# Improving GNSS PPP Convergence: The Case of Atmospheric-Constrained, Multi-GNSS PPP-AR
## 3 Concepts of Functional and Stochastic Atmospheric Constraints
### Methods of Constraint Application  
Though not a novel concept, it is well known that constraining an estimable parameter to a known value in a least squares or Kalman filter might aid in the optimization of the estimated solution. Applying atmospheric constraints functionally implies fixing the atmospheric parameters in the functional pseudorange and carrier-phase models. Equation (1) illustrates the well-known models with the parameters to be constrained:  
\[ \begin{array}{l}P_{r}^{s}=\rho_{r}^{s}+c(dt_{r}-dt^{s})+T_{r,constrained}^{s} +I_{r,constrained}^{s}+\varepsilon_{P}\\ \phi_{r}^{s}=\rho_{r}^{s}+c(dt_{r}-dt^{s})+T_{r,constrained}^{s}-I_{r,constrained }^{s}-\lambda N_{r}^{s}+\varepsilon_{\phi}\end{array}  \tag{1}\]  
where \(P_{r}^{s}\), \(\phi_{r}^{s}\) denote the pseudorange and carrier-phase observations from satellite (\(s\)) to receiver (\(r\)), respectively; \(\rho_{r}^{s}\) is the geometric range between the satellite and receiver antennas; \(dt_{r}\) and \(dt^{s}\) represent the receiver and satellite clock errors, respectively; \(T_{r,constrained}^{s}\) refers to the constrained tropospheric delay; \(I_{r,constrained}^{s}\) is the constrained ionospheric delay on the GNSS signal propagated; \(\lambda\) is the wavelength; \(N_{r}^{s}\) is the carrier-phase ambiguity including satellite and receiver phase instrumental delays and initial fractional phase bias; \(\varepsilon_{P}\) and \(\varepsilon_{\varphi}\) refer to a combination of observation noise, satellite and receiver instrumental delay s due to the transmitting and receiving hardware, multipath and unmodelled effects, respectively. By constraining atmospheric parameters in the functional models, the partial derivatives of these parameters, represented by the design matrix, must be deleted. Though this action eliminates the need to estimate the constrained parameters by using atmospheric products, there is the potential of residual bias in the solution due to time correlated errors. An simpler option presents itself through stochastic constraints. Using the ionospheric delay and tropospheric estimates, as well as their uncertainties provided in the atmospheric products, the covariance matrix can be adjusted to constrain the atmospheric parameters. It is key to note that there is no need to constrain every epoch given the potential of propagating time correlated errors. By constraining only the first epoch and \"freeing\" up the process noise to allow the filter to continue estimating the atmospheric parameters, the constraint initially applied should have a beneficial effect in the first few epochs, improving the solution accuracy in PPP processing. In a sequential least-squares implementation, updating and propagating the covariance information, \(C\hat{x}\), the first epoch requires the inclusion of appropriate process noise, \(C\varepsilon_{\Delta t}\), assuming a time interval of \(\Delta t\). This process can be represented by:  
\[Px^{0}=[C\hat{x}+C\varepsilon_{\Delta t}]^{-1} \tag{2}\]  
\(C\hat{x}\), representing a priori variance-covariance matrix, is given by:  
\[C\hat{x}=\left[ \begin{array}{cccccccc}\sigma_{x}^{2}&&&&\\ &\sigma_{y}^{2}&&&&\\ &&\sigma_{z}^{2}&&\\ &&&&\ddots&&\\ &&&&\sigma_{2 PD,constrained}^{2}&\\ &&&&\ddots&&\\ &&&&\sigma_{linear,unfitted}^{2}\\ \end{array} \right]\]\(C\epsilon_{\Delta t}\), defined as the process noise covariance matrix, is also given by:  
\[C\epsilon_{\Delta t}=\left[ \begin{array}{ccccccccc}C\epsilon(x)_{\Delta t}&&&&\\ &&C\epsilon(y)_{\Delta t}&&&&\\ &&C\epsilon(z)_{\Delta t}&&&&\\ &&&\ddots&&\\ &&&&C\epsilon(ZPD)_{\Delta t,consistent}&&\\ &&&&\ddots&&\\ &&&&C\epsilon(ino)_{\Delta t,consistent}&&\\ \end{array} \right]\]  
It is imperative to note that the choice of process noise value, as well as atmospheric constraint uncertainty, affects the effectiveness of the filtered solution in the first epochs of processing. To avoid the case of over-constraining, the uncertainties of the GIM delays should be inflated to accommodate any inherent errors in the generation of these ionospheric delays.