# A Sensitivity Study of POD Using Dual-Frequency GPS for CubeSats Data Limitation and Resources
## 2 Processing Strategy
The partial derivatives are computed with numerical integration of the variational equations [36; 37]. E is the expectation operator. The reduced-dynamic orbit can be interpolated for time epochs with higher sampling rates and produced for periods with data gaps. In this study, all orbits are resampled into time epochs with 10 s sampling interval for assessment, regardless of which duty-cycles and observation sampling rates are applied in the processing. 3. Phase preprocessing and orbit improvements. This step preprocesses the raw phase observations to detect cycle slips and mark bad observations. The preprocessing goes through several iterations to improve the LEO orbit quality. The orbit improvement is realized through estimation of stochastic velocity changes [38] in addition to the 15 parameters mentioned in the second step, and is performed in a least-squares adjustment making use of the IF combination of the phase observations. One set of the stochastic velocity changes is considered in each predefined time interval, e.g., every 15 min. The linearized phase observation equation at the epoch \(t_{i}\) can be expressed as: \[\mathrm{E}(\Delta\varphi_{IF})=\left[A_{lk},A_{ld},A_{lr}\right]\left[x_{k},x_ {d},x_{v}\right]^{T}+c\times\Delta t_{r}+\lambda_{IF}N_{IF},\] (7) With \[\lambda_{IF}=\frac{c}{f_{1}+f_{2}},\] (8) \[N_{IF}=\frac{f_{1}^{2}\lambda_{1}N_{1}-f_{2}^{2}\lambda_{2}N_{2}}{c(f_{1}-f_{ 2})},\] (9) where \(\Delta\varphi_{IF}\) represents the observed-minus-computed (O-C) term of the IF phase observations. \(c\) and \(\Delta t_{r}\) denote the speed of light and the receiver clock error, respectively. \(\lambda_{j},f_{j}\), and \(N_{j}\) represent the wavelength, the frequency, and the ambiguity on frequency \(j\) (\(j=1\), 2), respectively. Note that \(N_{IF}\) is not an integer. The receiver clock error is estimated epoch-wise independently, and the ambiguity is assumed constant before the detection of a cycle slip. Note that new ambiguities are setup for estimation at the beginning of each round of duty cycling. \(x_{v}\) stands for the vector containing all stochastic velocity changes in the RSW directions from the first to the current epoch and note that \(x_{v}\) is constrained to zero with a predefined a priori standard deviation. The design matrices \(A_{lk}\), \(A_{ld}\), and \(A_{lr}\) contain the partial derivatives of the O-C terms with respect to the \(x_{k}\), \(x_{d}\), and \(x_{v}\), respectively. To be estimated are the vector \([x_{k},x_{d},x_{v}]T\), the receiver clock offset \(\Delta t_{r}\), and the term \(N_{IF}\). Note that very little code observations are used to avoid the problem of matrix singularity between the receiver clock offset and the ambiguity terms. Note that the ambiguities on L1 and L2 are not attempted to be fixed in this study. 4. Generation of final orbits. With the preprocessed phase observations, the six Keplerian elements, the three constant dynamic parameters (\(a_{R0}\), \(a_{S0}\), and \(a_{W0}\)) are estimated together with stochastic accelerations in the RSW directions. The accelerations are set up in shorter time intervals compared to those in Step 3. The linearized phase observation equation is thus formulated as:  
\[a=a_{0}+a_{dyn},\]0  
where \(x_{d0}\) and \(x_{a}\) denote the increment vector of the three constant dynamic parameters and all the stochastic accelerations from the first to the current epoch, respectively. \(x_{a}\) is constrained to zero with a predefined a priori standard deviation (selected as \(5\times 10^{-9}\) m/s\({}^{2}\) in all the three directions based on the default setting for GRACE satellites in the Bernese software, as real data from the GRACE Follow-on mission is used for test purposes in this study, which will be explained later. This value may vary for satellites of other missions). \(A_{ld0}\) and \(A_{la}\) correspond to the partial derivatives of the phase O-C terms with respect to the \(x_{d0}\) and \(x_{a}\), respectively. Note that very little code observations are used to avoid singularity mentioned above.