# Improving Low Earth Orbit (LEO) Prediction with Accelerometer Data
## 2 LEO Orbit Prediction Procedure  
The procedures of LEO orbit prediction for real time applications will be clarified. The simplified demonstration and flowchart are shown as Figure 1.  
The downlink stations receive the LEO onboard GNSS observations and accelerometer data when LEO satellites pass over the downlink stations. LEO POD can be conducted in the analysis center with real time GNSS products from RTS. LEO initial conditions and the corresponding dynamic parameters can be obtained after LEO POD. Then these parameters could be uplinked to the corresponding LEO satellites for orbit prediction and real time LEO orbits can be broadcast to the users for real time applications. As is well known, the length of LEO orbit prediction determines the precision of real time  
Figure 1: Procedures of Low Earth Orbit (LEO) orbit prediction. The LEO onboard Global Navigation Satellite System (GNSS) and accelerometer data are downlinked and transferred to the analysis center and processed there. The estimated initial condition states of LEO as well as the dynamic parameters are then uplinked to LEO satellites and the predicted LEO orbit is integrated onboard. LEO orbits. However, the length of orbit prediction is fully dependent on the time gap of LEO onboard observation data collection and computation time of LEO POD. It takes about 1.5-1.7 h for one LEO satellite cycle with the altitude from 500 to 1000 km. For the computation time of LEO POD with a 24 h arc length, it takes about 10 min for one LEO satellite at the GFZ server. In general, LEO satellites fly overhead a ground station for 10-15 min. It is possible to download the observation data, calculate LEO orbit and inject the data to the corresponding LEO satellites in 15 min. Then after a half cycle of about 50 min, this procedure can be done again at another downlink and uplink station. Under this circumstance, one hour is enough for LEO orbit prediction and we took this length for our following analysis. For the sake of insurance, we also prepared two hours as backup if the orbital elements could not be uplinked in time and were delayed to be uplinked at the next uplink station. The timeline of data process is shown in Figure 2. If the time gap of LEO onboard observation data collection is shorter, the length of orbit prediction time can be even shorter. With these procedures, we proposed three strategies to fulfill LEO prediction. We took GRACE-A data for example with its data from Day Of Year (DOY) 010, 2007 to DOY 360, 2007. As aforementioned, the LEO initial conditions and dynamic parameters determinate the precision of predicted LEO orbits. The accelerometer measures all non-gravitational forces acting on the satellite. However, the measurement has to be calibrated in terms of bias and scale when they are used to process orbit determination. The bias and scale have to be estimated together with the LEO initial position and velocity. The calibration model can be written as  
\[\mathbf{a}_{ng}=\mathbf{B}+\mathbf{Ka} \tag{1}\]  
where \(\mathbf{a}_{ng}\) denotes the non-gravitational acceleration. \(\mathbf{B}=\left[ \begin{array}{cc}\mathrm{B}_{X}&\mathrm{B}_{Y}&\mathrm{B}_{Z} \end{array} \right]^{\mathrm{T}}\) denotes the bias vector. \(\mathbf{K}=\mathrm{diag}\big{(}\left[ \begin{array}{cc}\mathrm{K}_{X}&\mathrm{K}_ {Y}&\mathrm{K}_{Z}\end{array} \right]\big{)}\) denotes the scale matrix. \(\mathbf{a}\) denotes the non-gravitational measurement. Note both \(\mathbf{a}_{ng}\) and \(\mathbf{a}\) are in the GRACE Science Reference Frame (SRF), while LEO POD is conducted in the inertial system. Thus, one needs to transform \(\mathbf{a}_{ng}\) from SRF to the inertial frame, which is  
\[\mathbf{a}_{CIS}=\mathbf{Ma}_{ng}=\mathbf{M}(\mathbf{B}+\mathbf{Ka}) \tag{2}\]  
where \(\mathbf{M}\) is the transform matrix from SRF to the inertial frame. The proposed three strategies are mainly focused on the determination of dynamic parameters in LEO POD and they are as follows.