diff --git a/parts/F-multiFrameCrb.tex b/parts/F-multiFrameCrb.tex
index a9ae40b..040350b 100644
--- a/parts/F-multiFrameCrb.tex
+++ b/parts/F-multiFrameCrb.tex
@@ -78,15 +78,7 @@
   \in\mathbb C^{M_\ell T\times 1}.
 \end{equation}
 
-对角度参数 $x\in\{\alpha,\beta\}$，记导向矢量导数为
-%
-\begin{equation}
-  \mathbf a_{\ell,x}
-  \triangleq
-  \frac{\partial \mathbf a_\ell(\boldsymbol\xi)}{\partial x}.
-\end{equation}
-%
-由链式法则可得时间模板导数
+对角度参数 $x\in\{\alpha,\beta\}$，由链式法则可得时间模板导数
 %
 \begin{equation}
   \frac{\partial \mathbf q^{\mathrm{cp}}_{\ell,p}}{\partial x}
@@ -127,7 +119,7 @@
   \triangleq
   \frac{\partial \mathbf g^{\mathrm{cp}}_{\ell,p}}{\partial \alpha}
   =
-  \mathbf q^{\mathrm{cp}}_{\ell,p}\otimes \mathbf a_{\ell,\alpha}
+  \mathbf q^{\mathrm{cp}}_{\ell,p}\otimes \frac{\partial\mathbf a_\ell}{\partial\alpha}
   +
   \frac{\partial \mathbf q^{\mathrm{cp}}_{\ell,p}}{\partial \alpha}\otimes \mathbf a_\ell,
 \end{equation}
@@ -136,7 +128,7 @@
   \triangleq
   \frac{\partial \mathbf g^{\mathrm{cp}}_{\ell,p}}{\partial \beta}
   =
-  \mathbf q^{\mathrm{cp}}_{\ell,p}\otimes \mathbf a_{\ell,\beta}
+  \mathbf q^{\mathrm{cp}}_{\ell,p}\otimes \frac{\partial\mathbf a_\ell}{\partial\beta}
   +
   \frac{\partial \mathbf q^{\mathrm{cp}}_{\ell,p}}{\partial \beta}\otimes \mathbf a_\ell,
 \end{equation}
@@ -1170,15 +1162,7 @@ $\mathcal T_{\mathrm{sub}}=\{\tau_{p_1},\ldots,\tau_{p_Q}\}$，则
 %
 这里的 $\vartheta_{\ell,p}$ 是第 $(\ell,p)$ 个观测块的独立未知初始相位；与\gls{cp} 模型相比，跨帧绝对相位连续性已经被这些块级相位参数吸收。
 
-对角度参数 $x\in\{\alpha,\beta\}$，记
-%
-\begin{equation}
-  \mathbf a_{\ell,x}
-  \triangleq
-  \frac{\partial \mathbf a_\ell(\boldsymbol\xi)}{\partial x}.
-\end{equation}
-%
-由链式法则可得
+对角度参数 $x\in\{\alpha,\beta\}$，由链式法则可得
 %
 \begin{equation}
   \frac{\partial \mathbf q^{\mathrm{ip}}_{\ell,p}}{\partial x}
@@ -1235,7 +1219,7 @@ $\mathcal T_{\mathrm{sub}}=\{\tau_{p_1},\ldots,\tau_{p_Q}\}$，则
   \triangleq
   \frac{\partial \mathbf g^{\mathrm{ip}}_{\ell,p}}{\partial \alpha}
   =
-  \mathbf q^{\mathrm{ip}}_{\ell,p}\otimes \mathbf a_{\ell,\alpha}
+  \mathbf q^{\mathrm{ip}}_{\ell,p}\otimes \frac{\partial\mathbf a_\ell}{\partial\alpha}
   +
   \frac{\partial \mathbf q^{\mathrm{ip}}_{\ell,p}}{\partial \alpha}\otimes \mathbf a_\ell,
 \end{equation}
@@ -1244,7 +1228,7 @@ $\mathcal T_{\mathrm{sub}}=\{\tau_{p_1},\ldots,\tau_{p_Q}\}$，则
   \triangleq
   \frac{\partial \mathbf g^{\mathrm{ip}}_{\ell,p}}{\partial \beta}
   =
-  \mathbf q^{\mathrm{ip}}_{\ell,p}\otimes \mathbf a_{\ell,\beta}
+  \mathbf q^{\mathrm{ip}}_{\ell,p}\otimes \frac{\partial\mathbf a_\ell}{\partial\beta}
   +
   \frac{\partial \mathbf q^{\mathrm{ip}}_{\ell,p}}{\partial \beta}\otimes \mathbf a_\ell,
 \end{equation}