GNSS多普勒测速
某一时刻t测站i至卫星j的伪距观测方程为:
(1)ρ~ij(t)=ρij(t)+cdti−cdtj+Iij+Ti+ερ
\begin{aligned}
\tilde{\rho}_i^j(t) = \rho_i^j(t)+cdt_i-cdt^j+I_i^j+T_i+\varepsilon_{\rho} \tag{1}
\end{aligned}
ρ~ij(t)=ρij(t)+cdti−cdtj+Iij+Ti+ερ(1)式中,ρ~ij(t)\tilde{\rho}_i^j(t)ρ~ij(t)和ρij(t)\rho_i^j(t)ρij(t)分别为卫星j与接收机i在t时刻的测码伪距和几何距离;dtidt_idti和dtjdt^jdtj分别为接收机i和卫星j在t时刻的钟差;IijI_i^jIij和TiT_iTi分别为电离层和对流层延迟;ερ\varepsilon_{\rho}ερ为观测值噪声。对(1)式微分可得:
(2)ρ~ij(t)˙=ρij(t)˙+cdti˙−cdtj˙+I˙ij+T˙i
\begin{aligned}
\dot{\tilde{\rho}_i^j(t)} = \dot{\rho_i^j(t)}+c\dot{dt_i}-c\dot{dt^j}+\dot{I}_i^j+\dot{T}_i \tag{2}
\end{aligned}
ρ~ij(t)˙=ρij(t)˙+cdti˙−cdtj˙+I˙ij+T˙i(2)由此,可以得出多普勒测速观测方程,即:
(3)λDij=ρij(t)˙+cdti˙−cdtj˙+I˙ij+T˙i+εD
\begin{aligned}
\lambda D_i^j = \dot{\rho_i^j(t)}+c\dot{dt_i}-c\dot{dt^j}+\dot{I}_i^j+\dot{T}_i+\varepsilon_D \tag{3}
\end{aligned}
λDij=ρij(t)˙+cdti˙−cdtj˙+I˙ij+T˙i+εD(3)其中,
ρij(t)˙=[lij(t)mij(t)nij(t)][x˙j(t)−x˙i(t)y˙j(t)−y˙i(t)z˙j(t)−z˙i(t)]
\begin{aligned}
\dot{\rho_i^j(t)} = \begin{bmatrix}
l_i^j(t) & m_i^j(t) & n_i^j(t)
\end{bmatrix}\begin{bmatrix}
\dot{x}^j(t) - \dot{x}_i(t) \\
\dot{y}^j(t) - \dot{y}_i(t) \\
\dot{z}^j(t) - \dot{z}_i(t)
\end{bmatrix}
\end{aligned}
ρij(t)˙=[lij(t)mij(t)nij(t)]⎣⎡x˙j(t)−x˙i(t)y˙j(t)−y˙i(t)z˙j(t)−z˙i(t)⎦⎤式中,各“.”项为各相应变量的时间变化率;[lij(t)mij(t)nij(t)]\begin{bmatrix}l_i^j(t) & m_i^j(t) & n_i^j(t) \end{bmatrix}[lij(t)mij(t)nij(t)]为测站i与卫星j在t时刻的方向余弦;εD\varepsilon_DεD为观测噪声。