## 3 Inversion Method of Asteroid Spectra Reflectance Template  
The inversion method of the spectra reflectance template for celestial Doppler difference velocimetry uses EMD (empirical mode decomposition) to decompose the measured astronomical spectra, and the IMF obtained can be used to invert the measured reflectance template. The swarm intelligence algorithm is used to obtain the optimal combination of IMFs. The process is as follows:(1)  
Referential velocity difference  
The difference between the HE-SRV(HARPS ESO-Solar Radial Velocity) \(\overline{v_{s}}\) and the HE-ARV (HARPS ESO-Asteroid Radial Velocity) \(\overline{v}_{a}^{i}\) is the HE-SARDV(HARPS ESO-Solar and Asteroid Radial Difference Velocity) \(\overline{v}_{sa}^{i}\), whose expression is given as in Equation (13):  
\[\overline{v}_{sa}^{i}=\overline{v_{s}}-\overline{v}_{a}^{i}, \tag{13}\]  
where, \(i\) = 0,1,2,3, \(\ldots\), \(I-1\) is the serial number of the asteroid spectra, and \(I\) is the number of the asteroid spectra used. Since that asteroid is far from the earth, taking the earth as the observer, the motion of the asteroid approximates to a uniform linear motion in a short time. Assume that the OT (observation time) of the \(i\)-th asteroid spectra is \(t_{a}^{i}\). The relationship between the difference of OT \(t_{a}^{i}-t_{a}^{0}\) and \(\overline{v}_{sa}^{i}\) conforms to the Equation (14) model:  
\[\overline{v}_{sa}^{i}=\overline{v}_{sa}^{0}+a\left(t_{a}^{i}-t_{a}^{0}\right) +\omega^{i}-\omega^{0}, \tag{14}\]  
where \(a\) is the real value of acceleration, and \(\omega^{i}\) is the velocimetry noise of the \(i\)-th asteroid spectra. \(\omega^{i}-\omega^{0}\) is the difference in velocimetry noise of the asteroid spectra. From the Doppler difference velocimetry model [36], the Doppler difference velocimetry eliminates the Doppler velocity bias at the cost of the slight enlargement of the measurement noise, further improving navigation accuracy. Therefore, Doppler difference velocimetry can improve the navigation performance of autonomous navigation for integrated navigation. In order to improve the accuracy of the referential velocity difference, the least square method is used to fit OT and the velocity difference. The expression is shown as in Equation (15):  
\[\left[ \begin{matrix}\overline{v}_{sa}^{0}\\ \overline{a}\end{matrix} \right]=\left[ \begin{matrix}I&\sum\limits_{i=0}^{I-1} \left(t_{a}^{i}-t_{a}^{0}\right)\\ \sum\limits_{i=0}^{I-1}\left(t_{a}^{i}-t_{a}^{0}\right)&\sum\limits_{i=0}^{I-1 }\left(t_{a}^{i}-t_{a}^{0}\right)\end{matrix} \right]^{-1}\left[ \begin{matrix} \sum\limits_{i=0}^{I-1}\overline{v}_{sa}^{i}\\ \sum\limits_{i=0}^{I-1}\overline{v}_{sa}^{i}\left(t_{a}^{i}-t_{a}^{0}\right) \end{matrix} \right], \tag{15}\]  
where \(\overline{v}_{sa}^{0}\) is the referential velocity difference of the zeroth asteroid spectra and \(\overline{a}\) is the referential acceleration. \(\overline{v}_{sa}^{0}\) and \(a\) can be solved through Equation (15). Using them, we get the referential velocity difference \(\widehat{v}_{sa}^{i}\) of the \(i\)-th asteroid spectra in Equation (16):  
\[\widehat{v}_{sa}^{i}=\widehat{v}_{sa}^{0}+\widetilde{a}\left(t_{a}^{i}-t_{a}^ {0}\right), \tag{16}\]  
Due to the noise reduction capability of the least square method, the error of \(\widehat{v}_{sa}^{i}\) is \(1/\sqrt{I}\) times that of \(\overline{v}_{sa}^{i}\). (2) EMD  
The measured solar spectra \(S_{s}\) and the measured asteroid spectra \(S_{a}^{i}\) are decomposed by EMD to obtain the IMFs and their residuals. Assume that \(C_{s}\) and \(C_{a}^{i}\) are the residuals of the measured solar spectra and the measured asteroid spectra, respectively. \(h_{s}^{m}\) and \(h_{a}^{lim}(m\) = 1,2,3, \(\ldots\), \(M\)) are the IMFs of the measured solar spectra and the measured asteroid spectra, respectively, where \(i\) and \(I\) are the same as defined in \(\overline{v}_{sa}^{i}=\overline{v_{s}}-\overline{v}_{a}^{i}\). \(m\) is the number of levels of the IMFs, and \(M\) is the maximal number of levels.