v010\u2009sin\u2009\phi 010+7\Delta a3r02\u2009cos\u2009\phi 0v07\u2009sin\u2009\phi 08+9r0\u2009cos\u2009\phi 022v04\u2009sin\u2009\phi 06+r0(1\u22125\u2009cos\u2009\phi 02)2v04\u2009sin\u2009\phi 06.$(17)