## 3 New Adaptable Strategy for RT and NRT troposphere Monitoring
### Epoch-Wise Filtering vs. Batch Processing, PPP vs. Network Approach  
Nowadays, a common procedure of NRT analysis in E-GVAP is based on the least-squares adjustment (LSQ) analysis and a piece-wise linear function for modeling the tropospheric parameters within the processing interval. Figure 7 depicts the estimated tropospheric parameters by black dots and the corresponding piece-wise deterministic model by dashed lines connecting parameters within each hourly product update; not necessarily connected at update boundaries. According to E-GVAP conventions, the last product value is shifted by 1 min (HR:59) in order to avoid a duplicate value with the next hour product update. Although temporal resolution of estimated parameters can be increased in the LSQ analysis, it means a significant increase of size of normal equations, in particular for the network processing, and consequently extending the processing time due to the inversion of the large matrix of the solution. On the other hand, the Kalman filter is a powerful method for real-time applications because state vectors and covariance matrices are calculated recursively epoch by epoch. One of the consequences is that only previous observations contribute to a current estimate in the Kalman filter, and thus the solution needs a certain time to converge. An accuracy of parameters estimated during the initial convergence, or any later re-convergence, can be then improved using the backward smoothing algorithm [40]. This algorithm exploits all observations from past and future epochs for estimating the state vector at any epoch of the processed window. It improves significantly parameters such as kinematic coordinates or tropospheric parameters [41]. While the Kalman filter was designed for real-time signal processing, the backward smoothing filter was developed for the post-processing to achieve the accuracy of all parameters similar to those from the LSQ solution. Important advantage of data filtering is that the parameters are estimated epoch by epoch with applying individual stochastic properties and the temporal resolution can reach up to the original data sampling. For the real-time data processing, the PPP method has been implemented using the Extended Kalman filter in our software. In the first step of the filter, parameters are predicted via adding a particular amount of noise to diagonal elements of the variance-covariance matrix belonging to dynamic parameters. The state vector itself remains the same in this step. When new observations are available the state vector with its variance-covariance matrix is updated. Since the standard Kalman filter can suffer from numerical instabilities due to the round-off error we have implemented an alternative form of the filter using a square root of variance-covariance matrix instead of the original one. This modification guarantees that the estimated variance-covariance matrix is positive semidefinite. For the post-processing purposes, solutions from the Kalman filter are additionally  
Figure 7: Tropospheric parameters in NRT products using the deterministic model (dotted lines). smoothed by the filter running in the backward direction. Storing the current results from the forward filter is necessary for the backward smoothing. An accuracy of the state vector is improved using the special recurrent algorithm running from the last epoch with estimated parameters toward the beginning of the processing period. Numerical issues are also critical as in the case of the Kalman filter; therefore an advanced technique using singular value decomposition of the variance-covariance matrix was implemented [40]. By employing the PPP method, all GNSS data can be analyzed station by station, either at a central server with a separation into processing threads or through a decentralized manner including the processing at original observation sites. The PPP method effectively supports the utilization of multi-GNSS observations if consistent precise models and products are available for all satellites and systems.