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\begin{equation}  \delta(t) \approx \delta_k + \dot{\delta}_k t + \frac{1}{2}\ddot{\delta}_kt^2,  \end{equation}  where $\delta_k = \delta(t_k)$ is the time delay at time $t_k$, $\dot{\delta}_k = {\rm d}\delta(t_k)/{\rm d}t$ and $\ddot{\delta}_k = {\rm d}^2\delta(t_k)/{\rm d}t^2$. Therefore we can expand out the phase evolution to be  \begin{equation}  \phi(t) \approx 2\pi \sum_{l=0} \frac{f^{(l)}}{(l+1)!} \left(t(1+\dot{\delta}_k + \frac{\ddot{\delta}_k}{2}t)-t_0+\delta_k+t_k\right)^{(l+1)}.  \end{equation}